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

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 F size 12{F} {}

on a charge q size 12{q} {}

moving at a speed v size 12{v} {}

in a magnetic field of strength B size 12{B} {}

is given by

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

where θ size 12{θ} {}

is the angle between the directions of v

and B. size 12{B} {}

This force is often called the Lorentz force. In fact, this is how we define the magnetic field strength B size 12{B} {}

—in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength B 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=qvBsinθ size 12{F= ital "qvB""sin"θ} {}

for B size 12{B} {}

.

B=Fqvsinθ size 12{B= { {F} over { ital "qv""sin"θ} } } {}

Because sin θ size 12{θ} {}

is unitless, the tesla is

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} } } {}

(note that C/s = A).

Another smaller unit, called the gauss (G), where 1 G=10−4T size 12{1`G="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 5×10−5T size 12{5 times "10" rSup { size 8{ - 5} } `T} {}

, or 0.5 G.

The direction of the magnetic force F size 12{F} {}

is perpendicular to the plane formed by v size 12{v} {}

and B

, as determined by the right hand rule 1 (or RHR-1), which is illustrated in [link]. 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 v

, the fingers in the direction of B

, and a perpendicular to the palm points in the direction of F

. 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.

The right hand rule 1. An outstretched right hand rests palm up on a piece of paper on which a vector arrow v points to the right and a vector arrow B points toward the top of the paper. The thumb points to the right, in the direction of the v vector arrow. The fingers point in the direction of the B vector. B and v are in the same plane. The F vector points straight up, perpendicular to the plane of the paper, which is the plane made by B and v. The angle between B and v is theta. The magnitude of the magnetic force F equals q v B sine theta.

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.

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 [link].)

The effects of the Earth’s magnetic field on moving charges. Figure a shows a positive charge with a velocity vector due west, a magnetic field line B oriented due north, and a magnetic force vector F straight down. Figure b shows the right hand facing down, with the fingers pointing north with B, the thumb pointing west with v, and force down away from the hand.

Strategy

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

to find the force.

Solution

The magnetic force is

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

We see that sinθ=1 size 12{"sin"θ=1} {}

, since the angle between the velocity and the direction of the field is 90º 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. 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 "." {} } } {}

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

Conceptual Questions

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

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

figure a shows magnetic field line direction symbols with solid circles labeled B out; a velocity vector points down; figure b shows B vectors pointing right and v vector pointing up; figure c shows B in and v to the right; figure d shows B vector pointing right and v vector pointing left; figure e shows B vectors up and v vector into the page; figure f shows B vectors pointing left and v vectors out of the page

(a) Left (West)

(b) Into the page

(c) Up (North)

(d) No force

(e) Right (East)

(f) Down (South)

Repeat [link] for a negative charge.

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

Figure a shows the force vector pointing up and B out of the page. Figure b shows the F vector pointing up and the B vector pointing to the right. Figure c shows the F vector pointing to the left and the B vector pointing into the page.

(a) East (right)

(b) Into page

(c) South (down)

Repeat [link] for a positive charge.

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 B size 12{B} {}

is perpendicular to v size 12{v} {}

?

Figure a shows a force vector pointing toward the left and a velocity vector pointing up. Figure b shows the force vector pointing into the page and the velocity vector pointing down. Figure c shows the force vector pointing up and the velocity vector pointing to the left.

(a) Into page

(b) West (left)

(c) Out of page

Repeat [link] for a negative charge.

What is the maximum force on an aluminum rod with a 0.100-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?

7.50×10−7 N size 12{7 "." "50" times "10" rSup { size 8{ - 7} } " N"} {}

perpendicular to both the magnetic field lines and the velocity

(a) Aircraft sometimes acquire small static charges. Suppose a supersonic jet has a 0.500-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 8.00×10−5-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.

(a) A cosmic ray proton moving toward the Earth at 5.00×107m/s size 12{5 "." "00" times "10" rSup { size 8{7} } `"m/s"} {}

experiences a magnetic force of 1.70×10−16N size 12{1 "." "70" times "10" rSup { size 8{ - "16"} } `N} {}

. What is the strength of the magnetic field if there is a 45º 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.

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

(b) This is slightly less then the magnetic field strength of 5×10−5T size 12{5 times "10" rSup { size 8{ - 5} } `T} {}

at the surface of the Earth, so it is consistent.

An electron moving at 4.00×103m/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 1.40×10−16N 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.

(a) A physicist performing a sensitive measurement wants to limit the magnetic force on a moving charge in her equipment to less than 1.00×10−12N 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.

(a) 6.67×10−10 C

(taking the Earth’s field to be 5.00×10−5 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 v size 12{v} {}

and the fingers point in the direction of the magnetic field

B 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 NA⋅m
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=10–4T size 12{"1 G"="10" rSup { size 8{ - 4} } `T} {}
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