An electron with charge -e and mass m travels at a speed v in a plane
perpendicular to a magnetic field of magnitude B. The electron follows a
circular path of radius R. In a time, t, the electron travels halfway around the
circle. What is the amount of work done by the magnetic field?
Answers
Answer:
Magnetic force can cause a charged particle to move in a circular or spiral path. Cosmic rays are energetic charged particles in outer space, some of which approach the Earth. They can be forced into spiral paths by the Earth’s magnetic field. Protons in giant accelerators are kept in a circular path by magnetic force. The bubble chamber photograph in Figure 1 shows charged particles moving in such curved paths. The curved paths of charged particles in magnetic fields are the basis of a number of phenomena and can even be used analytically, such as in a mass spectrometer.
A drawing representing trails of bubbles in a bubble chamber.
Figure 1. Trails of bubbles are produced by high-energy charged particles moving through the superheated liquid hydrogen in this artist’s rendition of a bubble chamber. There is a strong magnetic field perpendicular to the page that causes the curved paths of the particles. The radius of the path can be used to find the mass, charge, and energy of the particle.
So does the magnetic force cause circular motion? 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 direction of motion is affected, but not the speed. This is typical of uniform circular motion. The simplest case occurs when a charged particle moves perpendicular to a uniform $latex \boldsymbol{B} $ -field, such as shown in Figure 2. (If this takes place in a vacuum, the magnetic field is the dominant factor determining the motion.) Here, the magnetic force supplies the centripetal force $latex \boldsymbol{F_c = mv^2/r} $. Noting that $latex \boldsymbol{\textbf{sin} \;\theta = 1} $, we see that $latex \boldsymbol{F = qvB} $.
Diagram showing an electrical charge moving clockwise in the plane of the page. Velocity vectors are tangent to the circular path. The magnetic field B is oriented into the page. Force vectors show that the force on the charge is toward the center of the charge’s circular path as the charge moves.
Figure 2. A negatively charged particle moves in the plane of the page in a region where the magnetic field is perpendicular into the page (represented by the small circles with x’s—like the tails of arrows). The magnetic force is perpendicular to the velocity, and so velocity changes in direction but not magnitude. Uniform circular motion results.
Because the magnetic force $latex \boldsymbol{F} $ supplies the centripetal force $latex \boldsymbol{F_c} $, we have
$latex \boldsymbol{qvB =} $ \boldsymbol{\frac{mv^2}{r}}.
Solving for $latex \boldsymbol{r} $ yields
$latex \boldsymbol{r =} $ \boldsymbol{\frac{mv}{qB}}.
Here, $latex \boldsymbol{r} $ is the radius of curvature of the path of a charged particle with mass $latex \boldsymbol{m} $ and charge $latex \boldsymbol{q} $, moving at a speed $latex \boldsymbol{v} $ perpendicular to a magnetic field of strength $latex \boldsymbol{B} $. If the velocity is not perpendicular to the magnetic field, then $latex \boldsymbol{v} $ is the component of the velocity perpendicular to the field. The component of the velocity parallel to the field is unaffected, since the magnetic force is zero for motion parallel to the field. This produces a spiral motion rather than a circular one.