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.

 

[Ambitwistor]

Originally posted by discoverer02
Using Gauss' law I can show that the electric field at the surface of the sphere is 0. Therefore the electron is in equilibrium.

That doesn't prove that the electron is in equilibrium, since the electron is not at the surface of the sphere: it is in the center.

Symmetry alone should tell you that the electron has to be in equilibrium at the center, but you can also formally derive that there is no force at the center using Gauss's law ... if you want to determine whether the equilibrium is stable, Gauss's law will tell you the sign of the field a small distance from the center, and thus the direction of the force. Of course, it will also tell you the exact force and the constant K as you note, but perhaps they are speaking qualitatively.

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

You mean the electric force, not the electric field. Your derivation of K is correct.

 

[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.

 

[Ambitwistor]

F = -Kr
E = (ke/R³)r
F = (-e)E

 

[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.

 

[Ambitwistor]

Originally posted by discoverer02
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?

You could do it by directly integrating differential force due to an infinitesimal charge over the volume of the sphere, by superposition. But any solution involving Gauss's law will solve for the field, from which you find the force, and it looks to me like they are asking for a Gauss law solution to (b).