Radiating Systems, Multipole Fields and Radiation — John David Jackson
Physics — E&M · synchrotron-radiation-energy-loss
Problem
A particle of mass m, charge q, moves in a plane perpendicular to a uniform, static, magnetic induction B.
(a) Calculate the total energy radiated per unit time, expressing it in terms of the constants already defined and the ratio γ of the particle's total energy to its rest energy.
(b) If at time t=0 the particle has a total energy E0=γ0mc2, show that it will have energy E=γmc2<E0 at a time t, where
t≈2q4B23m3c5(γ1−γ01)
provided γ≫1.
(c) If the particle is initially nonrelativistic and has a kinetic energy T0 at t=0, what is its kinetic energy at time t?
(d) If the particle is actually trapped in the magnetic dipole field of the earth and is spiraling back and forth along a line of force, does it radiate more energy while near the equator, or while near its turning points? Why? Make quantitative statements if you can.