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Lectures_Lect10

# Lectures_Lect10 - Physics 212 Lecture 10 Kirchhoff's Rules...

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Unformatted text preview: Physics 212 Lecture 10 Kirchhoff's Rules Physics 212 Lecture 10, Slide 1 Last Time Resistors in series: Current through is same. Voltage drop across is IRi Resistors in parallel: Voltage drop across is same. Current through is V/Ri Solved Circuits R1 R2 Reffective = R1 + R2 + R3 + ... 1 Reffective 1 1 1 = + + + ... R1 R2 R3 V R3 R4 = V I1234 R1234 Physics 212 Lecture 10, Slide 2 5 New Circuit R1 R3 V1 V2 R2 How Can We Solve This One? R1 V1 V2 R3 R2 = V I1234 R12 THE ANSWER: Kirchhoff's Rules Physics 212 Lecture 10, Slide 3 5 Kirchoff's Voltage Rule V i = 0 Kirchoff's Voltage Rule states that the sum of the voltage changes around a circuit must be zero. WHY? The potential difference between a point and itself is zero ! 2 1 - 3 r E l = V ( 1) - V ( 1) = 0 d loop Physics 212 Lecture 10, Slide 4 2 1 2 3 1 r 0 = E = dl + + loop 1 2 3 3 0 = ( V1 - V2 ) + ( V2 - V3 ) + ( V3 - V1 ) 0= loop Vi True for any loop Kirchoff's Current Rule I in = out I Kirchoff's Current Rule states that the sum of all currents entering any given point in a circuit must equal the sum of all currents leaving the same point. WHY? Electric charge is conserved and cannot accumulate at nodes. Node I1 I4 I2 I3 I in = I1 = out = I 2 + I 3 + I 4 I Physics 212 Lecture 10, Slide 6 Checkpoint 2 In the following circuit, consider the loop abc. The direction of the current through each resistor is indicated by black arrows. DROP GAIN N AI G 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? A A. drop, drop, drop B. gain, gain, gain C. drop, gain, gain B gain, drop, drop D. E. gain, gain, drop C D E With the current Against the current VOLTAGE DROP VOLTAGE GAIN Physics 212 Lecture 10, Slide 7 1 2 1 2V 1V 1V I2 Calculation In this circuit, assume Vi and Ri are known. What is I2 ?? Conceptual Analysis: Circuit behavior described by Kirchhoff's Rules: Strategic Analysis KVR: Vdrops = 0 KCR: Iin = Iout Write down Loop Equations (KVR) Write down Node Equations (KCR) Solve Physics 212 Lecture 10, Slide 8 (1) Label all currents Choose any direction Kirchhoff's Laws R1 A (2) Label +/ for all elements Current goes + (for resistors) Battery signs fixed! I1 + R2 I2 + R5 + V3 I3 R3 + + I 5 I4 R4 + + + V - 1 (3) Choose loops and directions (4) Write down voltage drops (5) Write down node equations Iin = Iout (6) Solve set of equations B V2 Physics 212 Lecture 10, Slide 9 17 R1 + + V1 - I1 I2 I3 Calculation In this circuit, assume Vi and Ri are known. What is I2 ?? R2 + - + V2 - R3 + + V3 - (1) Label and pick directions for each current (2) Label the + and side of each element This is easy for batteries For resistors, the "upstream" side is + Now write down loop and node equations Physics 212 Lecture 10, Slide 10 R1 + + V1 - I1 I2 I3 Calculation In this circuit, assume Vi and Ri are known. What is I2 ?? R2 + - + V2 - R3 + + V3 - How many equations do we need to write down in order to solve for I2? