# 21.3 Kirchhoff’s rules  (Page 6/10)

 Page 6 / 10

For this question, consider the circuit shown in the following figure.

1. Assuming that none of the three currents ( I 1 , I 2 , and I 3 ) are equal to zero, which of the following statements is false?

1. I 3 = I 1 + I 2 at point a .
2. I 2 = I 3 - I 1 at point e .
3. The current through R 3 is equal to the current through R 5.
4. The current through R 1 is equal to the current through R 5.
2. Which of the following statements is true?

1. E 1 + E 2 + I 1 R 1 - I 2 R 2 + I 1 r 1 - I 2 r 2 + I 1 R 5 = 0
2. - E 1 + E 2 + I 1 R 1 - I 2 R 2 + I 1 r 1 - I 2 r 2 - I 1 R 5 = 0
3. E 1 - E 2 - I 1 R 1 + I 2 R 2 - I 1 r 1 + I 2 r 2 - I 1 R 5 = 0
4. E 1 + E 2 - I 1 R 1 + I 2 R 2 - I 1 r 1 + I 2 r 2 + I 1 R 5 = 0
3. If I 1 = 5 A and I 3 = -2 A, which of the following statements is false?

1. The current through R 1 will flow from a to b and will be equal to 5 A.
2. The current through R 3 will flow from a to j and will be equal to 2 A.
3. The current through R 5 will flow from d to e and will be equal to 5 A.
4. None of the above.
4. If I 1 = 5 A and I 3 = -2 A, I 2 will be equal to

1. 3 A
2. -3 A
3. 7 A
4. -7 A

a. (c)

b. (c)

c. (d)

d. (d)

In an experiment this circuit is set up. Three ammeters are used to record the currents in the three vertical branches (with R 1 , R 2 , and E) . The readings of the ammeters in the resistor branches (i.e. currents in R 1 and R 2 ) are 2 A and 3 A respectively.

1. Find the equation obtained by applying Kirchhoff’s loop rule in the loop involving R 1 and R 2 .
2. What will be the reading of the third ammeter (i.e. the branch with E )? If E were replaced by 3 E , how would this reading change?
3. If the original circuit is modified by adding another voltage source (as shown in the following circuit), find the readings of the three ammeters.

In this circuit, assume the currents through R 1 , R 2 and R 3 are I 1 , I 2 and I 3 respectively and all are flowing in the clockwise direction.

1. Find the equation obtained by applying Kirchhoff’s junction rule at point A.
2. Find the equations obtained by applying Kirchhoff’s loop rule in the upper and lower loops.
3. Assume R 1 = R 2 = 6 Ω, R 3 = 12 Ω, r 1 = r 2 = 0 Ω, E 1 = 6 V and E 2 = 4 V. Calculate I 1 , I 2 and I 3 .
4. For the situation in which E 2 is replaced by a closed switch, repeat parts (a) and (b). Using the values for R 1 , R 2 , R 3 , r 1 and E 1 from part (c) calculate the currents through the three resistors.
5. For the circuit in part (d) calculate the output power of the voltage source and across all the resistors. Examine if energy is conserved in the circuit.
6. A student implemented the circuit of part (d) in the lab and measured the current though one of the resistors as 0.19 A. According to the results calculated in part (d) identify the resistor(s). Justify any difference in measured and calculated value.
1. I 1 + I 3 = I 2
2. E 1 - I 1 R 1 - I 2 R 2 - I 1 r 1 = 0; - E 2 + I 1 R 1 - I 3 R 3 - I 3 r 2 = 0
3. I 1 = 8/15 A, I 2 = 7/15 A and I 3 = - 1/15 A
4. I 1 = 2/5 A, I 2 = 3/5 A and I 3 = 1/5 A
5. P E1 = 18/5 W and P R1 = 24/25 W, P R2 = 54/25 W, P R3 = 12/25 W. Yes, P E1 = P R1 + P R2 + P R3
6. R 3, losses in the circuit

## Section summary

• Kirchhoff’s rules can be used to analyze any circuit, simple or complex.
• Kirchhoff’s first rule—the junction rule: The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
• Kirchhoff’s second rule—the loop rule: The algebraic sum of changes in potential around any closed circuit path (loop) must be zero.
• The two rules are based, respectively, on the laws of conservation of charge and energy.
• When calculating potential and current using Kirchhoff’s rules, a set of conventions must be followed for determining the correct signs of various terms.
• The simpler series and parallel rules are special cases of Kirchhoff’s rules.

## Conceptual questions

Can all of the currents going into the junction in [link] be positive? Explain.

Apply the junction rule to junction b in [link] . Is any new information gained by applying the junction rule at e? (In the figure, each emf is represented by script E.)

(a) What is the potential difference going from point a to point b in [link] ? (b) What is the potential difference going from c to b? (c) From e to g? (d) From e to d?

Apply the loop rule to loop afedcba in [link] .

Apply the loop rule to loops abgefa and cbgedc in [link] .

## Problem exercises

Apply the loop rule to loop abcdefgha in [link] .

$-{I}_{2}{R}_{2}+{\text{emf}}_{1}-{\text{I}}_{2}{r}_{1}+{\text{I}}_{3}{R}_{3}+{\text{I}}_{3}{r}_{2}-{\text{emf}}_{2}=\text{0}$

Apply the loop rule to loop aedcba in [link] .

Verify the second equation in [link] by substituting the values found for the currents ${I}_{1}$ and ${I}_{2}$ .

Verify the third equation in [link] by substituting the values found for the currents ${I}_{1}$ and ${I}_{3}$ .

Apply the junction rule at point a in [link] .

${I}_{3}={\text{I}}_{1}+{\text{I}}_{2}$

Apply the loop rule to loop abcdefghija in [link] .

Apply the loop rule to loop akledcba in [link] .

${\text{emf}}_{2}-{\text{I}}_{2}{r}_{2}-{\text{I}}_{2}{R}_{2}+{\text{I}}_{1}{R}_{5}+{I}_{1}{r}_{1}-{\text{emf}}_{1}+{\text{I}}_{1}{R}_{1}=0$

Find the currents flowing in the circuit in [link] . Explicitly show how you follow the steps in the Problem-Solving Strategies for Series and Parallel Resistors .

Solve [link] , but use loop abcdefgha instead of loop akledcba. Explicitly show how you follow the steps in the Problem-Solving Strategies for Series and Parallel Resistors .

(a) ${\text{I}}_{1}=\text{4.75 A}$

(b) ${\text{I}}_{\text{2}}=-3\text{.}\text{5 A}$ 

(c) ${\text{I}}_{3}=8\text{.}\text{25 A}$

Find the currents flowing in the circuit in [link] .

Unreasonable Results

Consider the circuit in [link] , and suppose that the emfs are unknown and the currents are given to be ${I}_{1}=5\text{.}\text{00 A}$ , ${I}_{2}=3\text{.0 A}$ , and ${I}_{3}=–2\text{.}\text{00 A}$ . (a) Could you find the emfs? (b) What is wrong with the assumptions?

(a) No, you would get inconsistent equations to solve.

(b) ${I}_{1}\ne {I}_{2}+{I}_{3}$ . The assumed currents violate the junction rule.

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