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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.