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 (A) (B) (C) (D) (E) Why?? We have 3 unknowns: I1, I2, and I3 We need 3 independent equations to solve for these unknowns (3) Choose loops and directions Physics 212 Lecture 10, Slide 11 R1 + + V1 - I1 I2 I3 Calculation In this circuit, assume Vi and Ri are known. What is I2 ? R2 + - + V2 - R3 + + V3 - Which of the following equations is NOT correct? (A) I2 = I1 + I3 (B) + V1 I1R1 + I3R3 V3 = 0 (C) + V3 I3R3 I2R2 V2 = 0 (D) + V2 + I2R2 I1R1 + V1 = 0 Why is (D) wrong ? Node Outer loop Bottom loop Top loop Start at negative terminal of V2 and go clockwise around top loop: (+V2) + (+I2R2) + (+I1R1) + (V1) = 0 This is not the same as answer (D). Physics 212 Lecture 10, Slide 12 R1 R2 R3 V1 V2 V3 I1 I2 I3 Calculation In this circuit, assume Vi and Ri are known. What is I2 ?? We need 3 equations: Which 3 should we use? A) Any 3 will do B) 1, 2, and 4 C) 2, 3, and 4 We have the following 4 equations: 1. I2 = I1 + I3 2. + V1 I1R1 + I3R3 V3 = 0 3. + V3 I3R3 I2R2 V2 = 0 3. 4. + V2 + I2R2 + I1R1 V1 = 0 4. Why?? We need 3 INDEPENDENT equations Equations 2, 3, and 4 are NOT INDEPENDENT Eqn 2 + Eqn 3 = Eqn 4 We must choose Equation 1 and any two of the remaining ( 2, 3, and 4) Physics 212 Lecture 10, Slide 13 R1 R2 R3 V1 V2 V3 I1 I2 I3 Calculation In this circuit, assume Vi and Ri are known. What is I2 ?? We have 3 equations and 3 unknowns. I2 = I1 + I3 V1 I1R1 + I3R3 V3 = 0 V2 + I2R2 + I1R1 V1 = 0 R 2R R 2V V V I1 I2 I3 (6) Solve the equations The solution will get very messy! Simplify: assume V2 = V3 = V V1 = 2V R1 = R3 = R R2 = 2R Physics 212 Lecture 10, Slide 14 Calculation: Simplify In this circuit, assume V and R are known. What is I2 ?? R 2R R 2V V V I1 I2 I3 current direction We have 3 equations and 3 unknowns. I2 = I1 + I3 +2V I1R + I3R V = 0 (outside) +V + I2(2R) + I1R 2V = 0 (top) With this simplification, you can verify: I2 = ( 1/5) V/R I1 = ( 3/5) V/R I3 = (2/5) V/R Physics 212 Lecture 10, Slide 15 R 2R a b 2V V V FollowUp I1 I2 I3 We know: R I2 = ( 1/5) V/R I1 = ( 3/5) V/R I3 = (2/5) V/R Suppose we short R3: What happens to Vab (voltage across R2?) R 2R a b c (A) Vab remains the same Why? (B) Vab changes sign Redraw: (C) Vab increases (D) Vab goes to zero Bottom Loop Equation: Vab + V V = 0 Vab = 0 2V V V I1 I2 I3 d Physics 212 Lecture 10, Slide 16 a b V R R Is there a current flowing between a and b ? A) Yes B) No A & B have the same potential Current flows from battery and splits at A No current flows between A & B Some current flows down Some current flows right Physics 212 Lecture 10, Slide 17 Checkpoint 3a Consider the circuit shown below. Note that this question is not identical to the similar looking one I1 I2 you answered in the prelecture. I1 I I2 I3 I4 Which of the following best describes the current flowing in the blue wire connecting points a and b? I1R I2 (2R) = 0 I4R I3 (2R) = 0 a: I1 = I + I3 b: I + I2 = I4 I2 = I1 I4 = 2 I3 I = +I3 Physics 212 Lecture 10, Slide 18 I1 I3 + I1 = 2I3 I1 = 2I3 Prelecture Checkpoint What is the same? Current flowing in and out of the battery 2R 3 2R 3 What is different? Current flowing from a to b Physics 212 Lecture 10, Slide 19 I 2 /3I a R 1 /3 I b 2R V 2 V/2 2R /3I R 1 /3 I 2 /3I 0 2 /3I Physics 212 Lecture 10, Slide 20 Consider the circuit shown below. Checkpoint 3b IA IB c c In which case is the current flowing in the blue wire connecting points a and b the largest? A. Case A B. Case B C. They are both the same Current will flow from left to right in both cases In both cases, Vac = V/2 I2R = 2I4R IA = IR I2R = IR 2I4R IB = IR I4R Physics 212 Lecture 10, Slide 21 Model for Real Battery: Internal Resistance + r V0 R VL V0 r R VL Usually can't supply too much current to the load without voltage "sagging" Physics 212 Lecture 10, Slide 22 ...
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