{"id":231,"date":"2023-02-22T16:50:48","date_gmt":"2023-02-22T21:50:48","guid":{"rendered":"https:\/\/opentextbc.ca\/foundationsofphysics\/?post_type=chapter&#038;p=231"},"modified":"2023-09-13T17:46:47","modified_gmt":"2023-09-13T21:46:47","slug":"electric-circuits","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/foundationsofphysics\/chapter\/electric-circuits\/","title":{"raw":"Electric Circuits","rendered":"Electric Circuits"},"content":{"raw":"<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Resources<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nVideo to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=YMNZ2oYu-qI\">Mechanical Universe - Episode 33 Electric Circuits<\/a>\r\n\r\nExtra Help: <a href=\"http:\/\/www.a-levelphysicstutor.com\/index-elect.php\">A-Level Physics Tutor<\/a>\r\n\r\n<\/div>\r\n<\/div>\r\nEquations Introduced and Used in this Topic:\r\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\r\n \t<li>Parallel Resistors:\r\n<ul style=\"list-style-type: none;\">\r\n \t<li>[latex]V_s=V_1=V_2=V_3=[\/latex] ...<\/li>\r\n \t<li>[latex]I_s=I_1+I_2+I_3+[\/latex] ...<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+ ...[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Series Resistors:\r\n<ul style=\"list-style-type: none;\">\r\n \t<li>[latex]I_s=I_1=I_2=I_3=[\/latex] ...<\/li>\r\n \t<li>[latex]V_s=V_1+V_2+V_3+[\/latex] ...<\/li>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nKirchoff\u2019s Circuit Laws:\r\n<p style=\"text-align: center;\">[latex]\\sum V_\\text{ source}=\\sum V_\\text{ drops}[\/latex]... (Conservation of Energy)<\/p>\r\n<p style=\"text-align: center;\">[latex]\\sum I_\\text{ into mode}=\\sum I_\\text{ out of mode}[\/latex]... (Conservation of Charge)<\/p>\r\nWhere\u2026\r\n<ul>\r\n \t<li>[latex]I[\/latex]\u00a0is the Electric Current, measured in amperes or amps (A)<\/li>\r\n \t<li>[latex]V[\/latex] is the voltage, measured in volts (V)<\/li>\r\n \t<li>[latex]R[\/latex] is the Resistance, measured in ohms (\u03a9)<\/li>\r\n<\/ul>\r\n<h1>23.1 Ohm\u2019s Law (Single Circuits)<\/h1>\r\nOhm\u2019s Law relates voltage, Current and Resistance given by the equation of\r\n<p style=\"text-align: center;\">[latex]V = IR[\/latex]<\/p>\r\nIn this last Chapter for this course, we analyze basic electric circuits using Ohm\u2019s Law and later Kirchoff\u2019s Laws in series, parallel and combined resistor arrangements. While there are a number of circuit symbols are used in circuit diagrams;\r\n<ul>\r\n \t<li>We use the symbol of jagged lines for electrical resistance <img class=\"alignnone wp-image-297 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/jagged-lines-e1678305914518.png\" alt=\"\" width=\"150\" height=\"33\" \/><\/li>\r\n \t<li>We use two open circles to represent the source voltage <img class=\"alignnone size-full wp-image-298\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circles.png\" alt=\"\" width=\"71\" height=\"27\" \/><\/li>\r\n<\/ul>\r\nCombined in a simple circuit these symbols look like:\r\n\r\n<img class=\"aligncenter wp-image-299\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circuit-e1678306053290.png\" alt=\"\" width=\"400\" height=\"329\" \/>\r\n\r\nThe following examples use the circuit diagram to solve for Ohm\u2019s Law questions.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 23.1.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the missing variable for each of the following circuits:\r\n<ol>\r\n \t<li><img class=\"size-full wp-image-301 aligncenter\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/1.png\" alt=\"\" width=\"291\" height=\"171\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]V = IR[\/latex]<\/li>\r\n \t<li>120 V = [latex]I[\/latex] (420 \u03a9)<\/li>\r\n \t<li>[latex]I [\/latex]= 120 V \u00f7 420 \u03a9\u00a0 or\u00a0 0.29 A<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><img class=\"size-full wp-image-302 aligncenter\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.png\" alt=\"\" width=\"286\" height=\"180\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]V = IR[\/latex]<\/li>\r\n \t<li>[latex]V[\/latex] = (32 \u00d7 10<sup>\u22123<\/sup> A)(240 \u03a9)<\/li>\r\n \t<li>[latex]V[\/latex] = 7.7 V<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><img class=\"size-full wp-image-303 aligncenter\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.png\" alt=\"\" width=\"290\" height=\"179\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]V = IR[\/latex]<\/li>\r\n \t<li>0.12 \u00d7 10<sup>\u22123<\/sup> V = (800 \u00d7 10<sup>\u22126<\/sup> A)([latex]R[\/latex])<\/li>\r\n \t<li>[latex]R[\/latex] = 0.12 \u00d7 10<sup>\u22123<\/sup> V \u00f7 800 \u00d7 10<sup>\u22126<\/sup> AR =\u00a0 0.15 \u03a9<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 23.1 Ohm\u2019s Law<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve for the unknown in each of the following single circuits.\r\n<ol class=\"twocolumn\">\r\n \t<li><img class=\"aligncenter size-full wp-image-305\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e1.png\" alt=\"\" width=\"260\" height=\"151\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-full wp-image-306\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e2.png\" alt=\"\" width=\"257\" height=\"153\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-307\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e3.png\" alt=\"\" width=\"257\" height=\"149\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-308\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e4.png\" alt=\"\" width=\"253\" height=\"147\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-888\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-5-e1680297396430.jpg\" alt=\"\" width=\"257\" height=\"150\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-full wp-image-310\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e6.png\" alt=\"\" width=\"257\" height=\"151\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-311\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e7.png\" alt=\"\" width=\"253\" height=\"148\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-889\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-8-e1680297461471.jpg\" alt=\"\" width=\"257\" height=\"149\" \/><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>23.2 Series Resistors<\/h1>\r\n<div>\r\n<div class=\"textbox\">\r\n\r\nExtra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/circuits\/Lesson-4\/Two-Types-of-Connections\">Two Types of Connections<\/a>\r\n\r\nExtra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/circuits\/Lesson-4\/Series-Circuits\">Series Circuits<\/a>\r\n\r\n<\/div>\r\n<\/div>\r\nEquations Introduced or Used for this Section:\r\n<p style=\"text-align: center;\">[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/p>\r\nThe equation we use to analyze both series and parallel resistors in combination can be found by using the original law developed for measuring the resistivity of resistors.\r\n\r\nIn general, this law follows the form:\r\n<p style=\"text-align: center;\">[latex]R=\\dfrac{\\rho L}{A}[\/latex]<\/p>\r\nWhere\u2026\r\n<ul>\r\n \t<li>[latex]R[\/latex] is the Resistance of the resistor measured in ohms (\u03a9)<\/li>\r\n \t<li>[latex]\\varrho[\/latex]\u00a0 is the resistivity of the material, which is a measure of how well the material conducts electricity<\/li>\r\n \t<li>[latex]L[\/latex] is the length of the conductor measured in metres (m)<\/li>\r\n \t<li>[latex]A[\/latex] is the cross sectional area of the conductor measured in square metres (m<sup>2<\/sup>)<\/li>\r\n<\/ul>\r\nResistors in series are simply resistors that are connected to each other and will look similar to:\r\n\r\n<img class=\"aligncenter wp-image-314 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resisters.png\" alt=\"\" width=\"661\" height=\"50\" \/>\r\n\r\nFrom the equation for calculating the resistivity of a resistor as shown above,\r\n<p style=\"text-align: center;\">[latex]R=\\dfrac{\\rho L}{A}[\/latex]<\/p>\r\nthe resultant of adding resistors one after another in a series effectively changes the length which can be easily envisioned by just considering adding identical resistors to each other which in simplest terms is just changing the length... We only need to add these resistances up.\r\n\r\nThis strategy works for all different kinds of resistors added to each other, simply add up the total resistance of each resistor, which yields:\r\n<p style=\"text-align: center;\">[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 23.2.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the total resistance of the following Resistors connected in Series.\r\n<ol>\r\n \t<li><img class=\"aligncenter size-medium wp-image-315\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.1-300x148.png\" alt=\"\" width=\"300\" height=\"148\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/li>\r\n \t<li>[latex]R_T[\/latex] = 240 \u03a9 + 220 \u03a9 + 280 \u03a9<\/li>\r\n \t<li>[latex]R_T[\/latex] = 740 \u03a9<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-316\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.2-300x148.png\" alt=\"\" width=\"300\" height=\"148\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/li>\r\n \t<li>[latex]R_T[\/latex] = 50 \u03a9 + 100 \u03a9 + 210 \u03a9<\/li>\r\n \t<li>[latex]R_T[\/latex] = 360 \u03a9<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-318\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.3-300x145.png\" alt=\"\" width=\"300\" height=\"145\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/li>\r\n \t<li>[latex]R_T[\/latex] = 0.6 k\u03a9 + 1.2 k\u03a9 + 0.8 k\u03a9<\/li>\r\n \t<li>[latex]R_T[\/latex] = 2.6 k\u03a9<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 23.2 Series Resistors<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the total resistance of each of the following circuits\r\n<ol class=\"twocolumn\">\r\n \t<li><img class=\"aligncenter wp-image-319 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.1-e1678319399151.png\" alt=\"\" width=\"290\" height=\"158\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-320 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.2-e1678319413808.png\" alt=\"\" width=\"290\" height=\"163\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-321 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.3-e1678319485574.png\" alt=\"\" width=\"290\" height=\"161\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-353 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.4-e1678319517593.png\" alt=\"\" width=\"290\" height=\"161\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-323 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.5-e1678319377803.png\" alt=\"\" width=\"290\" height=\"158\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-354 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.6-e1678319577360.png\" alt=\"\" width=\"290\" height=\"159\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-326 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.7-e1678319589151.png\" alt=\"\" width=\"290\" height=\"158\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-327 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.8-e1678319604363.png\" alt=\"\" width=\"290\" height=\"160\" \/><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>23.3 Parallel Resistors<\/h1>\r\n<div class=\"textbox\">Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/circuits\/Lesson-4\/Parallel-Circuits\">Parallel Circuits<\/a><\/div>\r\nEquations Introduced or Used for this Section:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/p>\r\nWhen resistors are combined in a parallel arrangement (side by side) the total resistance can once again be inferred by the original equation that is used to calculate the resistance of a resistor. From the equation for calculating the resistivity of a resistor as shown below,\r\n<p style=\"text-align: center;\">[latex]R=\\dfrac{\\varrho L}{A}[\/latex]<\/p>\r\n<img class=\"aligncenter size-full wp-image-329\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resistors-combined.png\" alt=\"\" width=\"301\" height=\"165\" \/>\r\n\r\nOne way of looking at resistors that are combined in parallel is that the cross sectional area of the resistor is increased.