Charged Particle in Uniform Magnetic Field

A nonrelativistic charged particle is orbiting in a uniform magnetic field of strength H₀ at the center of a large solenoid. The radius of the orbit is R₀. The field is changed slowly to H₁. What is the new radius R₁ of the orbit? If the field is suddenly changed back to H₂ what is the final radius R₂?

SOLUTION

Where we considered the adiabatic invariant for a mechanical system. Here, we have for the motion in the plane perpendicular to the magnetic field

where

is the projection of the generalized momentum on this plane, and the integral is taken over one period of motion in this plane, whose shape is a circle. (A is the vector potential and is the charge of the particle.)

(1)

Using Stokes’ theorem and substituting  we obtain

(2)

We used the fact that the absolute value of p_t is constant. The minus sign before the second term occurs since the line integral about the orbit is opposite to the velocity of the charge. After substituting p_t = eHr /c into (2), we obtain

(3)

So, for a slow change of magnetic field from H₀ to H₁, we find

or

Now, if the field changes suddenly from H₁ back to H₀, then the energy ε is conserved

where ω₁ and ω₀ are cyclotron frequencies, corresponding to magnetic fields H₁ and H₀, respectively:

where m is the mass of the particle. Therefore,