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A relativistic electron radiates while executing a nearly circular cyclotron motion in a uniform magnetic field B. Find how the function $\gamma(t) \equiv E(t)/mc^2$ decreases with time t starting from some initial value $\gamma(0)$. Assume that $\gamma(t) \gg 1$ and that the energy radiated during one period of cyclotron motion is small compared to the electron energy E(t).
Hint: the power W radiated by a relativistic electron can be written as
(1)where $p^{\mu}=(E/c,-\mathbf{P})$ and $p_{\mu}=(E/c,\mathbf{P})$ are contravariant and covariant 4-momenta, respectively, and $\tau$ is the proper time.
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Answer
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The relation between the proper and lab time is
(2)Therefore,
(3)For motion in magnetic field
(4)Hence,
(5)The energy balance equation becomes
(6)which has the solution
(7)