\u00a0 When put together these resistors combine to allow more electricity to flow in a circuit. This means that we can simply add up the increased areas to get some measure of how the total resistance changes.\r\n\r\nFrom the original equation, you will notice that the effect of changing the cross sectional area of the resistor is in the denominator or is the reciprocal variable in calculating the resistance for resistors combined in this fashion.\u00a0 This means that when combining resistors connected in parallel we are changing the cross sectional area in the denominator by adding new resistors that increase this cross sectional value.\r\n\r\nCombining resistors in parallel means that we add the reciprocals of all the resistors connected in parallel to arrive at the total resistance of the new circuit.\r\n\r\nThis equation looks like:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/p>\r\nThe following examples analyze the total resistance of resistors in parallel.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 23.3.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the total resistance of the following Resistors connected in Parallel.\r\n<ol>\r\n \t<li><img class=\"aligncenter size-medium wp-image-331\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.1-300x127.png\" alt=\"\" width=\"300\" height=\"127\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{20\\Omega}+\\dfrac{1}{30\\Omega}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}= 12\\Omega[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-332\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.2-300x145.png\" alt=\"\" width=\"300\" height=\"145\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{145\\Omega}+\\dfrac{1}{185\\Omega}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}= 81.3\\Omega[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-333\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.3-300x144.png\" alt=\"\" width=\"300\" height=\"144\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{10\\Omega}+\\dfrac{1}{12\\Omega}+\\dfrac{1}{16\\Omega}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}= 4.1\\Omega[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-334\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.4-300x143.png\" alt=\"\" width=\"300\" height=\"143\" \/>\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{15\\Omega}+\\dfrac{1}{12\\Omega}+\\dfrac{1}{12\\Omega}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}= 4.3\\Omega[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 23.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the total resistance of each of the following circuits.\r\n<ol class=\"twocolumn\">\r\n \t<li><img class=\"aligncenter wp-image-336\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.1.png\" alt=\"\" width=\"270\" height=\"149\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-337\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.2.png\" alt=\"\" width=\"270\" height=\"146\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-338\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.3-e1678374202289.png\" alt=\"\" width=\"270\" height=\"144\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-339\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.4.png\" alt=\"\" width=\"270\" height=\"153\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-340\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.5-e1678374225253.png\" alt=\"\" width=\"270\" height=\"147\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-341\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.6.png\" alt=\"\" width=\"273\" height=\"151\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-342\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.7.png\" alt=\"\" width=\"270\" height=\"141\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-343\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.8.png\" alt=\"\" width=\"270\" height=\"153\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-345\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.9.png\" alt=\"\" width=\"270\" height=\"149\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-344\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.10.png\" alt=\"\" width=\"270\" height=\"150\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-full wp-image-346\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.11.png\" alt=\"\" width=\"271\" height=\"158\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-347\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.12-e1678374363705.png\" alt=\"\" width=\"270\" height=\"142\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-348\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.13-e1678374389829.png\" alt=\"\" width=\"270\" height=\"152\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-349\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.14-e1678374412527.png\" alt=\"\" width=\"270\" height=\"143\" \/><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>23.4 Series &amp; Parallel Resistors<\/h1>\r\nEquations Introduced or Used for this Section:\r\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+...[\/latex]<\/li>\r\n<\/ul>\r\nWhen we try to find the total resistance of combined series and parallel circuits we generally reduce the circuits from the simplest combination to the most complex.\r\n\r\n<img class=\"aligncenter wp-image-355\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4-circut.png\" alt=\"\" width=\"500\" height=\"78\" \/>\r\n\r\nFor instance when we consider the following circuit arrangement...\r\n\r\nYou can see that we have a mix of a parallel resistor connected in series to a single resistor. The simplest way to reduce this is to reduce the parallel resistor to find a single resistor of equal value. After we have reduced the parallel branch we then just add the two resistor values together as if joined in series. This looks like:\r\n\r\n<img class=\"aligncenter size-full wp-image-356\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/parallel-resistor.png\" alt=\"\" width=\"312\" height=\"39\" \/>\r\n\r\nResistance of [latex]R_1 + R_2 + R_3 + ...[\/latex]\r\n\r\nIf we complicate this arrangement by adding another resistor in series to R<sub>1<\/sub>, we get the following:\r\n\r\n<img class=\"aligncenter wp-image-357\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r1_4.png\" alt=\"\" width=\"500\" height=\"83\" \/>\r\n\r\nFirst we reduce the top branch of R<sub>1<\/sub> &amp; R<sub>2<\/sub> to a single Resistor, we then continue to reduce as we did in the combination shown directly above.\r\n\r\n<img class=\"aligncenter wp-image-359\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r-rb.png\" alt=\"\" width=\"500\" height=\"75\" \/>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 23.4.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the total resistance of the following Resistors connected in Series and Parallel.\r\n\r\n<img class=\"aligncenter wp-image-360\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.1.png\" alt=\"\" width=\"500\" height=\"181\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n\r\nTo find the total resistance of this circuit you must identify and solve each part of the circuit... In this circuit, you have a parallel branch, which is combined in series.\r\n<ul>\r\n \t<li>First the parallel branch...\r\n<ul>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+ ...[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{50\\Omega}+\\dfrac{1}{50\\Omega}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}= 25\\Omega[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Second the series...\r\n<ul>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/li>\r\n \t<li>R<sub>T<\/sub> = 20 \u03a9 + 50 \u03a9 + 50 \u03a9<\/li>\r\n \t<li>R<sub>T<\/sub> = 125 \u03a9<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 23.4.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the total resistance of the following Resistors connected in Series and Parallel.\r\n\r\n<img class=\"aligncenter wp-image-361\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.2.png\" alt=\"\" width=\"500\" height=\"151\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n\r\nTo find the total resistance of this circuit you have three distinct parts... a series branch in a parallel branch which is finally combined in series.\r\n<ul>\r\n \t<li>First the series...\r\n<ul>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/li>\r\n \t<li>R<sub>T<\/sub> = 25 \u03a9 + 25 \u03a9<\/li>\r\n \t<li>R<sub>T<\/sub> = 50 \u03a9<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Second the parallel branch ...\r\n<ul>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{50\\Omega}+\\dfrac{1}{25\\Omega}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=16.7\\Omega[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Finally the total circuit...\r\n<ul>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/li>\r\n \t<li>R<sub>T<\/sub> = 25 \u03a9 + 16.7 \u03a9<\/li>\r\n \t<li>R<sub>T<\/sub> = 41.7 \u03a9 (\u2248 42 \u03a9)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 23.4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the total resistance of the following circuits\r\n<ol>\r\n \t<li><img class=\"aligncenter wp-image-362\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.1.png\" alt=\"\" width=\"500\" height=\"292\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-363\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.2.png\" alt=\"\" width=\"500\" height=\"139\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-364\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.3.png\" alt=\"\" width=\"500\" height=\"182\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-365\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.4.png\" alt=\"\" width=\"500\" height=\"155\" \/><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>23.5 Solving Series &amp; Parallel Resistor Circuits<\/h1>\r\n<div class=\"textbox\">Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/circuits\/Lesson-4\/Combination-Circuits\">Combination Circuits<\/a><\/div>\r\nKirchoff's Circuit Laws are extensions of the concept of conservation of energy and the law of conservation of charge applied to circuits.\r\n\r\nFrom the Conservation of Charge... The current entering any junction (node) of a circuit is equal to the current leaving that junction (node).\r\n<p style=\"text-align: center;\">[latex]I\\text{ entering}=I\\text{ leaving}[\/latex]<\/p>\r\n<img class=\"aligncenter wp-image-891\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1.png\" alt=\"\" width=\"600\" height=\"223\" \/>\r\n\r\nApplying Kirchoff\u2019s Law for the above circuit... [latex]I_0=I_1+I_2+I_3[\/latex]\r\n\r\nFrom the Conservation of Energy... The sum of all the voltages around the loop is equal to zero or for a simple circuit: The voltage at the source is equal to the voltage drops along the circuit.\r\n<p style=\"text-align: center;\">[latex]V\\text{source}=\\sum V\\text{drops}[\/latex]<\/p>\r\n<img class=\"aligncenter wp-image-368\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5-circuit.png\" alt=\"\" width=\"500\" height=\"252\" \/>\r\n\r\nApplying Kirchoff\u2019s Law for this circuit... [latex]V_s=V_1+V_2+V_3+V_4[\/latex]\r\n\r\nThe final part of this section includes adding Kirchoff\u2019s Laws to Ohm\u2019s Law and applying the techniques to reduce combined series and parallel circuits. Ultimately you will be looking to see if you can use any one of these to reduce some part of the circuit.\r\n\r\nSome of the rules that you must remember are:\r\n\r\nIn a series circuit:\r\n<ul>\r\n \t<li>The electric current remains constant throughout the circuit.<\/li>\r\n \t<li>The sum of all the voltage drops (or losses) will add to the original source voltage.