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## Homework Statement

In classical electromagnetism, an accelerated charge emits electromagnetic radiation. In non-relativistic

limit, where the velocity of the electron is smaller than c, the total power radiated is given by the

Larmor formula, to wit P=2/3*e

^{2}*a

^{2}/c

^{3}, where a denotes the acceleration of the electron. I am expected to use energy conservation, dE/dt=P, to show that in the adiabatic approximation in which the orbit remains nearly circular at all times, the radius of the electron evolves with time as:

r

^{3}(t)=r

^{3}(0)-4r

_{0}

^{2}ct, where r(0) is the initial radius at t=0 and r

_{0}=e

^{2}/(mc

^{2}) is the classical radius of the electron.

2. Homework Equations

2. Homework Equations

## The Attempt at a Solution

The general expression for energy in circular motion is:

E=1/2*m*ω

^{2}r

^{2}-e/r

^{2}

When I differentiate that wrt time and equate the result to P, I obtain the following:

md

^{2}r/dr

^{2}*dr/dt+2e(dr/dt)/r

^{3}=2/3*e*(d

^{2}r/dr

^{2})

^{2}/c

^{3}but I am not sure how to proceed. Any advice?