28.5:
Motion Of A Charged Particle In A Magnetic Field
Consider a point mass m of charge Q moving perpendicular to a uniform magnetic field B.
The Lorentz force's magnitude is constant and is always perpendicular to its velocity. So, it cannot change the velocity's magnitude, only its direction. Hence, the particle's path is a circle with a radius r.
Using Newton's second law of motion, the Lorentz force equals the centripetal force. Thus, the circle's radius is calculated.
The time period T of the particle's circular path is the circumference by speed. Substituting the value of r, T is obtained.
If the velocity is not perpendicular to the magnetic field, the component parallel to the magnetic field remains unaffected, thus producing a constant motion along the magnetic field. Hence, the particle moves in a helical path.
In the formula for the radius, the perpendicular component of the velocity is substituted. Since T is independent of the speed, it remains the same.
Its pitch is defined as the distance between adjacent turns, given by the product of the parallel component of the velocity and period.
28.5:
Motion Of A Charged Particle In A Magnetic Field
A charged particle experiences a force when moving through a magnetic field. Consider the field to be uniform and the charged particle to move perpendicular to it. If the field is in a vacuum, the magnetic field is the dominant factor determining the motion. Since the magnetic force is perpendicular to the direction of motion, a charged particle follows a curved path. The particle continues to follow this curved path until it forms a complete circle. Another way to look at this is that the magnetic force is always perpendicular to velocity, so that it does no work on the charged particle. The particle's kinetic energy and speed thus remain constant. The force affects the direction of motion but not the speed. Hence, the particle moves in a circle. The time for the charged particle to go around the circular path is defined as the period, which is the same as the distance traveled to make one complete cycle, or the circle's circumference, divided by the speed.
If the velocity is not perpendicular to the magnetic field, then each component of the velocity needs to be compared separately with the magnetic field. The component of the velocity perpendicular to the magnetic field produces a magnetic force perpendicular to both this velocity and the field. The component parallel to the magnetic field creates constant motion along the same direction as the magnetic field. The parallel motion determines the pitch of the helix, which is defined as the distance between adjacent turns. It is equal to the parallel component of the velocity times the period. The result is a helical motion.
While the charged particle travels in a helical path, it may enter a region where the magnetic field is not uniform. In particular, suppose a particle travels from a region of strong magnetic field to a region of weaker field, then back to a region of stronger field. The particle may reflect back before entering the stronger magnetic field region. This is similar to a wave on a string traveling from a very light, thin string to a hard wall and reflecting backward. If the reflection happens at both ends, the particle is trapped in a so-called magnetic bottle.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version], section 11.3, pages 501–506. Retrieved from Motion of a Charged Particle in a Magnetic Field