<\/li>\r\n<\/ul>\r\nIn a parallel circuit:\r\n<ul>\r\n \t<li>The voltage drop on any parallel branch will be identical to another.<\/li>\r\n \t<li>The electric current will split up in each of the branches. The sum of the electric current in each branch will equal the electric current going into the branches.<\/li>\r\n<\/ul>\r\nYou will Use Ohm\u2019s Law:\r\n<ul>\r\n \t<li>Whenever you know two of the three variables in any part of the circuit, you can apply Ohm's Law to that part.<\/li>\r\n<\/ul>\r\nSolving circuits can sometimes resemble a Puzzle:\r\n<ul>\r\n \t<li>Somewhere in the circuit is generally one small thing that you are able to solve. Solving this part will then lead to another part that you can solve. The challenge is that you will need to search for the one thing that you can solve in the circuit, which then allows you to solve another part.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_369\" align=\"aligncenter\" width=\"679\"]<img class=\"size-full wp-image-369\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/potential.png\" alt=\"\" width=\"679\" height=\"692\" \/> Potential difference for Gravitational Potential[\/caption]\r\n\r\nOne common way to visualize the voltage drops, as related to the source voltage, is to imagine a water pump leading to a series of water falls (Gravitational Potential Analogy). The source pump (source voltage) lifts the water up through a certain height, meaning that the potential difference allows the water to then fall through a series of water falls... Note for the parallel branch of [latex]V_3[\/latex] &amp; [latex]V_4[\/latex], both branches fall through the exact same parallel difference and are equal. What is unique about the parallel branch is the water splits into two currents, which then recombine back into a single current after the water has passed through both branches.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 23.5.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the missing variable for each of the following circuits:\r\n\r\n<img class=\"aligncenter wp-image-371 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.1-e1678374591611.png\" alt=\"\" width=\"500\" height=\"248\" \/>\r\n\r\nFirst... Find the total resistance of the circuit\r\n\r\nThe top branch is in series, so\r\n<ul>\r\n \t<li>First the series...\r\n<ul>\r\n \t<li>[latex]R_T=R_1+R_2+R_3+[\/latex] ...<\/li>\r\n \t<li>[latex]R_T[\/latex] = 20 \u03a9 + 20 \u03a9<\/li>\r\n \t<li>[latex]R_T[\/latex] = 40 \u03a9<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Second the parallel branch...\r\n<ul>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{50\\Omega}+\\dfrac{1}{25\\Omega}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{R_T}= 16.7\\omega[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Third apply Ohm's Law...\r\n<ul>\r\n \t<li>[latex]V=IR[\/latex]<\/li>\r\n \t<li>60 V = [latex]I[\/latex](20 \u03a9)<\/li>\r\n \t<li>[latex]I[\/latex] = 60 V \u00f7 20\u03a9 or 3.0 A<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nSince the resistance over the top and bottom branches are identical, the current splits evenly.\r\n\r\nTherefore... [latex]I_1 = I_2[\/latex] = 1.5 A\r\n\r\nCurrents [latex]I_1[\/latex] &amp; [latex]I_2[\/latex]\u00a0can also be found by using Ohm's Law for each branch.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 23.5.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the missing variable for each of the following circuits:\r\n\r\n<img class=\"aligncenter wp-image-372\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.2.png\" alt=\"\" width=\"500\" height=\"258\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n\r\nThis is solved in several steps...\r\n<ul class=\"twocolumn\">\r\n \t<li>[latex]R_1[\/latex]... [latex]V=IR_1[\/latex]\r\n<ul>\r\n \t<li>40 V = (1.2 A) [latex]R_1[\/latex]<\/li>\r\n \t<li>[latex]R_1[\/latex] = 40 V \u00f7 1.2 A<\/li>\r\n \t<li>[latex]R_1[\/latex]= 33.3 \u03a9 (\u2248 33 \u03a9)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>[latex]R_2[\/latex]... [latex]V=IR_2[\/latex]\r\n<ul>\r\n \t<li>80 V = (0.50 A) [latex]R_2[\/latex]<\/li>\r\n \t<li>[latex]R_2[\/latex] = 80 V \u00f7 0.50 A<\/li>\r\n \t<li>[latex]R_2[\/latex] = 160 \u03a9<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li style=\"break-inside: avoid;\">[latex]I_3[\/latex]... [latex]I_0=I_1+I_2+I_3[\/latex]\r\n<ul>\r\n \t<li>1.2 A = 0.50 A + I<sub>3<\/sub><\/li>\r\n \t<li>[latex]I_3[\/latex] = 1.2 A \u2212 0.50 A<\/li>\r\n \t<li>[latex]I_3[\/latex] = 0.70 A<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li style=\"break-inside: avoid;\">[latex]R_3[\/latex]... [latex]V=IR_3[\/latex]\r\n<ul>\r\n \t<li>80 V = (0.70 A) [latex]R_3[\/latex]<\/li>\r\n \t<li>[latex]R_3[\/latex] = 80 V \u00f7 0.70 A<\/li>\r\n \t<li>[latex]R_3[\/latex] = 114 \u03a9(\u2248 110 \u03a9)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>[latex]V_2[\/latex] &amp; [latex]V_3[\/latex] are identical...\r\n<ul>\r\n \t<li>[latex]V_2[\/latex] &amp; [latex]V_3[\/latex] = 120 V - 40 V<\/li>\r\n \t<li>[latex]V_2[\/latex] &amp; [latex]V_3[\/latex] = 80 V<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 23.5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve for all unknowns in the following circuits.\r\n<ol class=\"twocolumn\">\r\n \t<li><img class=\"alignnone wp-image-1198 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex-23.5-1.png\" alt=\"\" width=\"650\" height=\"414\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-full wp-image-375\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.2.png\" alt=\"\" width=\"278\" height=\"170\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-full wp-image-376\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.3.png\" alt=\"\" width=\"276\" height=\"171\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-377 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.4-e1678380405317.png\" alt=\"\" width=\"277\" height=\"169\" \/><\/li>\r\n<\/ol>\r\n<ol start=\"5\">\r\n \t<li><img class=\"aligncenter size-full wp-image-378\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.5.png\" alt=\"\" width=\"443\" height=\"210\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-full wp-image-379\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.6.png\" alt=\"\" width=\"552\" height=\"268\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-full wp-image-380\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.7.png\" alt=\"\" width=\"550\" height=\"265\" \/><\/li>\r\n \t<li><img class=\"aligncenter size-full wp-image-381\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.8.png\" alt=\"\" width=\"443\" height=\"208\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-382 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.9-e1678380663885.png\" alt=\"\" width=\"432\" height=\"218\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-383 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.10-e1678380061321.png\" alt=\"\" width=\"434\" height=\"203\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-384 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.11-e1678380082952.png\" alt=\"\" width=\"436\" height=\"202\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-385\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.12.png\" alt=\"\" width=\"580\" height=\"259\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-386 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.13-e1678380106687.png\" alt=\"\" width=\"580\" height=\"243\" \/><\/li>\r\n \t<li><img class=\"aligncenter wp-image-388\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.14-1.png\" alt=\"\" width=\"580\" height=\"249\" \/><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 23.6.1: Creating Different Resistor Combinations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUsing 3 identical resistors, how many circuits can you make with different total resistances? You should be able to create 4 combinations... There are actually more.\r\n\r\nSome unusual combinations???\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 23.6.2: A Trick Circuit Combination<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhich circuit shown below will have the greatest current flow?\r\n\r\n<img class=\"aligncenter size-full wp-image-390\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.6.2.png\" alt=\"\" width=\"564\" height=\"160\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Exercise Answers<\/h1>\r\n<h2>23.1 Ohm\u2019s Law (Single Circuits)<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>0.2 A<\/li>\r\n \t<li>360 V<\/li>\r\n \t<li>25 A<\/li>\r\n \t<li>3750 \u03a9<\/li>\r\n \t<li>216 V<\/li>\r\n \t<li>15 \u03a9<\/li>\r\n \t<li>0.05 A<\/li>\r\n \t<li>28.8 V<\/li>\r\n<\/ol>\r\n<h2>23.2 Series Resistors<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>32 \u03a9<\/li>\r\n \t<li>32 \u03a9<\/li>\r\n \t<li>10 \u03a9<\/li>\r\n \t<li>50 k\u03a9<\/li>\r\n \t<li>78 k\u03a9<\/li>\r\n \t<li>235 k\u03a9<\/li>\r\n \t<li>14.8 k\u03a9<\/li>\r\n \t<li>330 \u03a9<\/li>\r\n<\/ol>\r\n<h2>23.3 Parallel Resistors<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>3.75 \u03a9<\/li>\r\n \t<li>10.2 \u03a9<\/li>\r\n \t<li>7.47 \u03a9<\/li>\r\n \t<li>31.6 \u03a9<\/li>\r\n \t<li>40.5 \u03a9<\/li>\r\n \t<li>215 \u03a9<\/li>\r\n \t<li>25 \u03a9<\/li>\r\n \t<li>33.8 \u03a9<\/li>\r\n \t<li>70.6 \u03a9<\/li>\r\n \t<li>4.07 \u03a9<\/li>\r\n \t<li>48 \u03a9<\/li>\r\n \t<li>130 \u03a9<\/li>\r\n \t<li>35.7 \u03a9<\/li>\r\n \t<li>12 \u03a9<\/li>\r\n<\/ol>\r\n<h2>23.4 Series &amp; Parallel Resistors<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>6.3 \u03a9<\/li>\r\n \t<li>15 \u03a9<\/li>\r\n \t<li>25 \u03a9<\/li>\r\n \t<li>16.7 \u03a9<\/li>\r\n<\/ol>\r\n<h2>23.5 Solving Series &amp; Parallel Resistor Circuits<\/h2>\r\n<ol>\r\n \t<li>I<sub>1<\/sub> = 0.83 A, I<sub>2<\/sub> = 0.83 A, I = 1.66 A<\/li>\r\n \t<li>I<sub>1<\/sub> = 0.05 A, I<sub>2<\/sub> = 0.12 A, R<sub>2<\/sub> = 292 \u03a9<\/li>\r\n \t<li>I<sub>1<\/sub> = 1.2 A, I<sub>2<\/sub> = 2.0 A, I = 3.2 A<\/li>\r\n \t<li>V = 240 V, I<sub>1<\/sub> = 24 A, I = 44 A<\/li>\r\n \t<li>I<sub>2<\/sub> = 0.15 A, V<sub>1<\/sub> = 32 V, V<sub>3<\/sub> = 18 V, R<sub>1<\/sub> = 128 \u03a9, R<sub>3<\/sub> = 120 \u03a9<\/li>\r\n \t<li>V<sub>1<\/sub> = 96 V, V<sub>2<\/sub>= 84 V, R<sub>1<\/sub> = 2400 \u03a9, I<sub>3<\/sub> = 0.045 A = 45 mA, I = 85mA<\/li>\r\n \t<li>I<sub>1<\/sub> = 0.33 A, I<sub>2<\/sub> = 0.40 A, I 0.73 A<\/li>\r\n \t<li>I<sub>3<\/sub> = 0.38A, V<sub>2<\/sub> = 69 V, V<sub>3<\/sub> = 69 V, R<sub>1<\/sub> = 85 \u03a9, R<sub>2<\/sub> = 314 \u03a9, R<sub>3<\/sub> = 182 \u03a9<\/li>\r\n \t<li>I<sub>1\u00a0<\/sub> = 1.05 A, I<sub>2<\/sub> = 0.53 A, I = 1.58 A, V<sub>1<\/sub> = V<sub>2<\/sub> = 10.5 V, V<sub>3<\/sub> = 23.7 V, V<sub>4<\/sub> = 15.8 V<\/li>\r\n \t<li>I<sub>1<\/sub> = 0.120 A = 120 mA, I<sub>2<\/sub> = 0.080 A = 80 mA, I = 0.200 A = 200 mA,V<sub>2<\/sub> = V<sub>3 = <\/sub>14.4 V, V<sub>5<\/sub> = 10.8 V, R<sub>2<\/sub> = 120 \u03a9, R<sub>4<\/sub> = 84 \u03a9<\/li>\r\n \t<li>I<sub>1<\/sub> = 0.21 A, I = 0.46 A, V<sub>2<\/sub> = V<sub>3<\/sub> = 2.5 V, V<sub>4<\/sub> = 9.2 V, V<sub>5<\/sub> = 3.7 V, V = 15.4 V<\/li>\r\n \t<li>I<sub>1<\/sub> = 60 mA = 0.060 A, V<sub>1<\/sub> = 78 V, V<sub>3<\/sub> = V<sub>4<\/sub> = 180 V, R<sub>1<\/sub> = 1300 \u03a9 = 1.3. k\u03a9, R<sub>2<\/sub> = 1700 \u03a9 = 1.7 k\u03a9, R<sub>3<\/sub> = 2000 \u03a9 = 2.0 k\u03a9<\/li>\r\n \t<li>I<sub>2<\/sub> = 16.92 mA,\u00a0I = 24.12 mA, V<sub>1<\/sub> = 108 V, V<sub>3<\/sub> = 338.4 V, V<sub>4<\/sub> = 120.5 V, R<sub>2<\/sub> = 32.00\u00a0k\u03a9, R<sub>5<\/sub> = 11.94\u00a0k\u03a9<\/li>\r\n \t<li>I<sub>2<\/sub> = 0.30 A, V<sub>1<\/sub> =\u00a0V<sub>4<\/sub> = 6.0 V, V<sub>2<\/sub><span style=\"font-size: 14pt;\"> = 2.0 V<\/span><span style=\"font-size: 14pt;\">, V<\/span><sub>5<\/sub><span style=\"font-size: 14pt;\"> = 4.0 V, V = 26 V, R<\/span><sub>3<\/sub><span style=\"font-size: 14pt;\"> =\u00a0<\/span><span style=\"font-size: 14pt;\">R<\/span><sub>6 <\/sub><span style=\"font-size: 14pt;\">= <\/span><span style=\"font-size: 14pt;\">20 <\/span><span style=\"font-size: 14pt;\">\u03a9<\/span><\/li>\r\n<\/ol>\r\n<h2>23.6.1 Creating Different Resistor Combinations<\/h2>\r\nAssume each resistor = 2 W\r\n<ol type=\"i\">\r\n \t<li>All in series means R total = 6 W<\/li>\r\n \t<li>All in parallel means R total = 0.67 W<\/li>\r\n \t<li>One resistor followed by two in parallel means R total = 3 W<\/li>\r\n \t<li>One resistor in parallel with two in series means R total = 1.33 W<\/li>\r\n<\/ol>\r\n<h2>23.6.2 A Trick Circuit Combination<\/h2>\r\nThe hint was in the title.