Magnetic Force Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Magnetic Force.

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 force exerted on a moving charge or current-carrying conductor by a magnetic field.

A moving charge in a magnetic field feels a sideways push — perpendicular to both its motion and the field. It's like a cross-wind deflecting a moving ball.

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: Magnetic Force 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 magnetic force 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

easy

A proton (

q = 1.6 \times 10^{-19} \text{ C}

) moves at

5 \times 10^6 \text{ m/s}

perpendicular to a

0.3 \text{ T}

magnetic field. What is the magnetic force on the proton?

Answer

F = 2.4 \times 10^{- 13} N

First step

Use

F = qvB\sin\theta

with

\theta = 90°

Full solution

2
$F = 1.6 \times 10^{- 19} \times 5 \times 10^{6} \times 0.3 \times 1$
3
$F = 2.4 \times 10^{- 13} N$

The magnetic force on a moving charged particle is perpendicular to both the velocity and the magnetic field. It causes the particle to move in a circular path rather than speeding it up or slowing it down.

Example 2

medium

A wire carrying

8 \text{ A}

0.5 \text{ m}

long and placed at

60°

to a

0.4 \text{ T}

magnetic field. What force acts on the wire?

Example 3

medium

A proton (

q = 1.6 \times 10^{-19} \text{ C}

) moves at

3 \times 10^6 \text{ m/s}

perpendicular to a

0.5 \text{ T}

magnetic field. Find the magnetic force.

Example 4

medium

A proton (

q = 1.6\times 10^{-19} \text{ C}

) enters a

0.20 \text{ T}

field perpendicular to its velocity at

v = 2.0 \times 10^6 \text{ m/s}

. The proton's mass is

1.67\times 10^{-27} \text{ kg}

. Find the radius of its circular path.

Example 5

medium

Find the cyclotron period

T

of a proton (

m = 1.67\times 10^{-27} \text{ kg}

q = 1.6\times 10^{-19} \text{ C}

) in a

1.0 \text{ T}

magnetic field.

Example 6

medium

A positive charge moves in the

+x

direction through a region with

\vec{B}

in the

+y

direction. In which direction is the magnetic force, and what happens to the speed of the charge?

Example 7

hard

Two long parallel wires

0.40 \text{ m}

apart carry parallel currents

I_1 = 8 \text{ A}

and

I_2 = 3 \text{ A}

. Find the force per unit length between them. Use

\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}

Example 8

hard

A mass spectrometer uses a

0.50 \text{ T}

field. A singly ionized particle (

q = 1.6\times 10^{-19} \text{ C}

) is accelerated through

1000 \text{ V}

and travels in a circle of radius

0.10 \text{ m}

. Find the particle's mass.

Example 9

challenge

A rail gun has two parallel rails

L = 0.50 \text{ m}

apart connected by a sliding conducting bar of mass

m = 0.10 \text{ kg}

. A uniform vertical

B = 0.80 \text{ T}

field is applied. What current is required to accelerate the bar at

a = 20 \text{ m/s}^2

along the rails (ignore friction)?

Practice Problems

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

Example 1

medium

An electron (

q = 1.6 \times 10^{-19} \text{ C}

m = 9.11 \times 10^{-31} \text{ kg}

) moves at

2 \times 10^7 \text{ m/s}

perpendicular to a

0.01 \text{ T}

field. What is the radius of its circular path?

Example 2

hard

Two parallel wires

0.1 \text{ m}

apart each carry

10 \text{ A}

in the same direction. What is the force per meter between them? Is it attractive or repulsive? Use

\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}

Example 3

easy

A charge

q=3

C moves at

v=2

m/s perpendicular to

B=4

T. Find the force.

Example 4

easy

A charge moves parallel to a magnetic field. What is the magnetic force?

Example 5

easy

A wire of length

2

m carries

3

A perpendicular to

B=0.5

T. Find the force.

Example 6

easy

Is the magnetic force on a moving charge parallel or perpendicular to its velocity?

Example 7

easy

5

C charge at

1

m/s perpendicular to a field feels

10

N. Find the field strength.

Example 8

easy

For an electron versus a proton moving the same way in the same field, do the forces point the same or opposite way?

Example 9

easy

A wire feels force only because charges inside it are doing what?

Example 10

easy

2

C charge at

5

m/s perpendicular to

B=0.3

T. Find the force.

Example 11

medium

A charge

q=4

C at

v=3

m/s makes

30°

with

B=2

T. Find the force. (

\sin30°=0.5

Example 12

medium

A wire of length

0.4

m carries

5

A at

90°

to a field and feels

2

N. Find the field strength.

Example 13

medium

A charge

q=2

C, mass

m=4

kg, at

v=6

m/s perpendicular to

B=3

T. Find the radius of circular motion.

Example 14

medium

The maximum force on a charge in a field is

20

N (when perpendicular). What force results when it moves at

60°

? (

\sin60°\approx0.866

Example 15

medium

Two wires: wire 1 in field

B=0.5

T carries

4

A over

3

m; wire 2 carries

2

A over

3

m in

1

T, both perpendicular. Which feels more force?

Example 16

medium

A charge

q=1

C at

v=10

m/s feels

5

N in a

B=1

T field. Find the angle between velocity and field. (

\sin\theta=0.5

Example 17

medium

A wire carries

6

A over

0.5

m perpendicular to

B

and feels

9

N. Find

B

Example 18

challenge

A charge

q=2

C, mass

m=1

kg, circles at

r=5

m in a

B=0.4

T field. Find its speed.

