a) Express the total energy of an electron in the Coulomb potential of proton through the electron's angular momentum L and the shortest distance a between the proton and the electron's orbit. Hint: The electron's velocity is perpendicular to it's position vector whenever it is distance a away from the proton.

 

b) For a fixed L, minimize the expression found in (a) with respect to a. Show that the minimum corresponds to the case of a circular orbit. State the minimum value of the total energy for fixed L.

 

My only attempt so far has been:

E=U+KE => E= .5L*v/a - e^2/[4πε₀a]

I have little confidence in that as a solution. Either I've stopped short or I'm going in the wrong direction, I'm not sure. I really only want help with part (a), looking for a point in the right direction... Once I get that far I should be handle (b) somewhat easily, I just wanted to provide extra context for the problem.

You probably want to drop v from your equation. Can you express KE in terms of L, m and a?