\u00a0 This is a trick circuit.\u00a0 They have the same current flow out of the battery.\u00a0 Work it out from the far right.\u00a0 The 50 W in series with the 50 W at the far end gives you 100 W in parallel with 100 W for a total of 50 W, and so on.","rendered":"<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Resources<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=YMNZ2oYu-qI\">Mechanical Universe &#8211; Episode 33 Electric Circuits<\/a><\/p>\n<p>Extra Help: <a href=\"http:\/\/www.a-levelphysicstutor.com\/index-elect.php\">A-Level Physics Tutor<\/a><\/p>\n<\/div>\n<\/div>\n<p>Equations Introduced and Used in this Topic:<\/p>\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\n<li>Parallel Resistors:\n<ul style=\"list-style-type: none;\">\n<li>[latex]V_s=V_1=V_2=V_3=[\/latex] &#8230;<\/li>\n<li>[latex]I_s=I_1+I_2+I_3+[\/latex] &#8230;<\/li>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+ ...[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Series Resistors:\n<ul style=\"list-style-type: none;\">\n<li>[latex]I_s=I_1=I_2=I_3=[\/latex] &#8230;<\/li>\n<li>[latex]V_s=V_1+V_2+V_3+[\/latex] &#8230;<\/li>\n<li>[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>Kirchoff\u2019s Circuit Laws:<\/p>\n<p style=\"text-align: center;\">[latex]\\sum V_\\text{ source}=\\sum V_\\text{ drops}[\/latex]&#8230; (Conservation of Energy)<\/p>\n<p style=\"text-align: center;\">[latex]\\sum I_\\text{ into mode}=\\sum I_\\text{ out of mode}[\/latex]&#8230; (Conservation of Charge)<\/p>\n<p>Where\u2026<\/p>\n<ul>\n<li>[latex]I[\/latex]\u00a0is the Electric Current, measured in amperes or amps (A)<\/li>\n<li>[latex]V[\/latex] is the voltage, measured in volts (V)<\/li>\n<li>[latex]R[\/latex] is the Resistance, measured in ohms (\u03a9)<\/li>\n<\/ul>\n<h1>23.1 Ohm\u2019s Law (Single Circuits)<\/h1>\n<p>Ohm\u2019s Law relates voltage, Current and Resistance given by the equation of<\/p>\n<p style=\"text-align: center;\">[latex]V = IR[\/latex]<\/p>\n<p>In this last Chapter for this course, we analyze basic electric circuits using Ohm\u2019s Law and later Kirchoff\u2019s Laws in series, parallel and combined resistor arrangements. While there are a number of circuit symbols are used in circuit diagrams;<\/p>\n<ul>\n<li>We use the symbol of jagged lines for electrical resistance <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-297 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/jagged-lines-e1678305914518.png\" alt=\"\" width=\"150\" height=\"33\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/jagged-lines-e1678305914518.png 150w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/jagged-lines-e1678305914518-65x14.png 65w\" sizes=\"auto, (max-width: 150px) 100vw, 150px\" \/><\/li>\n<li>We use two open circles to represent the source voltage <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-298\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circles.png\" alt=\"\" width=\"71\" height=\"27\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circles.png 71w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circles-65x25.png 65w\" sizes=\"auto, (max-width: 71px) 100vw, 71px\" \/><\/li>\n<\/ul>\n<p>Combined in a simple circuit these symbols look like:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-299\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circuit-e1678306053290.png\" alt=\"\" width=\"400\" height=\"329\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circuit-e1678306053290.png 504w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circuit-e1678306053290-300x246.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circuit-e1678306053290-65x53.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circuit-e1678306053290-225x185.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/circuit-e1678306053290-350x288.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/p>\n<p>The following examples use the circuit diagram to solve for Ohm\u2019s Law questions.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 23.1.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the missing variable for each of the following circuits:<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-301 aligncenter\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/1.png\" alt=\"\" width=\"291\" height=\"171\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/1.png 291w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/1-65x38.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/1-225x132.png 225w\" sizes=\"auto, (max-width: 291px) 100vw, 291px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]V = IR[\/latex]<\/li>\n<li>120 V = [latex]I[\/latex] (420 \u03a9)<\/li>\n<li>[latex]I[\/latex]= 120 V \u00f7 420 \u03a9\u00a0 or\u00a0 0.29 A<\/li>\n<\/ul>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-302 aligncenter\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.png\" alt=\"\" width=\"286\" height=\"180\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.png 286w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2-65x41.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2-225x142.png 225w\" sizes=\"auto, (max-width: 286px) 100vw, 286px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]V = IR[\/latex]<\/li>\n<li>[latex]V[\/latex] = (32 \u00d7 10<sup>\u22123<\/sup> A)(240 \u03a9)<\/li>\n<li>[latex]V[\/latex] = 7.7 V<\/li>\n<\/ul>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-303 aligncenter\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.png\" alt=\"\" width=\"290\" height=\"179\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3-65x40.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3-225x139.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]V = IR[\/latex]<\/li>\n<li>0.12 \u00d7 10<sup>\u22123<\/sup> V = (800 \u00d7 10<sup>\u22126<\/sup> A)([latex]R[\/latex])<\/li>\n<li>[latex]R[\/latex] = 0.12 \u00d7 10<sup>\u22123<\/sup> V \u00f7 800 \u00d7 10<sup>\u22126<\/sup> AR =\u00a0 0.15 \u03a9<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 23.1 Ohm\u2019s Law<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve for the unknown in each of the following single circuits.<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-305\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e1.png\" alt=\"\" width=\"260\" height=\"151\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e1.png 260w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e1-65x38.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e1-225x131.png 225w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-306\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e2.png\" alt=\"\" width=\"257\" height=\"153\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e2.png 257w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e2-65x39.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e2-225x134.png 225w\" sizes=\"auto, (max-width: 257px) 100vw, 257px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-307\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e3.png\" alt=\"\" width=\"257\" height=\"149\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e3.png 257w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e3-65x38.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e3-225x130.png 225w\" sizes=\"auto, (max-width: 257px) 100vw, 257px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-308\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e4.png\" alt=\"\" width=\"253\" height=\"147\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e4.png 253w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e4-65x38.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e4-225x131.png 225w\" sizes=\"auto, (max-width: 253px) 100vw, 253px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-888\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-5-e1680297396430.jpg\" alt=\"\" width=\"257\" height=\"150\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-5-e1680297396430.jpg 546w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-5-e1680297396430-300x175.jpg 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-5-e1680297396430-65x38.jpg 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-5-e1680297396430-225x131.jpg 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-5-e1680297396430-350x204.jpg 350w\" sizes=\"auto, (max-width: 257px) 100vw, 257px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-310\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e6.png\" alt=\"\" width=\"257\" height=\"151\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e6.png 257w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e6-65x38.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e6-225x132.png 225w\" sizes=\"auto, (max-width: 257px) 100vw, 257px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-311\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e7.png\" alt=\"\" width=\"253\" height=\"148\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e7.png 253w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e7-65x38.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e7-225x132.png 225w\" sizes=\"auto, (max-width: 253px) 100vw, 253px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-889\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-8-e1680297461471.jpg\" alt=\"\" width=\"257\" height=\"149\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-8-e1680297461471.jpg 547w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-8-e1680297461471-300x174.jpg 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-8-e1680297461471-65x38.jpg 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-8-e1680297461471-225x131.jpg 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.1-8-e1680297461471-350x203.jpg 350w\" sizes=\"auto, (max-width: 257px) 100vw, 257px\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>23.2 Series Resistors<\/h1>\n<div>\n<div class=\"textbox\">\n<p>Extra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/circuits\/Lesson-4\/Two-Types-of-Connections\">Two Types of Connections<\/a><\/p>\n<p>Extra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/circuits\/Lesson-4\/Series-Circuits\">Series Circuits<\/a><\/p>\n<\/div>\n<\/div>\n<p>Equations Introduced or Used for this Section:<\/p>\n<p style=\"text-align: center;\">[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/p>\n<p>The equation we use to analyze both series and parallel resistors in combination can be found by using the original law developed for measuring the resistivity of resistors.<\/p>\n<p>In general, this law follows the form:<\/p>\n<p style=\"text-align: center;\">[latex]R=\\dfrac{\\rho L}{A}[\/latex]<\/p>\n<p>Where\u2026<\/p>\n<ul>\n<li>[latex]R[\/latex] is the Resistance of the resistor measured in ohms (\u03a9)<\/li>\n<li>[latex]\\varrho[\/latex]\u00a0 is the resistivity of the material, which is a measure of how well the material conducts electricity<\/li>\n<li>[latex]L[\/latex] is the length of the conductor measured in metres (m)<\/li>\n<li>[latex]A[\/latex] is the cross sectional area of the conductor measured in square metres (m<sup>2<\/sup>)<\/li>\n<\/ul>\n<p>Resistors in series are simply resistors that are connected to each other and will look similar to:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-314 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resisters.png\" alt=\"\" width=\"661\" height=\"50\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resisters.png 661w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resisters-300x23.