# OpenStax-CNX

You are here: Home » Content » College Physics » Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
Content endorsed by: OpenStax College

• Preface

• #### 34. Frontiers of Physics

• 35. Atomic Masses
• 37. Useful Information
• 38. Glossary of Key Symbols and Notation

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Endorsed by (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
• OpenStax College

This collection is included in aLens by: OpenStax College

Click the "OpenStax College" link to see all content they endorse.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Pierpont C & TC

This module is included inLens: Pierpont Community & Technical College's Lens
By: Pierpont Community & Technical CollegeAs a part of collection: "College Physics -- HLCA 1104"

Click the "Pierpont C & TC" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

• Featured Content

This collection is included inLens: Connexions Featured Content
By: Connexions

"This introductory, algebra-based, two-semester college physics book is grounded with real-world examples, illustrations, and explanations to help students grasp key, fundamental physics concepts. […]"

Click the "Featured Content" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Order printed collection

Inside Collection (Textbook):

Textbook by: OpenStax College. E-mail the author

# Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

Module by: OpenStax College. E-mail the author

Summary:

• Describe the effects of magnetic fields on moving charges.
• Use the right hand rule 1 to determine the velocity of a charge, the direction of the magnetic field, and the direction of the magnetic force on a moving charge.
• Calculate the magnetic force on a moving charge.

What is the mechanism by which one magnet exerts a force on another? The answer is related to the fact that all magnetism is caused by current, the flow of charge. Magnetic fields exert forces on moving charges, and so they exert forces on other magnets, all of which have moving charges.

## Right Hand Rule 1

The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force. The magnitude of the magnetic force FF size 12{F} {} on a charge qq size 12{q} {} moving at a speed vv size 12{v} {} in a magnetic field of strength BB size 12{B} {} is given by

F=qvBsinθ,F=qvBsinθ, size 12{F= ital "qvB""sin"θ} {}
(1)

where θθ size 12{θ} {} is the angle between the directions of vv and B.B. size 12{B} {} This force is often called the Lorentz force. In fact, this is how we define the magnetic field strength BB size 12{B} {}—in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength BB size 12{B} {} is called the tesla (T) after the eccentric but brilliant inventor Nikola Tesla (1856–1943). To determine how the tesla relates to other SI units, we solve F=qvBsinθF=qvBsinθ size 12{F= ital "qvB""sin"θ} {} for BB size 12{B} {}.

B=FqvsinθB=Fqvsinθ size 12{B= { {F} over { ital "qv""sin"θ} } } {}
(2)

Because sin θ sin θ size 12{θ} {} is unitless, the tesla is

1 T = 1 N C m/s = 1 N A m 1 T = 1 N C m/s = 1 N A m size 12{"1 T"= { {"1 N"} over {C cdot "m/s"} } = { {1" N"} over {A cdot m} } } {}
(3)

(note that C/s = A).

Another smaller unit, called the gauss (G), where 1 G=104T1 G=104T size 12{1G="10" rSup { size 8{ - 4} } T} {}, is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. The Earth’s magnetic field on its surface is only about 5×105T5×105T size 12{5 times "10" rSup { size 8{ - 5} } T} {}, or 0.5 G.

The direction of the magnetic force FF size 12{F} {} is perpendicular to the plane formed by vv size 12{v} {} and BB, as determined by the right hand rule 1 (or RHR-1), which is illustrated in Figure 1. RHR-1 states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of vv, the fingers in the direction of BB, and a perpendicular to the palm points in the direction of FF. One way to remember this is that there is one velocity, and so the thumb represents it. There are many field lines, and so the fingers represent them. The force is in the direction you would push with your palm. The force on a negative charge is in exactly the opposite direction to that on a positive charge.

### Making Connections: Charges and Magnets:

There is no magnetic force on static charges. However, there is a magnetic force on moving charges. When charges are stationary, their electric fields do not affect magnets. But, when charges move, they produce magnetic fields that exert forces on other magnets. When there is relative motion, a connection between electric and magnetic fields emerges—each affects the other.

