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

Learning Objectives

By the end of this section, you will be able to:

• 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.

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 $F$ on a charge $q$ moving at a speed $v$ in a magnetic field of strength $B$ is given by

$$F=\text{qvB}\text{sin}\theta \text{,}$$

where $\theta$ is the angle between the directions of $\mathbf{\text{v}}$ and $\mathbf{\text{B}}.$ This force is often called the Lorentz force. In fact, this is how we define the magnetic field strength $B$—in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength $B$ 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 $F=\text{qvB}\text{sin}\theta$ for $B$.

$$B=\frac{F}{\text{qv}\text{sin}\theta }$$

Because $\text{sin}\theta$ is unitless, the tesla is

$$\text{1 T}=\frac{\text{1 N}}{\mathrm{C}\cdot \text{m/s}}=\frac{\text{1 N}}{\mathrm{A}\cdot \mathrm{m}}$$

(note that C/s = A).

Another smaller unit, called the gauss (G), where $\mathrm{1\; G}={\text{10}}^{-4}\mathrm{T}$, 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 $5\times {\text{10}}^{-5}\mathrm{T}$, or 0.5 G.

The direction of the magnetic force $\mathbf{\text{F}}$ is perpendicular to the plane formed by $\mathbf{\text{v}}$ and $\mathbf{\text{B}}$, as determined by the right hand rule 1 (or RHR-1), which is illustrated in Figure 22.16. 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 $\mathbf{\text{v}}$, the fingers in the direction of $\mathbf{\text{B}}$, and a perpendicular to the palm points in the direction of $\mathbf{\text{F}}$. 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.

Test Prep for AP Courses

Section Summary

• Magnetic fields exert a force on a moving charge q, the magnitude of which is

  $$F=\text{qvB}\text{sin}\theta ,$$

  where $\theta$ is the angle between the directions of $v$ and $B$.
• The SI unit for magnetic field strength $B$ is the tesla (T), which is related to other units by

  $$\mathrm{1\; T}=\frac{\text{1 N}}{\mathrm{C}\cdot \text{m/s}}=\frac{\text{1 N}}{\mathrm{A}\cdot \mathrm{m}}.$$

• 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 $v$, the fingers in the direction of $B$, and a perpendicular to the palm points in the direction of $F$.
• The force is perpendicular to the plane formed by $\mathbf{\text{v}}$ and $\mathbf{\text{B}}$. Since the force is zero if $\mathbf{\text{v}}$ is parallel to $\mathbf{\text{B}}$, charged particles often follow magnetic field lines rather than cross them.

Conceptual Questions

Problems & Exercises

Exercise 22.1

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 22.18?

Exercise 22.3

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 22.19, assuming it moves perpendicular to $\mathbf{\text{B}}\mathrm{?}$

Exercise 22.5

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 $\mathbf{\text{B}}$ is perpendicular to $\mathbf{\text{v}}$?