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resisters-65x5.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resisters-225x17.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resisters-350x26.png 350w\" sizes=\"auto, (max-width: 661px) 100vw, 661px\" \/><\/p>\n<p>From the equation for calculating the resistivity of a resistor as shown above,<\/p>\n<p style=\"text-align: center;\">[latex]R=\\dfrac{\\rho L}{A}[\/latex]<\/p>\n<p>the resultant of adding resistors one after another in a series effectively changes the length which can be easily envisioned by just considering adding identical resistors to each other which in simplest terms is just changing the length&#8230; We only need to add these resistances up.<\/p>\n<p>This strategy works for all different kinds of resistors added to each other, simply add up the total resistance of each resistor, which yields:<\/p>\n<p style=\"text-align: center;\">[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 23.2.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the total resistance of the following Resistors connected in Series.<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-315\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.1-300x148.png\" alt=\"\" width=\"300\" height=\"148\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.1-300x148.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.1-65x32.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.1-225x111.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.1-350x173.png 350w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.1.png 384w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/li>\n<li>[latex]R_T[\/latex] = 240 \u03a9 + 220 \u03a9 + 280 \u03a9<\/li>\n<li>[latex]R_T[\/latex] = 740 \u03a9<\/li>\n<\/ul>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-316\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.2-300x148.png\" alt=\"\" width=\"300\" height=\"148\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.2-300x148.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.2-65x32.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.2-225x111.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.2-350x173.png 350w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.2.png 373w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/li>\n<li>[latex]R_T[\/latex] = 50 \u03a9 + 100 \u03a9 + 210 \u03a9<\/li>\n<li>[latex]R_T[\/latex] = 360 \u03a9<\/li>\n<\/ul>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-318\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.3-300x145.png\" alt=\"\" width=\"300\" height=\"145\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.3-300x145.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.3-65x31.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.3-225x109.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.3-350x169.png 350w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.3.png 383w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/li>\n<li>[latex]R_T[\/latex] = 0.6 k\u03a9 + 1.2 k\u03a9 + 0.8 k\u03a9<\/li>\n<li>[latex]R_T[\/latex] = 2.6 k\u03a9<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 23.2 Series Resistors<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the total resistance of each of the following circuits<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-319 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.1-e1678319399151.png\" alt=\"\" width=\"290\" height=\"158\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.1-e1678319399151.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.1-e1678319399151-65x35.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.1-e1678319399151-225x123.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-320 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.2-e1678319413808.png\" alt=\"\" width=\"290\" height=\"163\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.2-e1678319413808.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.2-e1678319413808-65x37.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.2-e1678319413808-225x126.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-321 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.3-e1678319485574.png\" alt=\"\" width=\"290\" height=\"161\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.3-e1678319485574.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.3-e1678319485574-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.3-e1678319485574-225x125.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-353 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.4-e1678319517593.png\" alt=\"\" width=\"290\" height=\"161\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.4-e1678319517593.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.4-e1678319517593-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.4-e1678319517593-225x125.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-323 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.5-e1678319377803.png\" alt=\"\" width=\"290\" height=\"158\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.5-e1678319377803.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.5-e1678319377803-65x35.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.5-e1678319377803-225x123.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-354 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.6-e1678319577360.png\" alt=\"\" width=\"290\" height=\"159\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.6-e1678319577360.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.6-e1678319577360-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.2.6-e1678319577360-225x123.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-326 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.7-e1678319589151.png\" alt=\"\" width=\"290\" height=\"158\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.7-e1678319589151.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.7-e1678319589151-65x35.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.7-e1678319589151-225x123.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-327 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.8-e1678319604363.png\" alt=\"\" width=\"290\" height=\"160\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.8-e1678319604363.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.8-e1678319604363-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/2.8-e1678319604363-225x124.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>23.3 Parallel Resistors<\/h1>\n<div class=\"textbox\">Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/circuits\/Lesson-4\/Parallel-Circuits\">Parallel Circuits<\/a><\/div>\n<p>Equations Introduced or Used for this Section:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/p>\n<p>When resistors are combined in a parallel arrangement (side by side) the total resistance can once again be inferred by the original equation that is used to calculate the resistance of a resistor. From the equation for calculating the resistivity of a resistor as shown below,<\/p>\n<p style=\"text-align: center;\">[latex]R=\\dfrac{\\varrho L}{A}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-329\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resistors-combined.png\" alt=\"\" width=\"301\" height=\"165\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resistors-combined.png 301w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resistors-combined-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/resistors-combined-225x123.png 225w\" sizes=\"auto, (max-width: 301px) 100vw, 301px\" \/><\/p>\n<p>One way of looking at resistors that are combined in parallel is that the cross sectional area of the resistor is increased.\u00a0 When put together these resistors combine to allow more electricity to flow in a circuit. This means that we can simply add up the increased areas to get some measure of how the total resistance changes.<\/p>\n<p>From the original equation, you will notice that the effect of changing the cross sectional area of the resistor is in the denominator or is the reciprocal variable in calculating the resistance for resistors combined in this fashion.\u00a0 This means that when combining resistors connected in parallel we are changing the cross sectional area in the denominator by adding new resistors that increase this cross sectional value.<\/p>\n<p>Combining resistors in parallel means that we add the reciprocals of all the resistors connected in parallel to arrive at the total resistance of the new circuit.<\/p>\n<p>This equation looks like:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/p>\n<p>The following examples analyze the total resistance of resistors in parallel.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 23.3.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the total resistance of the following Resistors connected in Parallel.<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-331\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.1-300x127.png\" alt=\"\" width=\"300\" height=\"127\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.1-300x127.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.1-65x28.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.1-225x95.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.1-350x148.png 350w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.1.png 390w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{20\\Omega}+\\dfrac{1}{30\\Omega}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}= 12\\Omega[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-332\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.2-300x145.png\" alt=\"\" width=\"300\" height=\"145\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.2-300x145.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.2-65x31.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.2-225x109.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.2.png 351w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{145\\Omega}+\\dfrac{1}{185\\Omega}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}= 81.3\\Omega[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-333\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.3-300x144.png\" alt=\"\" width=\"300\" height=\"144\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.3-300x144.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.3-65x31.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.3-225x108.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.3-350x168.png 350w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.3.png 393w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{10\\Omega}+\\dfrac{1}{12\\Omega}+\\dfrac{1}{16\\Omega}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}= 4.1\\Omega[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-334\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.4-300x143.png\" alt=\"\" width=\"300\" height=\"143\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.4-300x143.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.4-65x31.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.4-225x107.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.4-350x167.png 350w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/3.4.png 394w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{15\\Omega}+\\dfrac{1}{12\\Omega}+\\dfrac{1}{12\\Omega}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}= 4.3\\Omega[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 23.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the total resistance of each of the following circuits.<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-336\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.1.png\" alt=\"\" width=\"270\" height=\"149\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.1.png 289w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.1-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.1-225x124.