# SHM of electron inside a uniformly charged sphere

In Thomson’s model, an atom consisted of a sphere of positively charged material in which were embedded negatively charged electrons, like chocolate chips in a ball of cookie dough. Consider such an atom consisting of one electron with mass m and charge -e, which may be regarded as a point charge, and a uniformly charged sphere of charge +e and radius R. (a) Explain why the electron’s equilibrium position is at the center of the nucleus. (b) In Thomson’s model, it was assumed that the positive material provided little or no resistance to the electron’s motion. If the electron is displaced from equilibrium by a distance less than R, show that the resulting motion of the electron will be simple harmonic.

I was going through my text book's exercise and I am confused by the fact that the electron's motion will be simple harmonic. I tried to answer the part one of the question by visualizing a hollow sphere with uniformly distributed positive charge on its surface. Since at the center of the sphere, all the charges distributed on the hollow sphere will have same magnitude of electric field, they will cancel out each other and it will be the equilibrium where no net force exerted on electron.

But, In second part, I can't understand why does the electron oscillates. When it is displaced from the equilibrium, shouldn't the electron be accelerated towards the direction of displacement and collide with the sphere? Here I have tried to draw what I visualized.

Isn't the net force on the electron greater in the direction of displacement? How does another force acts on the electron to bring it back on its equilibrium position? I think I am missing something and not understanding the question properly. Any help or hints will be appreciated!

The shell theorem tells us that the part of the sphere from $$r$$ to $$R$$ exerts no force on the electron. The only force the electron feels is from the part between the centre and $$r$$ i.e. from the sphere of radius $$r$$.