Question

In class we discussed a simple model for an atom, in which a cloud of negative charge (the electrons) surrounds a fixed nucleus of positive charge. Develop the details of this model as follows: Suppose an atom has one electron with charge -e and a ucleus of charge e 1.6 x 10-19 C, and the electronic charge is uniformly distributed over a sphere with the Bohr radius a0 4 E0n°/(me2) 5.29% 10-11 m If the electron cloud is displaced a distance r, the magnitude of the attractive force between the cloud center and the nucleus is given by 2 where q is the portion of the electronic charge contained within a sphere of radius r. (This is easy to prove using Gauss's Law.) (a) Using this result, show that the restoring force is linear in and calculate the resulting oscillation frequency wo. The mass of the electron is 9.11 x 10-31 kg. (b) Numerically compare your results to the Lyman-a resonance of hydrogen, which has a wavelength of 122 nm.

Answer 1

Given that Electron charge - e & nucleus charge + e = 1.6 times 10^-19 c a_0 = 4 pi elementof_0 h^2/me^2 = 5.29 times 10^-11 m F_coulomb = 1 eq/4 pi elementof_0 x^2 In this problem we will co - relate restoring force to the coulomb force In the figure two positive charge one is attached to the spring and another charge Q is at fixed position these like charges exart force on each other this force will be F_coulomb = Qq/4 pi elementof_0 r^2 Elementof_0 is permittivity r is distance between two charges due to coulomb force spring compress 'x' distance from g/s natural length let string constant K restoring force by the spring F') = -k x At equilibrium position F = qQ/4 pi elementof_0 r^2 = -k x (2) we can write F = m d^2 x/dt^2 (or) F = ma m rightarrow mass of the charge m d^2 x/dt^2 = -kx d^2 x/d^2 x