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-337\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.2.png\" alt=\"\" width=\"270\" height=\"146\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.2.png 287w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.2-65x35.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.2-225x122.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-338\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.3-e1678374202289.png\" alt=\"\" width=\"270\" height=\"144\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.3-e1678374202289.png 291w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.3-e1678374202289-65x35.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.3-e1678374202289-225x120.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-339\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.4.png\" alt=\"\" width=\"270\" height=\"153\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.4.png 286w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.4-65x37.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.4-225x127.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-340\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.5-e1678374225253.png\" alt=\"\" width=\"270\" height=\"147\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.5-e1678374225253.png 290w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.5-e1678374225253-65x35.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.5-e1678374225253-225x123.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-341\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.6.png\" alt=\"\" width=\"273\" height=\"151\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-342\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.7.png\" alt=\"\" width=\"270\" height=\"141\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.7.png 285w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.7-65x34.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.7-225x118.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-343\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.8.png\" alt=\"\" width=\"270\" height=\"153\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.8.png 275w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.8-65x37.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.8-225x128.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-345\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.9.png\" alt=\"\" width=\"270\" height=\"149\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.9.png 279w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.9-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.9-225x124.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-344\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.10.png\" alt=\"\" width=\"270\" height=\"150\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.10.png 282w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.10-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.10-225x125.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-346\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.11.png\" alt=\"\" width=\"271\" height=\"158\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.11.png 271w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.11-65x38.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.11-225x131.png 225w\" sizes=\"auto, (max-width: 271px) 100vw, 271px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-347\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.12-e1678374363705.png\" alt=\"\" width=\"270\" height=\"142\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.12-e1678374363705.png 282w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.12-e1678374363705-65x34.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.12-e1678374363705-225x118.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-348\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.13-e1678374389829.png\" alt=\"\" width=\"270\" height=\"152\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.13-e1678374389829.png 278w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.13-e1678374389829-65x36.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.13-e1678374389829-225x126.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-349\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.14-e1678374412527.png\" alt=\"\" width=\"270\" height=\"143\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.14-e1678374412527.png 286w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.14-e1678374412527-65x35.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/233.14-e1678374412527-225x120.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>23.4 Series &amp; Parallel Resistors<\/h1>\n<p>Equations Introduced or Used for this Section:<\/p>\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\n<li>[latex]R_T=R_1+R_2+R_3+...[\/latex]<\/li>\n<\/ul>\n<p>When we try to find the total resistance of combined series and parallel circuits we generally reduce the circuits from the simplest combination to the most complex.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-355\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4-circut.png\" alt=\"\" width=\"500\" height=\"78\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4-circut.png 728w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4-circut-300x47.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4-circut-65x10.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4-circut-225x35.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4-circut-350x54.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p>For instance when we consider the following circuit arrangement&#8230;<\/p>\n<p>You can see that we have a mix of a parallel resistor connected in series to a single resistor. The simplest way to reduce this is to reduce the parallel resistor to find a single resistor of equal value. After we have reduced the parallel branch we then just add the two resistor values together as if joined in series. This looks like:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-356\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/parallel-resistor.png\" alt=\"\" width=\"312\" height=\"39\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/parallel-resistor.png 312w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/parallel-resistor-300x38.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/parallel-resistor-65x8.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/parallel-resistor-225x28.png 225w\" sizes=\"auto, (max-width: 312px) 100vw, 312px\" \/><\/p>\n<p>Resistance of [latex]R_1 + R_2 + R_3 + ...[\/latex]<\/p>\n<p>If we complicate this arrangement by adding another resistor in series to R<sub>1<\/sub>, we get the following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-357\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r1_4.png\" alt=\"\" width=\"500\" height=\"83\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r1_4.png 719w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r1_4-300x50.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r1_4-65x11.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r1_4-225x38.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r1_4-350x58.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p>First we reduce the top branch of R<sub>1<\/sub> &amp; R<sub>2<\/sub> to a single Resistor, we then continue to reduce as we did in the combination shown directly above.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-359\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r-rb.png\" alt=\"\" width=\"500\" height=\"75\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r-rb.png 719w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r-rb-300x45.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r-rb-65x10.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r-rb-225x34.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/r-rb-350x53.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 23.4.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the total resistance of the following Resistors connected in Series and Parallel.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-360\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.1.png\" alt=\"\" width=\"500\" height=\"181\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.1.png 731w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.1-300x108.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.1-65x23.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.1-225x81.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.1-350x126.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>To find the total resistance of this circuit you must identify and solve each part of the circuit&#8230; In this circuit, you have a parallel branch, which is combined in series.<\/p>\n<ul>\n<li>First the parallel branch&#8230;\n<ul>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+ ...[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{50\\Omega}+\\dfrac{1}{50\\Omega}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}= 25\\Omega[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Second the series&#8230;\n<ul>\n<li>[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/li>\n<li>R<sub>T<\/sub> = 20 \u03a9 + 50 \u03a9 + 50 \u03a9<\/li>\n<li>R<sub>T<\/sub> = 125 \u03a9<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 23.4.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the total resistance of the following Resistors connected in Series and Parallel.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-361\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.2.png\" alt=\"\" width=\"500\" height=\"151\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.2.png 664w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.2-300x91.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.2-65x20.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.2-225x68.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.4.2-350x106.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>To find the total resistance of this circuit you have three distinct parts&#8230; a series branch in a parallel branch which is finally combined in series.<\/p>\n<ul>\n<li>First the series&#8230;\n<ul>\n<li>[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/li>\n<li>R<sub>T<\/sub> = 25 \u03a9 + 25 \u03a9<\/li>\n<li>R<sub>T<\/sub> = 50 \u03a9<\/li>\n<\/ul>\n<\/li>\n<li>Second the parallel branch &#8230;\n<ul>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{50\\Omega}+\\dfrac{1}{25\\Omega}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}=16.7\\Omega[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Finally the total circuit&#8230;\n<ul>\n<li>[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/li>\n<li>R<sub>T<\/sub> = 25 \u03a9 + 16.7 \u03a9<\/li>\n<li>R<sub>T<\/sub> = 41.7 \u03a9 (\u2248 42 \u03a9)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 23.4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the total resistance of the following circuits<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-362\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.1.png\" alt=\"\" width=\"500\" height=\"292\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.1.png 789w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.1-300x175.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.1-768x448.png 768w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.1-65x38.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.1-225x131.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.1-350x204.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-363\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.2.png\" alt=\"\" width=\"500\" height=\"139\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.2.png 807w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.2-300x84.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.2-768x214.png 768w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.2-65x18.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.2-225x63.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.2-350x98.