This Class Resistors in series: Current through is same. Voltage drop across is IRi Resistors in series: Voltage drop across is same. Current through is V/Ri Resistors in parallel: Solve Circuits V R1 R2 R4 R3

Kirchhoff’s First Rule (“Loop Rule” or “Kirchhoff’s Voltage Law”) The algebraic sum of the changes in potential encountered in a complete traversal of any loop of circuit must be zero. e1 R1 R2 I e Move around circuit: 11/21/2018

Rules e1 e - + e - + e R1 R2 I Voltage Gains enter with a + sign. Voltage Drops enter with a - sign. - + e a b - + e b a I a b I b a Note: e always points from negative to positve 11/21/2018

Loop Example a d b e c f R1 I R2 R3 R4 e1 e + - 11/21/2018

Internal Resistance of an emf Device 11/21/2018

Resistors in Series *When a potential difference, V, is applied across resistances in series, the resistances have identical current. * Resistances in series can be replaced with an equivalent resistance, REQ, that has the same current I and the same total potential difference V as the actual resistances. I e1 - + R2 R3 R1 11/21/2018

Another (intuitive) way Consider two cylindrical resistors with lengths L1 and L2 V R1 R2 L2 L1 Put them together, end to end to make a longer one... 11/21/2018

Kirchhoff’s Second Rule (Junction Rule or “Kirchhoff’s Current Law”) The sum of the current entering any junction must be equal to the sum of the currents leaving that junction. 11/21/2018

How to use Kirchhoff’s Law: + - e - + R3 R2 R1 a b c d Analyze the circuit & identify all circuit nodes: Identify all independent loops & use Kirchhoff’s Voltage Law: 11/21/2018

Resistors in Parallel e R1 R3 R2 - + R3 R2 R1 *In parallel, the resistances all have the same potential differences. * Resistances in parallel can be replaced with an equivalent resistance, REQ, that has the same potential difference, V, and the same total current, I, as the actual resistances. 11/21/2018

Another (intuitive) way Consider two cylindrical resistors with cross-sectional areas A1 and A2 V R1 R2 A1 A2 Put them together, side by side … to make one “fatter”one, 11/21/2018

Example 1 Consider the ideal circuit shown: What is the relation between Va -Vd and Va -Vc ? 12V I1 I2 a b d c 50 20 80 (a) (Va -Vd) > (Va -Vc) (b) (Va -Vd) = (Va -Vc) (c) (Va -Vd) < (Va -Vc) 11/21/2018

Example 2 Consider the ideal circuit shown: 12V I1 I2 a b d c 50 20 80 What is the relationship between I1 and I2? (a) I1 > I2 (b) I1 = I2 (c) I1 < I2 11/21/2018

Summary of Resistor & Capacitor Combinations Resistors Capacitors Series Parallel 11/21/2018

Summary of Simple Circuits Resistors in series: Current through is same; Voltage drop across is IRi Resistors in parallel: Voltage drop across is same; Current thru is V /Ri 11/21/2018

Problem Solving Tips When you are given a circuit, you must first carefully analyze circuit topology. find the nodes and distinct branches and pick Linearly Independent subsets of each. assign branch currents Use Kirchhoff’s First Rule for all independent loops in the circuit. Use Kirchhoff’s Second Rule for all independent nodes in circuit. 11/21/2018

QUIZ 11: September 24, 2013 Three resistors are connected to a battery with emf V as shown. The resistances of the resistors are all the same, i.e. R1 = R2 = R3. Compare the current through R2 to the current through R3. I2 < I3 I2 = I3 I2 > I3 Current is the same

QUIZ lecture 11 Three resistors are connected to a battery with emf V as shown. The resistances of the resistors are all the same, i.e. R1 = R2 = R3. Compare the current through R2 to the current through R1. I2 < I1 I2 = I1 I2 > I1 Current is the same

Example 25-16 11/21/2018

Cramer’s Rule If then 11/21/2018

= New Circuit How Can We Solve This One? THE ANSWER: Kirchhoff’s Rules

QUIZ lecture 11 In the following circuit, consider the loop abc. The direction of the current through each resistor is indicated by black arrows. If we are to write Kirchoff's voltage equation for this loop in the clockwise direction starting from point a, what is the correct order of voltage gains/drops that we will encounter for resistors R1, R2 and R3? drop, drop, drop gain, gain, gain drop, gain, gain gain, drop, drop drop, drop, gain

QUIZ lecture 11 You are to connect resistors R1 and R2 (with R1>R2) to a battery, first individually, then in series and then in parallel. Rank those arrangements according to the amount of current through the battery, greatest first. Series, R1, R2, Parallel R2, Series, Parallel, R1 R1, Series, R2, Parallel Parallel, R2, R1, Series R1, Parallel, R2, Series 11/21/2018

DEMO 5B-06 For the same EMF source, the bulbs will burn equally bright when connected in Parallel Series 11/21/2018 http://www.how-to-draw-funny-cartoons.com/image-files/cartoon-light-7.gif

DEMO 5B-06 If the middle bulb is replaced with a 90 Watt bulb (15 Ohms), and connected to two 34 Watt (45 Ohms) in parallel, which bulb or bulbs will burn brightest? 11/21/2018 http://www.how-to-draw-funny-cartoons.com/image-files/cartoon-light-7.gif

Ammeter & Voltmeter I I Ammeter, A, inserted into the circuit. Voltmeter, V, across the circuit element. I 11/21/2018

Galvanometer Ig Rg = 11/21/2018

Make a 1.0 Ampere full scale deflection ammeter small parallel resistor (shunt) If Rg = 100  and Ig = 50 A, then To change the scale, change RP . 11/21/2018

Voltmeter Assume Rg = 100  and Ig = 50 A for full scale deflection (typical). Make a 100 V full scale deflection voltmeter. 100  negligible 11/21/2018

Ohmmeter 11/21/2018