21.3

The Motion of a Charged Particle in a Magnetic Field

| Comparing Particle Motion in Electric and Magnetic Fields

The motion of a charged particle in an electric field is noticeably different from the motion in a magnetic field. For example, Figure 21-10a shows a positive charge moving between the plates of a parallel plate capacitor. Initially, the charge is moving perpendicular to the direction of the electric field. Since the direction of the electric force on a positive charge is in the same direction as the electric field, the particle is deflected sideways. Part b of the drawing shows the same particle traveling initially at right angles to a magnetic field. An application of RHR-1 shows that when the charge enters the field, the charge is deflected upward (not sideways) by the magnetic force. As the charge moves upward, the direction of the magnetic force changes, always remaining perpendicular to both the magnetic field and the velocity. Conceptual Example 2 focuses on the difference in how electric and magnetic fields apply forces to a moving charge.

Figure 21-9 Right-Hand Rule No. 1 is illustrated. When the right hand is oriented so the fingers point along the magnetic field and the thumb points along the velocity of a positively charged particle, the palm faces in the direction of the magnetic force applied to the particle.

Figure 21-10 (a) The electric force that acts on a positive charge is parallel to the electric field and causes the particle’s trajectory to bend in a horizontal plane. (b) The magnetic force is perpendicular to both the magnetic field and the velocity and causes the particle’s trajectory to bend in a vertical plane.

| Conceptual Example 2 | | | A Velocity Selector

A velocity selector is a device for measuring the velocity of a charged particle. The device operates by applying electric and magnetic forces to the particle in such a way that these forces balance. Figure 21-11a shows a particle with a positive charge +q and a velocity , which is perpendicular to a constant magnetic field* . How should an electric field be directed so that the force it applies to the particle can balance the magnetic force produced by ?

Figure 21-11 (a) A particle with a positive charge q and velocity moves perpendicularly into a magnetic field . (b) A velocity selector is a tube in which an electric field is perpendicular to a magnetic field, and the field magnitudes are adjusted so that the electric and magnetic forces acting on the particle balance.

The physics of a velocity selector.

Reasoning and Solution If the electric and magnetic forces are to balance, they must have opposite directions. By applying RHR-1, we find that the magnetic force acting on the positively charged particle in Figure 21-11a is directed upward, toward the top of the page. Therefore, the electric force must be directed downward. But the force applied to a positive charge by an electric field has the same direction as the field itself. Therefore, the electric field must point downward if the electric force is to balance the magnetic force. In other words, the electric and magnetic fields are perpendicular. Figure 21-11b shows a velocity selector that uses perpendicular electric and magnetic fields. The device is a cylindrical tube located within the magnetic field . Inside the tube is a parallel plate capacitor that produces the electric field . The charged particle enters the left end of the tube perpendicular to both fields. If the strengths of the fields and are adjusted properly, the electric and magnetic forces acting on the particle will cancel each other. With no net force acting on the particle, the velocity remains unchanged, according to Newton’s first law. As a result, the particle moves in a straight line at a constant speed and exits at the right end of the tube. The magnitude of the velocity “selected” can be determined from a knowledge of the strengths of the electric and magnetic fields. Particles with velocities different from the one “selected” are deflected and do not exit at the right end of the tube.
Related Homework: Problems 19, 20

We have seen that a charged particle traveling in a magnetic field experiences a magnetic force that is always perpendicular to the field. In contrast, the force applied by an electric field is always parallel (or antiparallel) to the field direction. Because of the difference in the way that electric and magnetic fields exert forces, the work done on a charged particle by each field is different, as we now discuss.

| The Work Done on a Charged Particle Moving Through Electric and Magnetic Fields

In Figure 21-10a an electric field applies a force to a positively charged particle, and, consequently, the path of the particle bends in the direction of the force. Because there is a component of the particle’s displacement in the direction of the electric force, the force does work on the particle, according to Equation 6.1. This work increases the kinetic energy and, hence, the speed of the particle, in accord with the work–energy theorem (see Section 6.2). In contrast, the magnetic force in Figure 21-10b always acts in a direction that is perpendicular to the motion of the charge. Consequently, the displacement of the moving charge never has a component in the direction of the magnetic force. As a result, the magnetic force cannot do work and change the kinetic energy of the charged particle in Figure 21-10b. Thus, the speed of the particle does not change, although the force does alter the direction of the motion.

| The Circular Trajectory

To describe the motion of a charged particle in a constant magnetic field more completely, let’s discuss the special case in which the velocity of the particle is perpendicular to a uniform magnetic field. As Figure 21-12 illustrates, the magnetic force serves to move the particle in a circular path. To understand why, consider two points on the circumference labeled 1 and 2. When the positively charged particle is at point 1, the magnetic force is perpendicular to the velocity and points directly upward in the drawing. This force causes the trajectory to bend upward. When the particle reaches point 2, the magnetic force still remains perpendicular to the velocity but is now directed to the left in the drawing. Problem solving insight: The magnetic force always remains perpendicular to the velocity and is directed toward the center of the circular path.

Figure 21-12 A positively charged particle is moving perpendicular to a constant magnetic field. The magnetic force causes the particle to move on a circular path (R.H. = right hand).

