Faraday's Law Examples: 45 Problems with Answers

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

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

Concept Recap

The induced EMF in a circuit equals the negative rate of change of magnetic flux through the circuit.

The faster you change the magnetic field through a loop, the bigger the voltage you get. Faraday's law tells you exactly how much.

Read the full concept explanation →

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: Faraday's Law starts by naming the source, the object affected, and how the field or potential changes through space.

Common stuck point: Students often know a formula related to faraday's law but skip the recognition step: Am I using a field or potential to explain how one object influences another across space? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I using a field or potential to explain how one object influences another across space?

Worked Examples

Example 1

medium

A coil with

100

turns experiences a change in magnetic flux from

0.05 \text{ Wb}

0.02 \text{ Wb}

0.1 \text{ s}

. What is the induced EMF?

Answer

E = 30 V

First step

Faraday's law:

\mathcal{E} = -N\frac{\Delta \Phi}{\Delta t}

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Example 2

hard

A single loop of area

0.04 \text{ m}^2

is in a magnetic field that increases uniformly from

0

0.6 \text{ T}

0.3 \text{ s}

. What is the induced EMF?

Example 3

medium

A coil of

50

turns has a magnetic flux that changes from

0.02 \text{ Wb}

0.08 \text{ Wb}

0.1 \text{ s}

. Find the induced EMF.

Example 4

medium

A single rectangular loop of area

0.02 \text{ m}^2

lies perpendicular to a magnetic field that grows from

0.1 \text{ T}

0.5 \text{ T}

uniformly in

0.2 \text{ s}

. Find the induced EMF.

Example 5

medium

A bar of length

L = 0.3 \text{ m}

slides at

v = 4 \text{ m/s}

perpendicular to a

0.5 \text{ T}

field through rails completing the circuit. Find the motional EMF.

Example 6

medium

A coil has

N = 500

and self-inductance

L = 0.2 \text{ H}

. Current rises from

0

4 \text{ A}

0.1 \text{ s}

. Find the magnitude of the induced EMF.

Example 7

hard

10

-turn coil of area

0.02 \text{ m}^2

rotates at

50 \text{ rev/s}

in a

0.1 \text{ T}

field. Find the peak induced EMF.

Example 8

medium

1.0 \text{ m}^2

single-loop coil sits in a vertical field. The field changes from

+0.20 \text{ T}

(upward) to

-0.20 \text{ T}

(downward) uniformly in

0.5 \text{ s}

. Find

|\mathcal{E}|

Example 9

medium

30

-turn coil of area

0.01 \text{ m}^2

is in a field

B = (0.2 \text{ T/s})\, t

(linear growth). Find the induced EMF in the coil.

Example 10

challenge

A circular conducting loop of radius

0.05 \text{ m}

and resistance

0.02 \Omega

lies perpendicular to a magnetic field that decreases at

dB/dt = -3 \text{ T/s}

. Find the induced current and the power dissipated.

Practice Problems

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

Example 1

medium

A coil of

50

turns has an area of

0.01 \text{ m}^2

. The magnetic field through it decreases from

0.8 \text{ T}

0.2 \text{ T}

0.5 \text{ s}

. What is the average induced EMF?

Example 2

hard

A square coil with 200 turns and side length

0.1 \text{ m}

is in a magnetic field that decreases uniformly from

0.5 \text{ T}

0 \text{ T}

0.2 \text{ s}

. Calculate the induced EMF. In which direction does the induced current flow (clockwise or anticlockwise, if the field points into the page)?

Example 3

easy

State Faraday's law as a formula (with turns).

Example 4

easy

A single loop's flux changes at

5

Wb/s. Find the EMF magnitude.

Example 5

easy

10

-turn coil has flux changing at

2

Wb/s. Find the EMF.

Example 6

easy

What does the negative sign in Faraday's law represent?

Example 7

easy

20

-turn coil, flux changes by

0.5

Wb in

1

s. Find the EMF.

Example 8

easy

Flux

\Phi = BA\cos\theta

. At what angle is flux maximum?

Example 9

easy

Faster flux change gives a bigger or smaller EMF?

Example 10

easy

5

-turn coil has flux changing at

4

Wb/s. Find the EMF.

Example 11

medium

100

-turn coil of area

0.02

^2

sits in a field rising at

3

T/s (perpendicular). Find the EMF.

