Exploring the Larmor Formula and Energy Conservation for Electrons

[peripatein]

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*e²*a²/c³, 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:
r³(t)=r³(0)-4r₀²ct, where r(0) is the initial radius at t=0 and r₀=e²/(mc²) is the classical radius of the electron.

2. Homework Equations

The Attempt at a Solution

The general expression for energy in circular motion is:
E=1/2*m*ω²r²-e/r²
When I differentiate that wrt time and equate the result to P, I obtain the following:
md²r/dr²*dr/dt+2e(dr/dt)/r³=2/3*e*(d²r/dr²)²/c³ but I am not sure how to proceed. Any advice?

 

[STEMucator]

I'm not entirely sure, but using the fact $dE = P \space dt$, and $dE = m\omega^2r + \frac{2e}{r^3} \space dr$, I think you should solve:

$$\int P \space dt = \int [m\omega^2 + \frac{2e}{r^3}] \space dr$$

 

[BvU]

The general expression for energy in circular motion is:
E=1/2*m*ω2r2-e/r2

Doesn't look good to me.

Also, in this derivation you are supposed to make good use of a classical F = ma to give you an expression for a.