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I understand Rutherford proposed that electrons orbit around a central nucleus. However, since accelerating charges produce electromagnetic radiation, the orbiting electron should lose energy via E&M and spiral into the nucleus.

But my question is: How do I calculate the time it takes for the electron to spiral into the nucleus, given the rate of energy loss (as a function of acceleration) and the initial electron-nucleus distance?

The power loss equation is: P = (e^2 a^2 ) / (6 pi epsilon c^3)

So far I've thought of calculating the initial energy of the system and integrating the power, and then equating the lost energy to the initial energy; however the final energy is negative inifinity, so this doesn't seem to work.

Algebraic manipulation of circular motion equations didn't get me anywhere either; I'm not really sure how else to proceed now.