### Example 1: Calculating Magnetic Force: Earth’s Magnetic Field on a Charged Glass Rod

With the exception of compasses, you seldom see or personally experience forces due to the Earth’s small magnetic field. To illustrate this, suppose that in a physics lab you rub a glass rod with silk, placing a 20-nC positive charge on it. Calculate the force on the rod due to the Earth’s magnetic field, if you throw it with a horizontal velocity of 10 m/s due west in a place where the Earth’s field is due north parallel to the ground. (The direction of the force is determined with right hand rule 1 as shown in Figure 2.)

Strategy

We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation F=qvBsinθF=qvBsinθ size 12{F= ital "qvB""sin"θ} {} to find the force.

Solution

The magnetic force is

F=qvbsinθ.F=qvbsinθ. size 12{F= ital "qvb""sin"θ} {}
(4)

We see that sinθ=1sinθ=1 size 12{"sin"θ=1} {}, since the angle between the velocity and the direction of the field is 90º90º size 12{"90" rSup { size 8{ circ } } } {}. Entering the other given quantities yields

F = 20 × 10 –9 C 10 m/s 5 × 10 –5 T = 1 × 10 –11 C m/s N C m/s = 1 × 10 –11 N. F = 20 × 10 –9 C 10 m/s 5 × 10 –5 T = 1 × 10 –11 C m/s N C m/s = 1 × 10 –11 N. alignl { stack { size 12{F= left ("20" times "10" rSup { size 8{ - 9 } } C right ) left ("10""m/s" right ) left (5 times "10" rSup { size 8{ - 5} } T right )} {} # " "=1 times "10" rSup { size 8{ - "11"} }  left (C cdot "m/s" right ) left ( { {N} over {C cdot "m/s"} } right )=1 times "10" rSup { size 8{ - "11"} } N "." {} } } {}
(5)

Discussion

This force is completely negligible on any macroscopic object, consistent with experience. (It is calculated to only one digit, since the Earth’s field varies with location and is given to only one digit.) The Earth’s magnetic field, however, does produce very important effects, particularly on submicroscopic particles. Some of these are explored in Force on a Moving Charge in a Magnetic Field: Examples and Applications.

## Section Summary

• Magnetic fields exert a force on a moving charge q, the magnitude of which is
F=qvBsinθ,F=qvBsinθ, size 12{F= ital "qvB""sin"θ} {}
(6)
where θθ size 12{θ} {} is the angle between the directions of vv size 12{v} {} and BB size 12{B} {}.
• The SI unit for magnetic field strength BB size 12{B} {} is the tesla (T), which is related to other units by
1 T=1 NCm/s=1 NAm.1 T=1 NCm/s=1 NAm.
(7)
• The direction of the force on a moving charge is given by right hand rule 1 (RHR-1): Point the thumb of the right hand in the direction of vv size 12{v} {}, the fingers in the direction of BB size 12{B} {}, and a perpendicular to the palm points in the direction of FF size 12{F} {}.
• The force is perpendicular to the plane formed by vv and BB size 12{B} {}. Since the force is zero if vv size 12{v} {} is parallel to BB size 12{B} {}, charged particles often follow magnetic field lines rather than cross them.

## Conceptual Questions

### Exercise 1

If a charged particle moves in a straight line through some region of space, can you say that the magnetic field in that region is necessarily zero?

## Problems & Exercises

### Exercise 1

What is the direction of the magnetic force on a positive charge that moves as shown in each of the six cases shown in Figure 3?

#### Solution

(a) Left (West)

(b) Into the page

(c) Up (North)

(d) No force

(e) Right (East)

(f) Down (South)

### Exercise 2

Repeat Exercise 1 for a negative charge.

### Exercise 3

What is the direction of the velocity of a negative charge that experiences the magnetic force shown in each of the three cases in Figure 4, assuming it moves perpendicular to B?B? size 12{B?} {}

(a) East (right)

(b) Into page

(c) South (down)

### Exercise 4

Repeat Exercise 3 for a positive charge.

### Exercise 5

What is the direction of the magnetic field that produces the magnetic force on a positive charge as shown in each of the three cases in the figure below, assuming BB size 12{B} {} is perpendicular to vv size 12{v} {}?

