Finding equation for rate of change of distance of spiraling electron

[lausco]

Homework Statement

(This problem concerns an electron orbiting a proton. Ultimately we are trying to find the time for a classical electron to spiral into the nucleus of an atom, which will lead us to a discussion of why classical mechanics gives way to quantum mechanics when discussing things on very small scales or very high speeds, etc.
In this part of the problem, we're looking for an equation for the rate of change of the radius of the electron's orbit.)

The total energy of the electron is E= (-1/2)(k/r),
and it can be shown that when an electron accelerates it radiates energy at a rate
dE/dt = -(2ka^2)/(3c^3)
Assume the electron is always moving in a circular orbit but one whose radius r decreases as the electron loses energy. Find an equation for the rate of change dr/dt of the radius.

Homework Equations

F = k/r^2, the force on an electron from a proton. The force points toward the proton.
I found the acceleration by setting F=ma, and came up with a = 230/r^2 m/s^2

The Attempt at a Solution

For finding dr/dt, I think I need to first find an equation for the radius of the electron, but I'm not sure if just rearranging the given equation for E is the right way to go about that.
If that's correct, then I end up with r = (-1/2)(k/E).
Differentiating, I end up with dr/dt = (-1/2)(k/(dE/dt)) = 3c^3/4a^2.

I'm not totally sure if my approach is correct or not, and I think I might be neglecting a time dependence for k somehow. Any help you guys could offer is greatly appreciated!

 

[yands]

Are you sure about your equations ?

 

[haruspex]

Differentiating, I end up with dr/dt = (-1/2)(k/(dE/dt)) = 3c^3/4a^2.

The result looks dimensionally wrong (LT instead of L/T). I get a different expression. Please post your working.

 

[lausco]

Thanks for the quick response!
The first two given equations I'm sure about, and I've made sure I typed them correctly.

As for my differentiation, I rearranged the expression for E and got r = (-1/2)(k/E).
Differentiating both sides, I get dr/dt = (-1/2)(k/ (dE/dt)). [This is the bit I suspect could be wrong; this seems overly simple...]
Then I plug in dE/dt, and get dr/dt = (-1/2)(k/ (-3c^3/2ka^2)), and simplify.

 

[haruspex]

r = (-1/2)(k/E).
Differentiating both sides, I get dr/dt = (-1/2)(k/ (dE/dt)).

Yes, that's wrong. What is (d/dt)(1/E)? It isn't 1/(dE/dt).

Spoiler

1/E = E

-1

; apply the chain rule.

 

[lausco]

I understand that I should use the chain rule, my mistake; I'm not sure I understand what you've got in the spoiler tags. This class is more for philosophy students who want to understand quantum mechanics, so I wasn't expecting it to be so heavy on the calculus right off the bat . . .

If I call u = 1/E, then du = -1/E^2dt, I think. I'm not sure how to factor in the dE/dt that I've been given with that result. Am I still totally off?

 

[lausco]

Anyone out there that can help me understand this a bit better? I'm still not really sure how to substitute my given expression for dE/dt into the dr/dt that I solved for ...

Thanks in advance for any help you guys can offer!