Example 19

challenge

A charged particle enters a field region perpendicular to

B=0.2

T and follows a semicircle of radius

0.5

m before exiting. If

q/m = 10

C/kg, find its speed.

Example 20

challenge

A horizontal wire of mass

0.02

kg and length

0.5

m carries current

I

in a

0.4

T field so the magnetic force just balances gravity. Find

I

. (

g=10

m/s

^2

Example 21

medium

A wire of length

1.5

m carries

2

A perpendicular to

B=0.4

T. Find the force.

Example 22

medium

A charge

q=3

C moving at

v=4

m/s at

90°

feels

6

N. Find the field.

Example 23

easy

A charge of

0.20 \text{ C}

moves at

v = 50 \text{ m/s}

perpendicular to a

B = 0.40 \text{ T}

magnetic field. Find the force.

Example 24

easy

A straight wire of length

0.80 \text{ m}

carries

I = 5.0 \text{ A}

perpendicular to a

0.25 \text{ T}

field. Find the magnetic force on the wire.

Example 25

easy

Find the angle

\theta

between

\vec{v}

and

\vec{B}

if a charge

q = 1 \text{ C}

moving at

v = 2 \text{ m/s}

in a

B = 3 \text{ T}

field experiences

F = 3 \text{ N}

Example 26

medium

A wire of length

0.30 \text{ m}

carries

I = 6.0 \text{ A}

at an angle of

30°

to a

0.50 \text{ T}

magnetic field. Find the force on the wire.

Example 27

medium

An electron (

q = -1.6\times 10^{-19} \text{ C}

m = 9.11\times 10^{-31} \text{ kg}

) moves in a circle of radius

5.0 \text{ mm}

in a

0.02 \text{ T}

field. Find its speed.

Example 28

medium

A square loop of side

0.10 \text{ m}

lies in the

xy

-plane and carries

2.0 \text{ A}

counterclockwise as viewed from

+z

. A uniform

\vec{B} = 0.30 \text{ T}\,\hat{z}

field is applied. What is the net force on the loop?

Example 29

medium

Two long parallel wires

0.05 \text{ m}

apart carry currents

I_1 = 4 \text{ A}

and

I_2 = 6 \text{ A}

in opposite directions. Find the force per unit length between them. Use

\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}

Example 30

medium

A wire of mass per unit length

\lambda = 0.030 \text{ kg/m}

carries current

I

horizontally in a

B = 0.40 \text{ T}

horizontal field perpendicular to the wire. What current is required for the magnetic force to support the wire against gravity? (

g = 9.8 \text{ m/s}^2

)

Example 31

medium

A velocity selector uses crossed

\vec{E}

and

\vec{B}

fields. With

E = 2.0 \times 10^4 \text{ V/m}

and

B = 0.10 \text{ T}

, what speed passes through undeflected?

Example 32

hard

A wire carrying

I = 10 \text{ A}

has length vector

\vec{L} = (0.20\hat{x} + 0.10\hat{y}) \text{ m}

in a uniform field

\vec{B} = 0.30\hat{z} \text{ T}

. Find the force vector on the wire.

Example 33

hard

A proton enters a

0.50 \text{ T}

magnetic field with velocity

v = 3.0 \times 10^6 \text{ m/s}

60°

to the field. The proton's mass is

1.67\times 10^{-27} \text{ kg}

, charge

1.6\times 10^{-19} \text{ C}

. Find the pitch of the resulting helical path.

Example 34

hard

A rectangular loop with sides

a = 0.20 \text{ m}

and

b = 0.10 \text{ m}

carries

I = 3.0 \text{ A}

. The loop's normal makes

30°

with a uniform

\vec{B} = 0.40 \text{ T}

. Find the magnitude of the torque on the loop.

Example 35

hard

An electron moves in a circle in a

0.10 \text{ T}

field with kinetic energy

K = 1.0 \times 10^{-15} \text{ J}

. Find the radius. (

m_e = 9.11\times 10^{-31} \text{ kg}

e = 1.6\times 10^{-19} \text{ C}

, non-relativistic.)

Example 36

hard

A wire segment runs along

\vec{L} = 0.50\hat{x} \text{ m}

carrying

I = 4.0 \text{ A}

in a field

\vec{B} = (0.20\hat{y} + 0.30\hat{z}) \text{ T}

. Find the magnitude of the force.

Example 37

hard

Two long parallel wires carry equal currents

I

in the same direction. The wires are

0.10 \text{ m}

apart, and the force per meter is

2.0 \times 10^{-5} \text{ N/m}

(attractive). Find

I

. Use

\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}

Example 38

challenge

A particle (

q = +2.0\times 10^{-19} \text{ C}

m = 4.0\times 10^{-26} \text{ kg}

) enters a uniform

B = 0.30 \text{ T}

field at speed

v = 5.0\times 10^5 \text{ m/s}

perpendicular to the field. Find (a) the cyclotron radius and (b) the cyclotron frequency.

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Related Concepts

Magnetic Field Electric Current Force Electric Motor Electromagnetic Induction

Background Knowledge

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

magnetic fieldelectric currentforce