An electron moves in a circle of radius R in a uniform magnetic field 𝐁. The field is into the p

step 1 All right, so I have an electron moving in a magnetic field, which is going to cause it to spin

An electron moves in a circle of radius R in a uniform...

An electron moves in a circle of radius $R$ in a uniform magnetic field $\overrightarrow{\mathbf{B}}$. The field is into the page. (a) Does the electron move clockwise or counterclockwise? (b) How much time does the electron take to make one complete revolution? Derive an expression for the time, starting with the magnetic force on the electron. Your answer may include $R, B$, and any fundamental constants.

Key Concepts

Centripetal Motion

When the magnetic force acts as the centripetal force, it causes the charged particle to move in a circular path. Setting the magnetic force equal to the centripetal force (m*v^2/R) allows one to derive relationships among the particle’s speed, the radius of the circle, and the magnetic field strength. This concept is key to analyzing the dynamics of a particle under the influence of a magnetic field.

Cyclotron Frequency

Also known as the gyrofrequency, cyclotron frequency is defined as ? = |q|B/m, where q is the charge, B is the magnetic field strength, and m is the mass of the particle. This frequency characterizes the rate at which a charged particle orbits within a magnetic field and is derived by equating the magnetic force to the centripetal force required for circular motion.

Lorentz Force

This is the force experienced by a charged particle moving in a magnetic field. It is given by F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. The direction of the force is determined by the cross product and, in the case of a negatively charged electron, it requires careful application of the right?hand rule with a sign reversal. This force is central to understanding how a magnetic field can alter the motion of a charged particle.

Time Period of Revolution

The time period for one complete revolution is inversely related to the cyclotron frequency, given by T = 2?/?, which simplifies to T = (2?m)/(|q|B). This expression underlines how the mass, charge, and magnetic field directly influence the duration of one orbital cycle in a magnetic field.

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