Solving Gauss' Law Problem with J.J. Thomson Model of Hydrogen Atom

[discoverer02]

I'm having trouble with the following problem:

An early (incorrect) model of the hydrogen atom, suggested by J.J. Thomson, proposed that a positive cloud of charge +e was uniformly distributed throughout the volume of a sphere of radius R, with the electron an equal-magnitude negative point charge -e at the center. (a) Using Gauss' Law, show that the electron would be in equilibrium at the center and, if displaced from the center a distance r < R, would experience a restoring force of the form F = -Kr, where K is a constant. (b) show that K = ke^2/R^3. (c) Find an expression for the frequency f of simple harmonic oscillations that an electron of mass m would undergo if displaced a short distance (<R) from the center and released. (d) Calculate a numerical value for R that would result in a frequency of electron vibration of 2.47 x 10^15 Hz, the frequency of the light in the most intense line in the hydrogen spectrum.

The second half of Part a) is where I'm having most of my trouble.

a) Using Gauss' law I can show that the electric field at the surface of the sphere is 0. Therefore the electron is in equilibrium. I'm not sure how to show that F restoring = -Kr without first doing part b) (see below)

b) If I move the electron to position r. I can show that the E-field from the positive cloud of charge at that point is (ke^2/R^3)r so the K in part a) = ke^2/R^3

Any comments or suggestions would be greatly appreciated.

Thanks.

 

[discoverer02]

You're right, since r = 0 at the center of the positively charged cloud the E-field of the cloud there is zero so the -e is in equilibrium there.

I'm now kind of confused by part b). At a distance r from the center the pos. cloud's E-field is (ke/R^3)r and not (ke^2/R^3), so I guess I'm not seeing how K becomes ke^2/R^3.

 

[HallsofIvy]

"
a) Using Gauss' law I can show that the electric field at the surface of the sphere is 0. " Was that a typo? Surely you meant to say "at the center of the sphere"?

Move the electron from the center of the sphere. The total force from the part of the sphere beyond the electron is 0 (the field at any point inside a hollow sphere is 0) so you can ignore all of the charge past r. What is the total charge inside r? Now think of that charge as concetrated at the center.

 

[discoverer02]

Yes, I made a mistake. I should only be concerned with what's going on at the center of the sphere.

Is there some way to show that F = -Kr without first finding the E-field of the positively charged cloud at r and then finding F = -eE?
It sounds like the problem is looking for something like this.

Anyway, other than that, I've got the rest of the problem figured out.

Thanks to all for the help.