(a) Into page

(b) West (left)

(c) Out of page

### Exercise 6

Repeat Exercise 5 for a negative charge.

### Exercise 7

What is the maximum force on an aluminum rod with a 0.100-μC0.100-μC size 12{0 "." "100""-μC"} {} charge that you pass between the poles of a 1.50-T permanent magnet at a speed of 5.00 m/s? In what direction is the force?

#### Solution

7.50×107 N7.50×107 N size 12{7 "." "50" times "10" rSup { size 8{ - 7} } " N"} {} perpendicular to both the magnetic field lines and the velocity

### Exercise 8

(a) Aircraft sometimes acquire small static charges. Suppose a supersonic jet has a 0.500-μC0.500-μC size 12{0 "." "500""-μC"} {} charge and flies due west at a speed of 660 m/s over the Earth’s south magnetic pole, where the 8.00×105-T8.00×105-T size 12{8 "." "00" times "10" rSup { size 8{ - 5} } "-T"} {} magnetic field points straight up. What are the direction and the magnitude of the magnetic force on the plane? (b) Discuss whether the value obtained in part (a) implies this is a significant or negligible effect.

### Exercise 9

(a) A cosmic ray proton moving toward the Earth at 5.00×107m/s5.00×107m/s size 12{5 "." "00" times "10" rSup { size 8{7} } "m/s"} {} experiences a magnetic force of 1.70×1016N1.70×1016N size 12{1 "." "70" times "10" rSup { size 8{ - "16"} } N} {}. What is the strength of the magnetic field if there is a 45º45º size 12{"45" rSup { size 8{ circ } } } {} angle between it and the proton’s velocity? (b) Is the value obtained in part (a) consistent with the known strength of the Earth’s magnetic field on its surface? Discuss.

#### Solution

(a) 3.01×105 T3.01×105 T size 12{3 "." "01" times "10" rSup { size 8{ - 5} } " T"} {}

(b) This is slightly less then the magnetic field strength of 5×105T5×105T size 12{5 times "10" rSup { size 8{ - 5} } T} {} at the surface of the Earth, so it is consistent.

### Exercise 10

An electron moving at 4.00×103m/s4.00×103m/s size 12{4 "." "00" times "10" rSup { size 8{3} } "m/s"} {} in a 1.25-T magnetic field experiences a magnetic force of 1.40×1016N1.40×1016N size 12{1 "." "40" times "10" rSup { size 8{ - "16"} } N} {}. What angle does the velocity of the electron make with the magnetic field? There are two answers.

### Exercise 11

(a) A physicist performing a sensitive measurement wants to limit the magnetic force on a moving charge in her equipment to less than 1.00×1012N1.00×1012N size 12{1 "." "00" times "10" rSup { size 8{ - "12"} } N} {}. What is the greatest the charge can be if it moves at a maximum speed of 30.0 m/s in the Earth’s field? (b) Discuss whether it would be difficult to limit the charge to less than the value found in (a) by comparing it with typical static electricity and noting that static is often absent.

#### Solution

(a) 6.67×1010 C6.67×1010 C (taking the Earth’s field to be 5.00×105 T5.00×105 T size 12{5 "." "00" times "10" rSup { size 8{ - 5} } " T"} {})

(b) Less than typical static, therefore difficult

## Glossary

right hand rule 1 (RHR-1):
the rule to determine the direction of the magnetic force on a positive moving charge: when the thumb of the right hand points in the direction of the charge’s velocity vv size 12{v} {} and the fingers point in the direction of the magnetic field BB size 12{B} {}, then the force on the charge is perpendicular and away from the palm; the force on a negative charge is perpendicular and into the palm
Lorentz force:
the force on a charge moving in a magnetic field
tesla:
T, the SI unit of the magnetic field strength; 1 T=1 NAm1 T=1 NAm
magnetic force:
the force on a charge produced by its motion through a magnetic field; the Lorentz force
gauss:
G, the unit of the magnetic field strength; 1 G=10–4T1 G=10–4T size 12{"1 G"="10" rSup { size 8{ - 4} } `T} {}

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks