Question

By Thomson's time, it was known that excited atoms emit light waves of only certain frequencies. In his model, the frequency of emitted light is the same as the oscillation frequency of the electron or electrons in the atom. What would the radius of a Thomson-model atom have to be for it to produce red light of frequency 4.63×10¹⁴ Hz ? (see Appendix F from the textbook for data about the electron)

Any solution with an explanation for this question would be greatly appreciated.

Thank you

Answer 1

If Q is the total charge, then charge density p = Q/4/3 pi R^3 R is the radius of atom If E is the electric field at distance r from the center then E middot 4 pi r^2 = q_enc/epsilon_0 (Gauss law) E = 4/3 pi r^3 p/4 pi epsilon_0 r^2 = p r/3 epsilon_0 force on electron F = -e E F = -e p r/3 epsilon_0 r implies Acceleration = F/m alpha = -e p/3 epsilon_0 m r on comparing with standard equation of SHM i.e alpha = -w^2 r w^2 = e p/3 epsilon_0 m implies f = 1/2 pi squareroot p e/3 m epsilon_0 = 1/2 pi squareroot Q e/4 pi m epsilon_0 R^3 For H-atom Q = e so f = 1/2 pi squareroot e^2/4 pi m epsilon_0 R^3 Given f = 4.63 times 10^14 Hz R = {(e^2/4 pi m epsilon_0)1/4 pi^2 f^2}^1/3 = {(1.6 times 10^-19)^2/16 pi^3 times 9.1 times 10^-31 epsilon_0 f^2}^1/3