22 Magnetism

# 171 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

- 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

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} {}.

Because

sinθsinθ size 12{θ} {}

is unitless, the tesla is

(note that C/s = A).

Another smaller unit, called the gauss (G), where 1 G=10−4T1 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 5×10−5T5×10−5T 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 [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 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.

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.

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].)

**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

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

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

- Magnetic fields exert a force on a moving charge
*q*, the magnitude of which isF=qvBsinθ,F=qvBsinθ, size 12{F= ital “qvB””sin”θ} {}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 NC⋅m/s=1 NA⋅m.1 T=1 NC⋅m/s=1 NA⋅m.
- 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

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]?

(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?B? size 12{B?} {}

(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 BB size 12{B} {} is perpendicular to vv size 12{v} {}?

(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-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?

7.50×10−7 N7.50×10−7 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-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 8.00×10−5-T8.00×10−5-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/s5.00×107m/s size 12{5 “.” “00” times “10” rSup { size 8{7} } `”m/s”} {} experiences a magnetic force of 1.70×10−16N1.70×10−16N size 12{1 “.” “70” times “10” rSup { size 8{ – “16”} } `N} {}. What is the strength of the magnetic field if there is a 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.

(a) 3.01×10−5 T3.01×10−5 T size 12{3 “.” “01” times “10” rSup { size 8{ – 5} } ” T”} {}

(b) This is slightly less then the magnetic field strength of 5×10−5T5×10−5T 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/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 1.40×10−16N1.40×10−16N 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 1.00×10−12N1.00×10−12N 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 C6.67×10−10 C (taking the Earth’s field to be 5.00×10−5 T5.00×10−5 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 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 NA⋅m1 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=10–4T1 G=10–4T size 12{“1 G”=”10” rSup { size 8{ – 4} } `T} {}