In the previous section, you learned that a magnetic field exerts a force on a moving electric charge. Because this force is always perpendicular to the particle's velocity, it changes the direction of motion without changing the particle's speed. This property allows magnetic fields to guide charged particles along curved paths.

Charged particles moving through magnetic fields are found throughout nature and modern technology. High-energy particles from space often follow spiral paths as they interact with Earth's magnetic field. In research laboratories, powerful electromagnets keep beams of protons and other particles traveling along curved paths inside particle accelerators. Similar principles are used in medical technology, including proton therapy systems and mass spectrometers that identify biological molecules by measuring the paths of charged ions.

Circular Motion in a Magnetic Field

Why does a magnetic field produce circular motion? Recall that the magnetic force on a moving charge is always perpendicular to the particle's velocity. Since work is done only by the component of force parallel to displacement, the magnetic force does no work on the particle. The particle's kinetic energy and speed therefore remain constant, while the direction of its velocity changes continuously.

This is the same behavior observed in uniform circular motion, where speed remains constant but velocity changes direction. If a charged particle moves perpendicular to a uniform magnetic field, as shown in Figure 38.2, the magnetic force continuously points toward the center of the circular path. The magnetic force therefore acts as the centripetal force.

For motion perpendicular to the magnetic field, the angle between the velocity and magnetic field is [latex]90^\circ[/latex], so [latex]\sin\theta=1[/latex]. The magnetic force therefore has its maximum magnitude:

Equating the magnetic force to the centripetal force gives

Solving for the radius of the circular path yields

The magnitude of the charge, [latex]|q|[/latex], is used because the sign of the charge determines the direction of curvature, not the radius.

The radius of curvature depends on four quantities:

• Mass ([latex]m[/latex]): More massive particles are harder to bend and therefore follow paths with larger radii.
• Speed ([latex]v[/latex]): Faster particles also follow paths with larger radii.
• Charge magnitude ([latex]|q|[/latex]): Particles with larger charge magnitudes experience stronger magnetic forces and follow tighter curves.
• Magnetic field strength ([latex]B[/latex]): Stronger magnetic fields bend particle trajectories more sharply, producing smaller radii.

Example 38.1: Calculating the Radius of an Electron's Circular Path

One early application of magnetic deflection was in cathode-ray tube (CRT) televisions and computer monitors. These devices produced images by directing a beam of electrons onto a phosphorescent screen. A nearby magnet could bend the electron beam and distort the image. Modern LCD and OLED displays do not use electron beams and are not affected in this way.

An electron travels at [latex]6.00\times10^7\ \text{m/s}[/latex] and enters a uniform magnetic field of strength [latex]0.500\ \text{T}[/latex] perpendicular to its direction of motion. What is the radius of its circular path?

Strategy

Because the electron moves perpendicular to the magnetic field, the magnetic force provides the centripetal force. Use

[latex]r=\frac{mv}{|q|B}[/latex]

The electron's mass and charge magnitude are known constants, and its speed and the magnetic field strength are given.

Solution

Substitute the known values:

[latex]\begin{aligned} r &=\frac{mv}{|q|B}\\[4pt] &=\frac{\left(9.11\times10^{-31}\ \text{kg}\right) \left(6.00\times10^7\ \text{m/s}\right)} {\left(1.60\times10^{-19}\ \text{C}\right) \left(0.500\ \text{T}\right)}\\[4pt] &=6.83\times10^{-4}\ \text{m} \end{aligned}[/latex]

Therefore,

[latex]r=0.683\ \text{mm}[/latex]

Discussion

The calculated radius is less than one millimeter, showing that even a moderately strong magnetic field can dramatically change the path of an electron.

The same principle is used to steer charged particles in particle accelerators, separate ions in mass spectrometers, and guide proton beams in some forms of cancer therapy. In each case, the radius depends on the particle's mass, charge, speed, and the magnetic field strength.

Helical Motion and Magnetic Mirrors

The circular-path equation applies when a particle's velocity is perpendicular to the magnetic field. If the velocity also has a component parallel to the field, that component experiences no magnetic force and remains unchanged. The perpendicular component produces circular motion, while the parallel component carries the particle forward along the magnetic field. The combined motion is a helical path.

The radius of the helical path depends on the perpendicular component of velocity:

If the particle enters a region where the magnetic field becomes stronger, the spiral becomes tighter. Under appropriate conditions, the particle's motion parallel to the field can reverse, causing the particle to travel back along the field line. This phenomenon is called a magnetic mirror and is important in space physics and fusion research.

Charged Particles in Earth's Magnetic Field

The motion of charged particles in magnetic fields explains many natural phenomena. Energetic charged particles emitted by the Sun and arriving from deep space interact continuously with Earth's magnetic field. Rather than traveling directly through the atmosphere, many spiral along magnetic field lines toward the polar regions.

When these particles collide with atoms and molecules in the upper atmosphere, they transfer energy that is later emitted as visible light. This produces the aurora borealis in the Northern Hemisphere and the aurora australis in the Southern Hemisphere.

Some incoming particles become trapped for long periods within two toroidal regions surrounding Earth called the Van Allen radiation belts. Within these regions, charged particles spiral along field lines while repeatedly reflecting between the Northern and Southern Hemispheres because of the magnetic mirror effect.

The Van Allen belts contain radiation levels high enough to pose hazards to astronauts and spacecraft electronics. Satellites are therefore designed to withstand radiation exposure or placed in orbits that reduce the time spent inside the belts.

Controlling Charged Particles

Scientists use magnetic fields to control charged particles in the laboratory. Particle accelerators employ carefully designed electromagnets to bend, focus, and steer beams of charged particles moving at high speeds. These machines have played a central role in discoveries about the particles and forces that make up matter.

Magnetic confinement is also one approach to controlled nuclear fusion, the process that powers the Sun. In a device called a tokamak, an extremely hot electrically charged gas—or plasma—is confined by strong magnetic fields so that it does not touch the walls of the reactor.

Mass spectrometry, one of the most important practical applications of this principle, is discussed in detail in a following section: More Applications of Magnetism.