Example 12

medium

A coil produces

12

V with flux changing at

0.6

Wb/s. Find the number of turns.

Example 13

medium

40

-turn coil of area

0.1

^2

in a

2

T field is flipped from

\theta=0

\theta=90°

0.2

s. Find the average EMF.

Example 14

medium

A loop's flux is

\Phi(t) = 3t

(in Wb,

t

in s). Find the EMF in a single loop.

Example 15

medium

30

-turn coil produces an average EMF of

9

V over

0.3

s. Find the flux change per turn.

Example 16

medium

A coil of

N=50

turns and area

0.04

^2

has the field change from

1

T to

5

T in

0.5

s (perpendicular). Find the EMF.

Example 17

challenge

A coil rotates so its flux is

\Phi(t) = 0.5\cos(10t)

Wb per turn, with

N=20

turns. Find the peak EMF.

Example 18

challenge

200

-turn coil of area

0.05

^2

in a

0.8

T field is yanked completely out of the field in

0.02

s (starting perpendicular). Find the EMF.

Example 19

challenge

A square loop of side

0.3

m and

10

turns rotates at

\omega = 20

rad/s in a

0.4

T field. Find the peak EMF.

Example 20

medium

25

-turn coil has flux changing at

0.8

Wb/s. Find the EMF.

Example 21

medium

60

-turn coil of area

0.01

^2

in a field rising at

5

T/s (perpendicular). Find the EMF.

Example 22

medium

A coil produces

8

V with flux changing at

0.4

Wb/s. Find the number of turns.

Example 23

easy

A coil with

40

turns has flux changing at

0.5 \text{ Wb/s}

. Find the magnitude of the induced EMF.

Example 24

medium

200

-turn coil has area

0.005 \text{ m}^2

. The magnetic field through it decreases from

0.4 \text{ T}

0

0.1 \text{ s}

. Find the magnitude of the induced EMF.

Example 25

medium

A coil with

N = 50

has area

0.01 \text{ m}^2

. The field through it rotates uniformly so

\cos\theta

goes from

1

0

0.04 \text{ s}

while

B = 0.2 \text{ T}

is held constant. Find the magnitude of the induced EMF.

Example 26

easy

A coil's flux changes by

0.02 \text{ Wb}

0.005 \text{ s}

. If

N = 1

, find

|\mathcal{E}|

Example 27

hard

25

-turn circular coil of radius

0.10 \text{ m}

lies in a field that changes uniformly from

0.30 \text{ T}

0.90 \text{ T}

0.5 \text{ s}

. Find the magnitude of the induced EMF.

Example 28

medium

100

-turn coil has flux per turn changing from

0.03 \text{ Wb}

0.01 \text{ Wb}

0.05 \text{ s}

. Find

|\mathcal{E}|

Example 29

easy

5

-turn coil's flux per turn changes at

1.2 \text{ Wb/s}

. Find the magnitude of the induced EMF.

Example 30

medium

A rectangular loop

0.2 \text{ m}\times 0.3 \text{ m}

moves with velocity

5 \text{ m/s}

perpendicular to its short side into a region with

B = 0.4 \text{ T}

(perpendicular to the loop). Find the EMF while the short side enters the field.

Example 31

medium

A coil has

N = 200

, area

0.008 \text{ m}^2

. A field perpendicular to the coil changes from

0.5 \text{ T}

0.2 \text{ T}

0.06 \text{ s}

. Find

|\mathcal{E}|

Example 32

easy

A coil has

N = 1

and area

0.05 \text{ m}^2

in a

0.4 \text{ T}

field. If the angle between field and normal goes from

0°

90°

0.2 \text{ s}

, find

|\mathcal{E}|

Example 33

medium

A bar slides at

v

perpendicular to a

B

field on rails, length

L

. Express the induced EMF and the magnitude of the induced current if the total circuit resistance is

R

Example 34

hard

0.2 \text{ m}

bar slides at

3 \text{ m/s}

perpendicular to a

0.5 \text{ T}

field on rails of total resistance

2 \Omega

. Find the magnitude of the induced current and the magnetic force on the bar.

Example 35

medium

A coil with

N = 80

has flux

\Phi = (0.005 \text{ Wb/s}^2)\, t^2

per turn. Find the induced EMF magnitude at

t = 2 \text{ s}

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Related Concepts

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Background Knowledge

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

electromagnetic induction