# Does a particle's rotational kinetic energy change when it enters a uniform magnetic field?

Consider a charged particle entering a uniform magnetic field with some constant velocity perpendicular to the magnetic field. I know that the particle will follow a circular path within the magnetic field, and the magnitude of the velocity will stay constant. Hence, the particle will have a uniform circular motion. Since the particle starts with zero angular velocity and ends with some nonzero angular velocity, does the particle's rotational kinetic energy change when it enters the uniform magnetic field even though the work done on the particle by the external magnetic field is zero?

• “I know that the particle will follow a circular path within the magnetic field and the magnitude of the velocity will stay constant.” That is wrong. The electron emits photons, follows a spiral path, exhausted its kinetic energy and came to standstill. Aug 11 '19 at 20:52
• @HolgerFiedler Thank you for that comment. Although, I am considering classical mechanics here so the charged particle will not emit photons. Aug 12 '19 at 11:18

Your mistake is in saying that the initial angular velocity is $$0$$. When the particle enters the magnetic field it already has a non-zero angular velocity about the center of the circular path it will start following. Therefore, it's rotational kinetic energy does not change. This makes sense, since there is no torque about the center of the circle acting on the particle.

• Thanks Aaron, you've given me the insight I needed. I calculated the angular momentum before and after the charged particle entered the magnetic field and found that the change in angular momentum is zero. Therefore, the net external torque on the charged particle is zero. Hence, the rotational kinetic energy does not change. Aug 12 '19 at 11:23
• How did it obtain this initial angular velocity even? This seems strange as torque acting on it even during the moment when it entered the magnetic field's region is 0 Apr 7 '20 at 3:11
• @SchwarzKugelblitz It got it from whatever gave it its initial velocity to begin with Apr 7 '20 at 3:15
• let's say it was projected straight by giving it an initial impulse into the magnetic field and it travelled some distance in a straight line before entering the magnetic field. Now upon entering the magnetic field, it gains an angular velocity. Now what would be the cause of the gain in angular velocity? Apr 7 '20 at 3:19
• @SchwarzKugelblitz It has angular velocity before entering the field. An object doesn't need to be moving in a circle to have angular velocity about some point. Apr 7 '20 at 6:45

I am not sure if I understand what you mean by particle's rotational kinetic energy. If by particle, you mean an electron, the rotational kinetic energy is meaningless. If by particle you mean a conducting or non conducting sphere (hallow or solid), then I can answer the question.

If the sphere rotating as if it is attached by a string (or spring!) and some one is rotating it, at each moment you get an additional magnetic force component which is added to the force of the string. This can increase or decrease the angular velocity and radius of rotation. But changing the radius also changes moment of inertial and thus the total change in the rotational kinetic energy remains zero.

If by rotating you mean spin the answer will be as follow: Conducting sphere: before the sphere enters the field, the electrons are rotating. As the sphere enters the field the electrons enter the field at different directions of velocity. So they will move on the surface of the conductor based in the direction of their velocities. If the sphere is very heavy and the magnetic field is weak the sphere moves on a straight line with some surface current. if the sphere is light and the magnetic field is very strong the sphere will stray from the straight line.

Non conducting sphere: since the electrons cannot move (at least in small magnetic field) nothing happens and the sphere moves on a straight line. What happens to the electrons and atoms inside the non conducting sphere depends on the material and the bound of the atoms. Again if the sphere is light and the magnetic field is strong enough the sphere will stray from the straight line.

• The OP is referring to motion about the center of the circular path. Not the spinning of the particle itself. Aug 11 '19 at 4:00