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 = kee2/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.
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 = kee2/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.
Step by step
Solved in 4 steps with 6 images
