The instability of Rutherford's atomic model

[GuitarDean]

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.

 

[Meir Achuz]

You need a differential equation for the radius R.
The P you give is dE/dt.
Use the Bohr formula for E in terms of R, and use a=v^2/R.

 

[GuitarDean]

I realized a mistake in my earlier analysis; when the electron enters the nucleus, r is not 0 but rather on the order or 10^-14 - this means when the electron enters the nucleus, the electric potential energy does not diverge to negtive inifity like I first thought - so I integrated P from initial r to the nucleus radius and found the total energy loss.

Then I found the average power loss by dividing the power integral by the interval I integrated over (r final - r initial); for the hydrogen atom I came up with time = 10^-9 which seems about right.

Does my analysis make sense though? I haven't had much experience with in this particular part of physics and I'm not sure if I just came up with a reasonable answer by a wrong route.

 

[Gokul43201]

Your approach is probably good for an approximation, but is not correct for getting the desired value.