To find the radius of the path in Figure 21-12, we use the concept of centripetal force from Section 5.3. The centripetal force is the net force, directed toward the center of the circle, that is needed to keep a particle moving along a circular path. The magnitude Fc of this force depends on the speed v and mass m of the particle, as well as the radius r of the circle:

| (5.3)

In the present situation, the magnetic force furnishes the centripetal force. Being perpendicular to the velocity, the magnetic force does no work in keeping the charge +q on the circular path. According to Equation 21.1, the magnitude of the magnetic force is , so or

| (21.2)

This result shows that the radius of the circle is inversely proportional to the magnitude of the magnetic field, with stronger fields producing “tighter” circular paths. Example 3 illustrates an application of Equation 21.2.

| Example 3 | | | The Motion of a Proton

A proton is released from rest at point A, which is located next to the positive plate of a parallel plate capacitor (see Figure 21-13). The proton then accelerates toward the negative plate, leaving the capacitor at point B through a small hole in the plate. The electric potential of the positive plate is 2100 V greater than that of the negative plate, so . Once outside the capacitor, the proton travels at a constant velocity until it enters a region of constant magnetic field of magnitude 0.10 T. The velocity is perpendicular to the magnetic field, which is directed out of the page in Figure 21-13. Find (a) the speed vB of the proton when it leaves the negative plate of the capacitor, and (b) the radius r of the circular path on which the proton moves in the magnetic field.

Figure 21-13 A proton, starting from rest at the positive plate of the capacitor, accelerates toward the negative plate. After leaving the capacitor, the proton enters a magnetic field, where it moves on a circular path of radius r.

Reasoning The only force that acts on the proton (charge = +e) while it is between the capacitor plates is the conservative electric force. Thus, we can use the conservation of energy to find the speed of the proton when it leaves the negative plate. The total energy of the proton is the sum of its kinetic energy, , and its electric potential energy, EPE. Following Example 4 in Chapter 19, we set the total energy at point B equal to the total energy at point A:

We note that vA = 0 m/s, since the proton starts from rest, and use Equation 19.4 to set . Then the conservation of energy reduces to . Solving for vB gives . The proton enters the magnetic field with this speed and moves on a circular path with a radius that is given by Equation 21.2.
Solution

a. | The speed of the proton is

b. | When the proton moves in the magnetic field, the radius of the circular path is

| (21.2)

Need more practice?

Interactive LearningWare21.1
A particle that has a charge of moves in a 0.20-T magnetic field. It travels on a circular path that has a radius of 0.18 m. The centripetal acceleration of the particle is . What is the magnitude of the centripetal force that acts on the particle?

One of the exciting areas in physics today is the study of elementary particles, which are the basic building blocks from which all matter is constructed. Important information about an elementary particle can be obtained from its motion in a magnetic field, with the aid of a device known as a bubble chamber. A bubble chamber contains a superheated liquid such as hydrogen, which will boil and form bubbles readily. When an electrically charged particle passes through the chamber, a thin track of bubbles is left in its wake. This track can be photographed to show how a magnetic field affects the particle motion. Conceptual Example 4 illustrates how physicists deduce information from such photographs.

| Conceptual Example 4 | | | Particle Tracks in a Bubble Chamber

Figure 21-14a shows the bubble-chamber tracks resulting from an event that begins at point A. At this point a gamma ray (emitted by certain radioactive substances), traveling in from the left, spontaneously transforms into two charged particles. There is no track from the gamma ray itself. These particles move away from point A, producing the two spiral tracks. A third charged particle is knocked out of a hydrogen atom and moves forward, producing the long track with the slight upward curvature. Each of the three particles has the same mass and carries a charge of the same magnitude. A uniform magnetic field is directed out of the paper toward you. Guided by RHR-1 and Equation 21.2, deduce the sign of the charge carried by each particle, identify which particle is moving most rapidly, and account for the two spiral paths.

Figure 21-14 (a) A photograph of tracks in a bubble chamber. A magnetic field is directed out of the paper. At point A a gamma ray (not visible) spontaneously transforms into two charged particles, and a third charged particle is knocked out of a hydrogen atom in the chamber. (Lawrence Berkeley Laboratory/Photo Researchers) (b) In accord with RHR-1, the magnetic field applies a downward force to a positively charged particle that moves to the right.

Reasoning and Solution To help in our reasoning, Figure 21-14b shows a positively charged particle traveling with a velocity that is perpendicular to a magnetic field. The field is directed out of the paper. RHR-1 indicates that the magnetic force points downward. Therefore, downward-curving tracks in the photograph indicate a positive charge, while upward-curving tracks indicate a negative charge. Particles 1 and 3, then, must carry a negative charge. They are, in fact, electrons (e−). In contrast, particle 2 must have a positive charge. It is called a positron (e+), an elementary particle that has the same mass as an electron but an opposite charge.
In Equation 21.2 the mass m, the charge magnitude |q|, and the magnetic field strength B are the same for each particle. Therefore, the radius r is proportional to the speed v, and a greater radius means a greater speed. The track for particle 3 has the greatest radius, so particle 3 has the greatest speed.
Each spiral indicates that the radius is decreasing as the particle moves. Since the radius is proportional to the speed, the speeds of particles 1 and 2 must be decreasing as these particles move. Correspondingly, the kinetic energies of these particles must be decreasing. Particles 1 and 2 are losing energy each time they collide with a hydrogen atom in the bubble chamber.
Related Homework: Conceptual Question 5, Problem 24

Check Your Understanding 2

Three particles have identical charges and masses. They enter a constant magnetic field and follow the paths shown in the drawing. Rank the speeds of the particles, largest to smallest. (The answer is given at the end of the book.)

Background: When a charged particle enters a magnetic field, the particle experiences a magnetic force that causes it to move in a circular path, the radius of which depends on the particle’s speed, mass, and charge, and also on the magnitude of the field.
For similar questions (including calculational counterparts), consult Self-Assessment Test 21.1, which is described at the end of Section 21.4.

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