Many important applications of magnetism involve the motion of charged particles in a magnetic field. Older television picture tubes use magnetic fields to steer electrons through a vacuum from the electron gun to the screen. Microwave generators, which are used in applications ranging from ovens to radar, use a device called a magnetron in which electrons oscillate rapidly in a magnetic field.

You’ve just seen that there is no force on a charge that has velocity $\vec{v}$ parallel or antiparallel to a magnetic field. Consequently, a magnetic field has no effect on a charge moving parallel or antiparallel to the field. To understand the motion of charged particles in magnetic fields, we need to consider only motion perpendicular to the field.

FIGURE 29.37 shows a positive charge q moving with a velocity $\vec{v}$ in a plane that is perpendicular to a uniform magnetic field $\vec{B}$. According to the right-hand rule, the magnetic force on this particle is perpendicular to the velocity $\vec{v}$. A force that is always perpendicular to $\vec{v}$ changes the direction of motion, by deflecting the particle sideways, but it cannot change the particle’s speed. Thus a particle moving perpendicular to a uniform magnetic field undergoes uniform circular motion at constant speed. This motion is called the cyclotron motion of a charged particle in a magnetic field.

FIGURE 29.37 Cyclotron motion of a charged particle moving in a uniform magnetic field.

You’ve seen many analogies to cyclotron motion earlier in this text. For a mass moving in a circle at the end of a string, the tension force is always perpendicular to $\vec{v}$. For a satellite moving in a circular orbit, the gravitational force is always perpendicular to $\vec{v}$. Now, for a charged particle moving in a magnetic field, it is the magnetic force of strength $F=qvB$ that points toward the center of the circle and causes the particle to have a centripetal acceleration.

Newton’s second law for circular motion, which you learned in Chapter 8, is

Thus the radius of the cyclotron orbit is

The inverse dependence on B indicates that the size of the orbit can be decreased by increasing the magnetic field strength.

We can also determine the frequency of the cyclotron motion. Recall from your earlier study of circular motion that the frequency of revolution f is related to the speed and radius by $f=v\text{/}2\pi r$. A rearrangement of Equation 29.20 gives the cyclotron frequency:

where the ratio $q\text{/}m$ is the particle’s charge-to-mass ratio. Notice that the cyclotron frequency depends on the charge-to-mass ratio and the magnetic field strength but not on the charge’s speed.

EXAMPLE 29.11 The radius of cyclotron motion

In FIGURE 29.38, an electron is accelerated from rest through a potential difference of 500 V, then injected into a uniform magnetic field. Once in the magnetic field, it completes half a revolution in 2.0 ns. What is the radius of its orbit?

FIGURE 29.38 An electron is accelerated, then injected into a magnetic field.

FIGURE 29.38 Full Alternative Text

MODEL Energy is conserved as the electron is accelerated by the potential difference. The electron then undergoes cyclotron motion in the magnetic field, although it completes only half a revolution before hitting the back of the acceleration electrode.

SOLVE The electron accelerates from rest $(v_{\text{i}}=0\text{m}\text{/}\text{s})$ at $V_{\text{i}}=0\text{V}$ to speed $v_{\text{f}}$ at $V_{\text{f}}=500\text{V}$. We can use conservation of energy $K_{\text{f}}+q{V}_{\text{f}}=K_{\text{i}}+q{V}_{\text{i}}$ to find the speed $v_{\text{f}}$ with which it enters the magnetic field:

$$\begin{array}{l}{\displaystyle \frac{1}{2}}m{v}_{\text{f}}^2+(-e)V_{\text{f}}=0+0\\ v_{\text{f}}=\sqrt{{\displaystyle \frac{2e{V}_{\text{f}}}{m}}}=\sqrt{{\displaystyle \frac{2(1.60\times {10}^{-19}\text{\hspace{0.17em}C})(500\text{V})}{9.11\times {10}^{-31}\text{\hspace{0.17em}kg}}}}\\ =1.33\times {10}^7\text{m}\text{/}\text{s}\end{array}$$

The cyclotron radius in the magnetic field is $r_{\text{cyc}}=mv\text{/}eB$, but we first need to determine the field strength. Were it not for the electrode, the electron would undergo circular motion with period $T=4.0\text{ns}$. Hence the cyclotron frequency is $f=1\text{/}\text{T}=2.5\times {10}^8\text{Hz}$. We can use the cyclotron frequency to determine that the magnetic field strength is

$$\begin{array}{ccc}\hfill B& \hfill =\hfill & {\displaystyle \frac{2\pi m{f}_{\text{cyc}}}{e}}={\displaystyle \frac{2\pi (9.11\times {10}^{-31}\text{kg})(2.50\times {10}^8\text{Hz})}{1.60\times {10}^{-19}\text{C}}}\hfill \\ \hfill & \hfill =\hfill & 8.94\times {10}^{-3}\text{T}\hfill \end{array}$$

Thus the radius of the electron’s orbit is

$$r_{\text{cyc}}={\displaystyle \frac{mv}{qB}}=8.5\times {10}^{-3}\text{m}=8.5\text{mm}$$

REVIEW A 17-mm-diameter orbit is similar to what is seen in the photo just before this example, so this seems to be a typical size for electrons moving in modest magnetic fields.

FIGURE 29.39a shows a more general situation in which the charged particle’s velocity $\vec{v}$ is neither parallel nor perpendicular to $\vec{B}$. The component of $\vec{v}$ parallel to $\vec{B}$ is not affected by the field, so the charged particle spirals around the magnetic field lines in a helical trajectory. The radius of the helix is determined by $\vec{v}{}_1$, the component of $\vec{v}$ perpendicular to $\vec{B}$.

FIGURE 29.39 In general, charged particles spiral along helical trajectories around the magnetic field lines. This motion is responsible for the earth’s aurora.

Full Alternative Text

Full Alternative Text

The motion of charged particles in a magnetic field is responsible for the earth’s aurora. High-energy particles and radiation streaming out from the sun, called the solar wind, create ions and electrons as they strike molecules high in the atmosphere. Some of these charged particles become trapped in the earth’s magnetic field, creating what is known as the Van Allen radiation belt.

As FIGURE 29.39b shows, the electrons spiral along the magnetic field lines until the field leads them into the atmosphere. The shape of the earth’s magnetic field is such that most electrons enter the atmosphere near the magnetic poles. There they collide with oxygen and nitrogen atoms, exciting the atoms and causing them to emit auroral light, as seen in FIGURE 29.39c.