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-364\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.3.png\" alt=\"\" width=\"500\" height=\"182\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.3.png 812w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.3-300x109.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.3-768x280.png 768w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.3-65x24.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.3-225x82.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.3-350x128.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-365\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.4.png\" alt=\"\" width=\"500\" height=\"155\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.4.png 731w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.4-300x93.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.4-65x20.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.4-225x70.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.4.4-350x108.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>23.5 Solving Series &amp; Parallel Resistor Circuits<\/h1>\n<div class=\"textbox\">Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/circuits\/Lesson-4\/Combination-Circuits\">Combination Circuits<\/a><\/div>\n<p>Kirchoff&#8217;s Circuit Laws are extensions of the concept of conservation of energy and the law of conservation of charge applied to circuits.<\/p>\n<p>From the Conservation of Charge&#8230; The current entering any junction (node) of a circuit is equal to the current leaving that junction (node).<\/p>\n<p style=\"text-align: center;\">[latex]I\\text{ entering}=I\\text{ leaving}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-891\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1.png\" alt=\"\" width=\"600\" height=\"223\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1.png 1824w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1-300x112.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1-1024x381.png 1024w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1-768x286.png 768w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1-1536x572.png 1536w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1-65x24.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1-225x84.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/node-1-350x130.png 350w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/p>\n<p>Applying Kirchoff\u2019s Law for the above circuit&#8230; [latex]I_0=I_1+I_2+I_3[\/latex]<\/p>\n<p>From the Conservation of Energy&#8230; The sum of all the voltages around the loop is equal to zero or for a simple circuit: The voltage at the source is equal to the voltage drops along the circuit.<\/p>\n<p style=\"text-align: center;\">[latex]V\\text{source}=\\sum V\\text{drops}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-368\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5-circuit.png\" alt=\"\" width=\"500\" height=\"252\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5-circuit.png 721w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5-circuit-300x151.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5-circuit-65x33.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5-circuit-225x113.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5-circuit-350x176.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p>Applying Kirchoff\u2019s Law for this circuit&#8230; [latex]V_s=V_1+V_2+V_3+V_4[\/latex]<\/p>\n<p>The final part of this section includes adding Kirchoff\u2019s Laws to Ohm\u2019s Law and applying the techniques to reduce combined series and parallel circuits. Ultimately you will be looking to see if you can use any one of these to reduce some part of the circuit.<\/p>\n<p>Some of the rules that you must remember are:<\/p>\n<p>In a series circuit:<\/p>\n<ul>\n<li>The electric current remains constant throughout the circuit.<\/li>\n<li>The sum of all the voltage drops (or losses) will add to the original source voltage.<\/li>\n<\/ul>\n<p>In a parallel circuit:<\/p>\n<ul>\n<li>The voltage drop on any parallel branch will be identical to another.<\/li>\n<li>The electric current will split up in each of the branches. The sum of the electric current in each branch will equal the electric current going into the branches.<\/li>\n<\/ul>\n<p>You will Use Ohm\u2019s Law:<\/p>\n<ul>\n<li>Whenever you know two of the three variables in any part of the circuit, you can apply Ohm&#8217;s Law to that part.<\/li>\n<\/ul>\n<p>Solving circuits can sometimes resemble a Puzzle:<\/p>\n<ul>\n<li>Somewhere in the circuit is generally one small thing that you are able to solve. Solving this part will then lead to another part that you can solve. The challenge is that you will need to search for the one thing that you can solve in the circuit, which then allows you to solve another part.<\/li>\n<\/ul>\n<figure id=\"attachment_369\" aria-describedby=\"caption-attachment-369\" style=\"width: 679px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-369\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/potential.png\" alt=\"\" width=\"679\" height=\"692\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/potential.png 679w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/potential-294x300.png 294w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/potential-65x66.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/potential-225x229.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/potential-350x357.png 350w\" sizes=\"auto, (max-width: 679px) 100vw, 679px\" \/><figcaption id=\"caption-attachment-369\" class=\"wp-caption-text\">Potential difference for Gravitational Potential<\/figcaption><\/figure>\n<p>One common way to visualize the voltage drops, as related to the source voltage, is to imagine a water pump leading to a series of water falls (Gravitational Potential Analogy). The source pump (source voltage) lifts the water up through a certain height, meaning that the potential difference allows the water to then fall through a series of water falls&#8230; Note for the parallel branch of [latex]V_3[\/latex] &amp; [latex]V_4[\/latex], both branches fall through the exact same parallel difference and are equal. What is unique about the parallel branch is the water splits into two currents, which then recombine back into a single current after the water has passed through both branches.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 23.5.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the missing variable for each of the following circuits:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-371 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.1-e1678374591611.png\" alt=\"\" width=\"500\" height=\"248\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.1-e1678374591611.png 500w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.1-e1678374591611-300x149.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.1-e1678374591611-65x32.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.1-e1678374591611-225x112.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.1-e1678374591611-350x174.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p>First&#8230; Find the total resistance of the circuit<\/p>\n<p>The top branch is in series, so<\/p>\n<ul>\n<li>First the series&#8230;\n<ul>\n<li>[latex]R_T=R_1+R_2+R_3+[\/latex] &#8230;<\/li>\n<li>[latex]R_T[\/latex] = 20 \u03a9 + 20 \u03a9<\/li>\n<li>[latex]R_T[\/latex] = 40 \u03a9<\/li>\n<\/ul>\n<\/li>\n<li>Second the parallel branch&#8230;\n<ul>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{R_1}+\\dfrac{1}{R_2}+\\dfrac{1}{R_3}+...[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}=\\dfrac{1}{50\\Omega}+\\dfrac{1}{25\\Omega}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{R_T}= 16.7\\omega[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Third apply Ohm&#8217;s Law&#8230;\n<ul>\n<li>[latex]V=IR[\/latex]<\/li>\n<li>60 V = [latex]I[\/latex](20 \u03a9)<\/li>\n<li>[latex]I[\/latex] = 60 V \u00f7 20\u03a9 or 3.0 A<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>Since the resistance over the top and bottom branches are identical, the current splits evenly.<\/p>\n<p>Therefore&#8230; [latex]I_1 = I_2[\/latex] = 1.5 A<\/p>\n<p>Currents [latex]I_1[\/latex] &amp; [latex]I_2[\/latex]\u00a0can also be found by using Ohm&#8217;s Law for each branch.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 23.5.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the missing variable for each of the following circuits:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-372\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.2.png\" alt=\"\" width=\"500\" height=\"258\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.2.png 650w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.2-300x155.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.2-65x34.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.2-225x116.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex23.5.2-350x181.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>This is solved in several steps&#8230;<\/p>\n<ul class=\"twocolumn\">\n<li>[latex]R_1[\/latex]&#8230; [latex]V=IR_1[\/latex]\n<ul>\n<li>40 V = (1.2 A) [latex]R_1[\/latex]<\/li>\n<li>[latex]R_1[\/latex] = 40 V \u00f7 1.2 A<\/li>\n<li>[latex]R_1[\/latex]= 33.3 \u03a9 (\u2248 33 \u03a9)<\/li>\n<\/ul>\n<\/li>\n<li>[latex]R_2[\/latex]&#8230; [latex]V=IR_2[\/latex]\n<ul>\n<li>80 V = (0.50 A) [latex]R_2[\/latex]<\/li>\n<li>[latex]R_2[\/latex] = 80 V \u00f7 0.50 A<\/li>\n<li>[latex]R_2[\/latex] = 160 \u03a9<\/li>\n<\/ul>\n<\/li>\n<li style=\"break-inside: avoid;\">[latex]I_3[\/latex]&#8230; [latex]I_0=I_1+I_2+I_3[\/latex]\n<ul>\n<li>1.2 A = 0.50 A + I<sub>3<\/sub><\/li>\n<li>[latex]I_3[\/latex] = 1.2 A \u2212 0.50 A<\/li>\n<li>[latex]I_3[\/latex] = 0.70 A<\/li>\n<\/ul>\n<\/li>\n<li style=\"break-inside: avoid;\">[latex]R_3[\/latex]&#8230; [latex]V=IR_3[\/latex]\n<ul>\n<li>80 V = (0.70 A) [latex]R_3[\/latex]<\/li>\n<li>[latex]R_3[\/latex] = 80 V \u00f7 0.70 A<\/li>\n<li>[latex]R_3[\/latex] = 114 \u03a9(\u2248 110 \u03a9)<\/li>\n<\/ul>\n<\/li>\n<li>[latex]V_2[\/latex] &amp; [latex]V_3[\/latex] are identical&#8230;\n<ul>\n<li>[latex]V_2[\/latex] &amp; [latex]V_3[\/latex] = 120 V &#8211; 40 V<\/li>\n<li>[latex]V_2[\/latex] &amp; [latex]V_3[\/latex] = 80 V<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 23.5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve for all unknowns in the following circuits.<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1198 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex-23.5-1.png\" alt=\"\" width=\"650\" height=\"414\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex-23.5-1.png 650w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex-23.5-1-300x191.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex-23.5-1-65x41.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex-23.5-1-225x143.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex-23.5-1-350x223.png 350w\" sizes=\"auto, (max-width: 650px) 100vw, 650px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-375\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.2.png\" alt=\"\" width=\"278\" height=\"170\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.2.png 278w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.2-65x40.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.2-225x138.png 225w\" sizes=\"auto, (max-width: 278px) 100vw, 278px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-376\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.3.png\" alt=\"\" width=\"276\" height=\"171\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.3.png 276w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.3-65x40.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.3-225x139.png 225w\" sizes=\"auto, (max-width: 276px) 100vw, 276px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-377 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.4-e1678380405317.png\" alt=\"\" width=\"277\" height=\"169\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.4-e1678380405317.png 277w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.4-e1678380405317-65x40.