Finding an expression for the energy in terms of the fundamental constants.

Question

I've been stuck on this question in which you have find an expression for $E$ in terms of the fundamental constants $m$, $e$, $ϵ_0$, $h$ and the integer $n$. You are given three equations: $$E=\frac{1}{2}mv^2-\frac{e^2}{4πϵ_0r}$$ $$\frac{e^2}{4πϵ_0r^2}=\frac{mv^2}{r}$$ $$mvr=\frac{nh}{2π}$$ I understand that you have to eliminate $v$ and $r$, but I have no clue where to begin. Do I perhaps rearrange the bottom equation for $r$ and substitute it into the second equation? How do I then eliminate $r$? I know that none of you will do my homework for me as it were, but I would appreciate any guidance or hint. ​

Answer 1

The first two equations combine to tell us that $$E=-\frac 12\times \frac {e^2}{4\pi\epsilon_0 r}$$

So we need to use the third to clear the dependence on $r$.

Note that the second tells us that $$mv^2r=\frac {e^2}{4\pi\epsilon_0 }$$

And the third tells us that $$m^2v^2r^2=\frac {(nh)^2}{(2\pi)^2}$$

Dividing the the third by the second gets you an expression for $r$ alone.

I'll leave you the details.