Kirchhoff's Laws Examples in Physics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Kirchhoff's Laws.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Kirchhoff's laws are two rules for analyzing circuits. Kirchhoff's current law (junction rule) says the total current into any junction equals the total current out of it, because charge is conserved. Kirchhoff's voltage law (loop rule) says the sum of the voltage rises and drops around any closed loop is zero, because energy per unit charge is conserved.

Charge cannot pile up at a junction, and energy per unit charge must balance around a complete loop.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Kirchhoff's Laws asks students to follow the circuit path and identify what quantity changes at each component.

Common stuck point: Students often know a formula related to kirchhoff's laws but skip the recognition step: Can I identify the circuit path, what quantity is flowing or changing, and which electrical rule links the quantities? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Can I identify the circuit path, what quantity is flowing or changing, and which electrical rule links the quantities?

Worked Examples

Example 1

medium

In a loop with

20\,\text{V}

battery and three resistors (

3\,\Omega

5\,\Omega

2\,\Omega

in series), find the current and the drop across each resistor.

KVL: I = 2 A; drops are 6 V, 10 V, 4 V

Answer

I = 2\ \text{A},\ V_1 = 6,\ V_2 = 10,\ V_3 = 4\ \text{V}

First step

Total resistance

R = 3 + 5 + 2 = 10\,\Omega

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Example 2

medium

Apply KVL to a loop:

\varepsilon - I R_1 - I R_2 - I R_3 = 0

. Given

\varepsilon = 12\,\text{V}

R_1 = 1\,\Omega

R_2 = 2\,\Omega

R_3 = 3\,\Omega

, find

I

KVL: I = 12/(1+2+3) = 2 A

Example 3

hard

In a two-mesh circuit, mesh 1 has KVL

12 = 4 I_1 + 2(I_1 - I_2)

and mesh 2 has

-6 = 3 I_2 + 2(I_2 - I_1)

. Solve for

I_1

and

I_2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

State Kirchhoff's current law (KCL).

Example 2

easy

State Kirchhoff's voltage law (KVL).

Example 3

easy

At a junction,

5

A flows in and

2

A flows out one branch. What flows out the only other branch?

Apply KCL: 5 A in = 2 A + I

Example 4

easy

In a single loop, a

9

V battery and two resistors drop

4

V and

V_2

. Find

V_2

using KVL.

KVL: 4 V + V₂ = 9 V, find V₂

Example 5

easy

Why can't current accumulate at a junction?

Example 6

easy

What physical principle underlies Kirchhoff's voltage law?

Example 7

easy

A node has currents

3

A and

4

A entering. What total current must leave?

KCL: I_out = 3 A + 4 A = 7 A

Example 8

easy

In a closed loop, the only source is

12

V and there is one resistor. What is the voltage drop across the resistor?

KVL: one resistor drops the full 12 V

Example 9

medium

A junction has

6

A in, and two branches out: one carries

2.5

A. Find the other branch current.

KCL: 6 A = 2.5 A + I

Example 10

medium

A loop has a

15

V battery and resistors dropping

3

6

V, and

V_3

. Find

V_3

KVL: 3 + 6 + V₃ = 15, find V₃

Example 11

medium

At a node,

4

A and

3

A enter while

2

A leaves one branch. Find the current in the remaining branch.

KCL: (4+3) A in = 2 A + I

Example 12

medium

A loop has two batteries: a

12

V rise and a

4

V rise, plus a single resistor. What is the voltage drop across the resistor?

Example 13

medium

Two batteries oppose each other in a loop:

10

V rise and

6

V drop, with a single resistor. Find the resistor's drop.

Example 14

medium

In a single loop, a

20

V battery drives a

2\,\Omega

and a

3\,\Omega

resistor. Use KVL to find the current.

KVL: 20 = I(2+3), find I = 4 A

Example 15

medium

A node analysis assumes a

3

A branch leaves, but balancing gives

-3

A. What does the sign mean?

Example 16

challenge

A two-loop circuit shares a middle branch. The left loop forces

5

A down the middle and the right loop forces

2

A up the middle. What net current flows in the middle branch and in which direction?

Example 17

challenge

A loop contains a

24

V battery, a

4\,\Omega

resistor, and a second loop branch carrying a known

2

A through a

3\,\Omega

resistor shared with this loop (drop aiding). If the loop current is

I

, KVL reads

24 = 4I + 3(I+2)

. Solve for

I

Example 18

challenge

At a node, three currents are defined as leaving:

I_1 = 4

I_2 = -3

I_3 = ?

, with KCL

I_1 + I_2 + I_3 = 0

. Find

I_3

and interpret its sign.