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.4-e1678380405317-225x137.png 225w\" sizes=\"auto, (max-width: 277px) 100vw, 277px\" \/><\/li>\n<\/ol>\n<ol start=\"5\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-378\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.5.png\" alt=\"\" width=\"443\" height=\"210\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.5.png 443w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.5-300x142.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.5-65x31.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.5-225x107.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.5-350x166.png 350w\" sizes=\"auto, (max-width: 443px) 100vw, 443px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-379\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.6.png\" alt=\"\" width=\"552\" height=\"268\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.6.png 552w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.6-300x146.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.6-65x32.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.6-225x109.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.6-350x170.png 350w\" sizes=\"auto, (max-width: 552px) 100vw, 552px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-380\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.7.png\" alt=\"\" width=\"550\" height=\"265\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.7.png 550w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.7-300x145.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.7-65x31.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.7-225x108.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.7-350x169.png 350w\" sizes=\"auto, (max-width: 550px) 100vw, 550px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-381\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.8.png\" alt=\"\" width=\"443\" height=\"208\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.8.png 443w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.8-300x141.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.8-65x31.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.8-225x106.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.8-350x164.png 350w\" sizes=\"auto, (max-width: 443px) 100vw, 443px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-382 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.9-e1678380663885.png\" alt=\"\" width=\"432\" height=\"218\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.9-e1678380663885.png 432w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.9-e1678380663885-300x151.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.9-e1678380663885-65x33.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.9-e1678380663885-225x114.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.9-e1678380663885-350x177.png 350w\" sizes=\"auto, (max-width: 432px) 100vw, 432px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-383 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.10-e1678380061321.png\" alt=\"\" width=\"434\" height=\"203\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.10-e1678380061321.png 434w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.10-e1678380061321-300x140.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.10-e1678380061321-65x30.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.10-e1678380061321-225x105.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.10-e1678380061321-350x164.png 350w\" sizes=\"auto, (max-width: 434px) 100vw, 434px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-384 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.11-e1678380082952.png\" alt=\"\" width=\"436\" height=\"202\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.11-e1678380082952.png 436w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.11-e1678380082952-300x139.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.11-e1678380082952-65x30.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.11-e1678380082952-225x104.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.11-e1678380082952-350x162.png 350w\" sizes=\"auto, (max-width: 436px) 100vw, 436px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-385\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.12.png\" alt=\"\" width=\"580\" height=\"259\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.12.png 594w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.12-300x134.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.12-65x29.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.12-225x100.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.12-350x156.png 350w\" sizes=\"auto, (max-width: 580px) 100vw, 580px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-386 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.13-e1678380106687.png\" alt=\"\" width=\"580\" height=\"243\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.13-e1678380106687.png 580w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.13-e1678380106687-300x126.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.13-e1678380106687-65x27.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.13-e1678380106687-225x94.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.13-e1678380106687-350x147.png 350w\" sizes=\"auto, (max-width: 580px) 100vw, 580px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-388\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.14-1.png\" alt=\"\" width=\"580\" height=\"249\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.14-1.png 602w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.14-1-300x129.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.14-1-65x28.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.14-1-225x96.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.5.14-1-350x150.png 350w\" sizes=\"auto, (max-width: 580px) 100vw, 580px\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 23.6.1: Creating Different Resistor Combinations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Using 3 identical resistors, how many circuits can you make with different total resistances? You should be able to create 4 combinations&#8230; There are actually more.<\/p>\n<p>Some unusual combinations???<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 23.6.2: A Trick Circuit Combination<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Which circuit shown below will have the greatest current flow?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-390\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.6.2.png\" alt=\"\" width=\"564\" height=\"160\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.6.2.png 564w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.6.2-300x85.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.6.2-65x18.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.6.2-225x64.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/23.6.2-350x99.png 350w\" sizes=\"auto, (max-width: 564px) 100vw, 564px\" \/><\/p>\n<\/div>\n<\/div>\n<h1>Exercise Answers<\/h1>\n<h2>23.1 Ohm\u2019s Law (Single Circuits)<\/h2>\n<ol class=\"twocolumn\">\n<li>0.2 A<\/li>\n<li>360 V<\/li>\n<li>25 A<\/li>\n<li>3750 \u03a9<\/li>\n<li>216 V<\/li>\n<li>15 \u03a9<\/li>\n<li>0.05 A<\/li>\n<li>28.8 V<\/li>\n<\/ol>\n<h2>23.2 Series Resistors<\/h2>\n<ol class=\"twocolumn\">\n<li>32 \u03a9<\/li>\n<li>32 \u03a9<\/li>\n<li>10 \u03a9<\/li>\n<li>50 k\u03a9<\/li>\n<li>78 k\u03a9<\/li>\n<li>235 k\u03a9<\/li>\n<li>14.8 k\u03a9<\/li>\n<li>330 \u03a9<\/li>\n<\/ol>\n<h2>23.3 Parallel Resistors<\/h2>\n<ol class=\"twocolumn\">\n<li>3.75 \u03a9<\/li>\n<li>10.2 \u03a9<\/li>\n<li>7.47 \u03a9<\/li>\n<li>31.6 \u03a9<\/li>\n<li>40.5 \u03a9<\/li>\n<li>215 \u03a9<\/li>\n<li>25 \u03a9<\/li>\n<li>33.8 \u03a9<\/li>\n<li>70.6 \u03a9<\/li>\n<li>4.07 \u03a9<\/li>\n<li>48 \u03a9<\/li>\n<li>130 \u03a9<\/li>\n<li>35.7 \u03a9<\/li>\n<li>12 \u03a9<\/li>\n<\/ol>\n<h2>23.4 Series &amp; Parallel Resistors<\/h2>\n<ol class=\"twocolumn\">\n<li>6.3 \u03a9<\/li>\n<li>15 \u03a9<\/li>\n<li>25 \u03a9<\/li>\n<li>16.7 \u03a9<\/li>\n<\/ol>\n<h2>23.5 Solving Series &amp; Parallel Resistor Circuits<\/h2>\n<ol>\n<li>I<sub>1<\/sub> = 0.83 A, I<sub>2<\/sub> = 0.83 A, I = 1.66 A<\/li>\n<li>I<sub>1<\/sub> = 0.05 A, I<sub>2<\/sub> = 0.12 A, R<sub>2<\/sub> = 292 \u03a9<\/li>\n<li>I<sub>1<\/sub> = 1.2 A, I<sub>2<\/sub> = 2.0 A, I = 3.2 A<\/li>\n<li>V = 240 V, I<sub>1<\/sub> = 24 A, I = 44 A<\/li>\n<li>I<sub>2<\/sub> = 0.15 A, V<sub>1<\/sub> = 32 V, V<sub>3<\/sub> = 18 V, R<sub>1<\/sub> = 128 \u03a9, R<sub>3<\/sub> = 120 \u03a9<\/li>\n<li>V<sub>1<\/sub> = 96 V, V<sub>2<\/sub>= 84 V, R<sub>1<\/sub> = 2400 \u03a9, I<sub>3<\/sub> = 0.045 A = 45 mA, I = 85mA<\/li>\n<li>I<sub>1<\/sub> = 0.33 A, I<sub>2<\/sub> = 0.40 A, I 0.73 A<\/li>\n<li>I<sub>3<\/sub> = 0.38A, V<sub>2<\/sub> = 69 V, V<sub>3<\/sub> = 69 V, R<sub>1<\/sub> = 85 \u03a9, R<sub>2<\/sub> = 314 \u03a9, R<sub>3<\/sub> = 182 \u03a9<\/li>\n<li>I<sub>1\u00a0<\/sub> = 1.05 A, I<sub>2<\/sub> = 0.53 A, I = 1.58 A, V<sub>1<\/sub> = V<sub>2<\/sub> = 10.5 V, V<sub>3<\/sub> = 23.7 V, V<sub>4<\/sub> = 15.8 V<\/li>\n<li>I<sub>1<\/sub> = 0.120 A = 120 mA, I<sub>2<\/sub> = 0.080 A = 80 mA, I = 0.200 A = 200 mA,V<sub>2<\/sub> = V<sub>3 = <\/sub>14.4 V, V<sub>5<\/sub> = 10.8 V, R<sub>2<\/sub> = 120 \u03a9, R<sub>4<\/sub> = 84 \u03a9<\/li>\n<li>I<sub>1<\/sub> = 0.21 A, I = 0.46 A, V<sub>2<\/sub> = V<sub>3<\/sub> = 2.5 V, V<sub>4<\/sub> = 9.2 V, V<sub>5<\/sub> = 3.7 V, V = 15.4 V<\/li>\n<li>I<sub>1<\/sub> = 60 mA = 0.060 A, V<sub>1<\/sub> = 78 V, V<sub>3<\/sub> = V<sub>4<\/sub> = 180 V, R<sub>1<\/sub> = 1300 \u03a9 = 1.3. k\u03a9, R<sub>2<\/sub> = 1700 \u03a9 = 1.7 k\u03a9, R<sub>3<\/sub> = 2000 \u03a9 = 2.0 k\u03a9<\/li>\n<li>I<sub>2<\/sub> = 16.92 mA,\u00a0I = 24.12 mA, V<sub>1<\/sub> = 108 V, V<sub>3<\/sub> = 338.4 V, V<sub>4<\/sub> = 120.5 V, R<sub>2<\/sub> = 32.00\u00a0k\u03a9, R<sub>5<\/sub> = 11.94\u00a0k\u03a9<\/li>\n<li>I<sub>2<\/sub> = 0.30 A, V<sub>1<\/sub> =\u00a0V<sub>4<\/sub> = 6.0 V, V<sub>2<\/sub><span style=\"font-size: 14pt;\"> = 2.0 V<\/span><span style=\"font-size: 14pt;\">, V<\/span><sub>5<\/sub><span style=\"font-size: 14pt;\"> = 4.0 V, V = 26 V, R<\/span><sub>3<\/sub><span style=\"font-size: 14pt;\"> =\u00a0<\/span><span style=\"font-size: 14pt;\">R<\/span><sub>6 <\/sub><span style=\"font-size: 14pt;\">= <\/span><span style=\"font-size: 14pt;\">20 <\/span><span style=\"font-size: 14pt;\">\u03a9<\/span><\/li>\n<\/ol>\n<h2>23.6.1 Creating Different Resistor Combinations<\/h2>\n<p>Assume each resistor = 2 W<\/p>\n<ol type=\"i\">\n<li>All in series means R total = 6 W<\/li>\n<li>All in parallel means R total = 0.67 W<\/li>\n<li>One resistor followed by two in parallel means R total = 3 W<\/li>\n<li>One resistor in parallel with two in series means R total = 1.33 W<\/li>\n<\/ol>\n<h2>23.6.2 A Trick Circuit Combination<\/h2>\n<p>The hint was in the title.\u00a0 This is a trick circuit.\u00a0 They have the same current flow out of the battery.\u00a0 Work it out from the far right.\u00a0 The 50 W in series with the 50 W at the far end gives you 100 W in parallel with 100 W for a total of 50 W, and so on.<\/p>\n","protected":false},"author":125,"menu_order":23,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-231","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/231","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":25,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/231\/revisions"}],"predecessor-version":[{"id":1200,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/231\/revisions\/1200"}],"part":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/231\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/media?parent=231"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapter-type?post=231"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/contributor?post=231"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/license?post=231"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}