Question

2. (10 points) The English physicists J.J. Thompson proposed a model in which the atoms consisted of a sphere of positively charged material into which were embedded negatively charged electrons. Consider such an atom consisting of one electron of mass m, having a charge -e which is considered as a point charge, embedded in a sphere having total charge e uniformly distributed over a sphere of radius R a) Find the nuclear electric field inside and outside the sphere. (Ignore the electron.) b) Find the electrostatic potential inside and outside the sphere, ignoring again the electron c) Show that the potential V has an extremurn at r = 0, where r is the radial distance from the centre of the sphere d) If the electron where to evolve in the potential of the sphere, always staying inside the sphere, show that the electronic motion would be harmnic. You can do this by showing that the electric force on the electron satisfies Hooke's law: F, where k is a constant to be determined from the data of your problem

Answer 1

From Gauss law, contourintegral E vector middot dr vector = q_enclosed/epsilon_0 (a) For r < R, E contourintegral dr = (e/R^3 r^3)/epsilon_0 E (4 pi r^2) = er^3/epsilon_0 R^3 rightwardsdoublearrow E = 1/4 pi epsilon_0 e/R^3 r For r > R, E contourintegral ds = e/epsilon_0 E (4 pi r^2) = e/epsilon_0 rightwardsdoublearrow E = 1/4 pi epsilon)0 e/R^2 (b) we know that E = -dv/dr For r < R, integral_R^r dv = -integral_R^r E dr = rightwardsdoublearrow v(r) - v(R) = -e/4 pi epsilon_0 R^3 integral_R^r r dR v(r) = 1/4 pi epsilon_0 e/R = -e/4 pi epsilon_0 R^3 (r^2/2 - R^2/2) v(r) = e/4 pi epsilon_0 [- r^2/2R^2 + 1/2R + 1/R) (v(r) = e/4 pi epsilon_0 integral 3/2R - r^2/2R^3) For r > R, integral_R^r dv = -integral E dr. rightwardsdoublearrow v(r) = e/4 pi epsilon_0 1/r (c) At r = 0 v(r) = e/4 pi epsilon_0 (3/2R - 0) v = 3e/4 pi epsilon_0 R