Example 19

medium

A node has

8

A in and branches of

3

A and

I

out. Find

I

KCL: 8 A = 3 A + I

Example 20

medium

A loop has a

30

V battery and drops of

12

8

V, and

V_3

. Find

V_3

KVL: 12 + 8 + V₃ = 30, find V₃ = 10 V

Example 21

easy

A junction has

2\,\text{A}

and

3\,\text{A}

entering and

I

leaving (only one outgoing branch). Find

I

KCL: I = 2 A + 3 A = 5 A

Example 22

easy

In a single loop with a

24\,\text{V}

battery, one resistor drops

9\,\text{V}

and another drops

V_2

. Find

V_2

KVL: 9 V + V₂ = 24 V, find V₂ = 15 V

Example 23

easy

A node has

7\,\text{A}

entering and two branches leaving with

4\,\text{A}

and

I

. Find

I

KCL: 7 A = 4 A + I, find I = 3 A

Example 24

easy

In a single loop a

9\,\text{V}

battery drives a

3\,\Omega

resistor. Use KVL to find the current.

KVL: 9 = I × 3, find I = 3 A

Example 25

medium

A node has

9\,\text{A}

in and three branches out carrying

2\,\text{A}

3\,\text{A}

, and

I

. Find

I

KCL: 9 = 2 + 3 + I, find I = 4 A

Example 26

medium

A single-loop circuit contains a

12\,\text{V}

battery and resistors of

2\,\Omega

and

4\,\Omega

in series. Use KVL to find the current.

KVL: 12 = I(2+4), find I = 2 A

Example 27

medium

Two batteries in a loop oppose each other: a

15\,\text{V}

rise and a

9\,\text{V}

rise in the opposite sense (i.e.,

9\,\text{V}

drop in the loop direction). One

3\,\Omega

resistor closes the loop. Find the current.

Example 28

medium

A node has

3\,\text{A}

5\,\text{A}

entering and

2\,\text{A}

I

leaving. Find

I

KCL: 3+5 = 2+I, find I = 6 A

Example 29

medium

In a loop with a

9\,\text{V}

battery and resistors

1\,\Omega

and

R

in series, the current is

1\,\text{A}

. Find

R

KVL: 9 = 1·1 + 1·R, find R = 8 Ω

Example 30

medium

A loop has a

36\,\text{V}

battery, a

4\,\Omega

resistor, and a

2\,\Omega

resistor in series. Use KVL to find the voltage drop across the

4\,\Omega

resistor.

I = 6 A; V across 4 Ω = 24 V

Example 31

medium

A node has currents

I_1 = 3\,\text{A}

in,

I_2 = 1\,\text{A}

out, and

I_3

in. The only fourth branch carries

5\,\text{A}

out. Find

I_3

KCL: 3 + I₃ = 1 + 5, find I₃ = 3 A

Example 32

medium

In a single-loop circuit, the sum of resistor drops is

18\,\text{V}

. What is the EMF of the only battery, if it is the only source?

KVL: EMF = total drop = 18 V

Example 33

medium

A node has

4\,\text{A}

entering and three branches out:

1\,\text{A}

1\,\text{A}

I

. Find

I

KCL: 4 = 1+1+I, find I = 2 A

Example 34

hard

A two-loop circuit shares a

5\,\Omega

branch. Loop 1 drives

3\,\text{A}

down through the shared branch; loop 2 drives

1\,\text{A}

up. Find the net current in the shared branch and its direction.

Example 35

hard

A loop contains a

24\,\text{V}

battery, a

4\,\Omega

resistor in the outer loop, and an inner branch with

2\,\Omega

that also carries a known

1\,\text{A}

from a neighboring mesh, aiding the loop current. The loop KVL reads

24 = 4 I + 2(I + 1)

. Find

I

Example 36

hard

Two batteries in series aid each other (

6\,\text{V}

and

9\,\text{V}

) with three resistors in series (

1\,\Omega

2\,\Omega

3\,\Omega

). Use KVL to find the loop current.

KVL: I = 15/(1+2+3) = 2.5 A

Example 37

hard

At a node

I_1 + I_2 - I_3 - I_4 = 0

with

I_1 = 4\,\text{A}

I_2 = 2\,\text{A}

I_3 = 3\,\text{A}

. Find

I_4

Example 38

hard

A loop contains a

10\,\text{V}

battery and three parallel resistors (

2\,\Omega

4\,\Omega

4\,\Omega

) connected across it. Using KCL at one node, find the total current drawn from the battery.

KCL: I_total = 5+2.5+2.5 = 10 A

Example 39

challenge

In a bridge-like circuit, KCL at the top node gives

I_1 = I_2 + I_3

with

I_1 = 8\,\text{A}

. A KVL loop along

R_2 = 2\,\Omega

and

R_3 = 4\,\Omega

gives

2 I_2 = 4 I_3

. Find

I_2

and

I_3

Example 40

challenge

A two-mesh circuit shares a

5\,\Omega

resistor. With clockwise mesh currents

I_1

and

I_2

, KVL gives

20 = 10 I_1 - 5 I_2

and

0 = -5 I_1 + 8 I_2

. Find

I_1

and

I_2

Example 41

challenge

A loop contains a

12\,\text{V}

battery and an internal resistance

r = 0.5\,\Omega

, plus an external

5.5\,\Omega

resistor. Use KVL to find the current and the terminal voltage.

KVL: I = 12/(0.5+5.5) = 2 A; V_term = 11 V

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Related Concepts

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Background Knowledge

These ideas may be useful before you work through the harder examples.

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