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’s 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 e 2/R3.
(c) Find an expression for the frequency f of simple harmonic oscillations that an electron of mass me would undergo if displaced a small distance (-R) from the center and released.
(d) Calculate a numerical value for R that would result in a frequency of 2.47 x 1015 Hz, the frequency of the light radiated in the most intense line in the hydrogen spectrum.