Charged Particle in Uniform Magnetic Field
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 74
8-8-2016
1636
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 pt 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 pt = 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,
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