1. Physics 212
Lecture 10
Kirchhoff’s Rules

2. = Last Time 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:
Solved Circuits V R1 R2 R4 R3 V
R1234
I1234 = 5

3. = New Circuit How Can We Solve This One? THE ANSWER: Kirchhoff’s Rules5

4. Kirchoff’s Voltage Rule
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 ! 1 2 3

5. The voltage “gains” , ( V2-V1) + (V3-V2) + (V1-V3) = 0
True for any loop
Add the voltage “drops” ( V1-V2) + (V2-V3) + (V3-V1) = 0 or
The voltage “gains” , ( V2-V1) + (V3-V2) + (V1-V3) = 0

6. Kirchoff’s Current Rule
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 I2 I4 I3

7. Checkpoint 2 With the current VOLTAGE DROP Against the current
In the following circuit, consider the loop abc. The direction of the current through each resistor is indicated by black arrows.
DROP
GAIN
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. drop, drop, drop B. gain, gain, gain C. drop, gain, gain
D. gain, drop, drop E. gain, gain, drop A B C D E
With the current
VOLTAGE DROP
Against the current
VOLTAGE GAIN

8. In this circuit, assume Vi and Ri are known.
Calculation V1 I1 I3 I2
(1) Label and pick directions for each current R1
For resistors, the “upstream” side is +.
This is easy for batteries.
In this circuit, assume Vi and Ri are known.
What is I2 ? R2 V2 R3 V3
(2) Label the + and – side of each element
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.
Now write down loop and node equations.

9. In this circuit, assume Vi and Ri are known.
Calculation V1 R1 I1 R2 V2 I2
In this circuit, assume Vi and Ri are known.
What is I2? R3 V3 I3
How many equations do we need to write down in order to solve for I2?
(A) (B) (C) (D) (E) 5
Why?
We have 3 unknowns: I1, I2, and I3
We need 3 independent equations to solve for these unknowns
(3) Choose loops and directions
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

10. In this circuit, assume Vi and Ri are known.
Calculation V1 R1 I1 R2 V2 I2
In this circuit, assume Vi and Ri are known.
What is I2 ? R3 V3 I3
Which of the following equations is NOT correct?
I2 = I1 + I3
+ V1 - I1R1 + I3R3 - V3 = 0
+ V3 - I3R3 - I2R2 - V2 = 0
+ V2 + I2R2 - I1R1 + V1 = 0
Node
Outer loop
Bottom loop
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.
Top loop
Why is (D) wrong ?
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).

11. In this circuit, assume Vi and Ri are known.
Calculation V1 R1 I1 R2 V2 I2
In this circuit, assume Vi and Ri are known.
What is I2 ? R3 V3 I3
We have the following 4 equations:
We need 3 equations:
Which 3 should we use?
1. I2 = I1 + I3
2. + V1 - I1R1 + I3R3 - V3 = 0
3. + V3 - I3R3 - I2R2 - V2 = 0
4. + V2 + I2R2 + I1R1 - V1 = 0
Node
Outer
Bottom
Top
A) Any 3 will do
B) 1, 2, and 4
C) 2, 3, and 4
Do the addition of equations
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)

12. In this circuit, assume Vi and Ri are known.
Calculation V1 R1 I1
In this circuit, assume Vi and Ri are known.
What is I2? R2 V2 I2 R3 V3 I3
We have 3 equations and 3 unknowns.
I2 = I1 + I3
V1 - I1R1 + I3R3 - V3 = 0
V2 + I2R2 + I1R1 - V1 = 0 2V R 2R V I1 I3 I2
(6) Solve the equations
The solution will get very messy!
Simplify: assume V2 = V3 = V
V1 = 2V
R1 = R3 = R
R2 = 2R
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

13. Calculation: Simplify
In this circuit, assume V and R are known What is I2 ? 2V R 2R V I1 I3 I2
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
current direction
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

14. Follow-Up 2V R 2R V I1 I3 I2
We know:
I2 = ( 1/5) V/R
I1 = ( 3/5) V/R
I3 = (-2/5) V/R a b
Suppose we short R3: What happens to Vab (voltage across R2?)
Vab remains the same
Vab changes sign
Vab increases
Vab goes to zero 2V R 2R V I1 I3 I2 a b c d
Why?
Redraw:
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.
(Va - Vb ) + V - V = 0
Bottom Loop Equation:
Vab = 0

15. V Is there a current flowing between a and b ? a b Yes No
Current flows from battery and splits at A
Some current flows right
Some current flows down V R R
Is there a current flowing between a and b ?
Yes No
No current flows between A & B
A & B have the same potential

16. Checkpoint 3a
Consider the circuit shown below. Note that this question is not identical to the similar looking one you answered in the prelecture. I2 2I I I I 2I
Which of the following best describes the current flowing in the blue wire connecting points a and b?
1. Same voltage across R, 2R in parallel so IR = 2 I2R .
2. R, 2R combinations are identical so voltage drop in each case is V/2.
Therefore total current in top and bottom must be 3 I.
3. Satisfy Kirchhoff current law at node a so Iab = I

17. What is different? Current flowing from a to b
Prelecture
Checkpoint
What is the same? Current flowing in and out of the battery 2R 3
What is different? Current flowing from a to b

18. I
2/3I
1/3 I R 2R V
2/3I a b
V/2 R
2/3I 2R I
1/3

19. Checkpoint 3b Current will flow from left to right in both cases
Consider the circuit shown below.
Checkpoint 3b IA IB
Current will flow from left to right in both cases
In both cases, Vac = V/2 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
I2R = 2I4R
IA = IR – I2R
= IR – 2I4R
IB = IR – I4R

20. + r r V0 VL R R VL V0 Model for Real Battery: Internal Resistance
Usually can’t supply too much current to the load without voltage “sagging”

21. r V0 R VL Voltage divider I = V0 / (r + R) VL = I R
VL = V0 R / (r + R)
VL V0
R >> r