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Question

# Suppose that you have an electron moving with speed comparable to the speed of light in a circular orbit of radius r in a large region of uniform magnetic field B. Now the uniform magnetic field begins to increase with time: $B=B_{0}+b t,$ where $B_{0}$ and b are positive constants. In one orbit, how much does the energy of the electron increase, assuming that in one orbit the radius doesn't change very much?

Solution

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To get the energy of the electron, we can use the relation

\begin{align*} E=p c \end{align*}

where $p$ is the relativistic momentum and $c$ is the speed of light. We established that $p=e B r$, and it is stated in the problem that $B=B_0+bt$, hence

\begin{align*} E&= e B r c \\ E&= e r c (B_0 + bt) \\ E&= B_0 e r c + b t e r c \\ E&= (B_0 e c + b t e c) r \end{align*}

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