This is an interesting question.
Think of the Hydrogen atom with the nucleus at the centre.
Think of the orbit as a circular wire around it.
The Bohr quantization condition gives us the angular momentum L
$\displaystyle L\ =\ m\ \omega\ r^2\ =\ \frac{n\ h}{2\ \pi}$ .
Thus it can be seen that omega is proportional to n . But
$\displaystyle \omega\ =\ 2\ \pi \ f$ where f is the frequency.
So f is proportional to n
Thus, if we look at any point on the "orbit wire" , it is as though f electrons pass it each sec ( actually it is the same one !)
Thus a charge of f . e coulombs passes any point on the wire per sec.
e here is the charge of the electron.
This corresponds to a current of $\displaystyle I\ =\ \frac{f\ e}{1}$ .
Thus I is also proportional to n.
Now look up the formula for the magnetic field at the centre of a current carrying loop and find what power of I it corresponds to.
It should be the same for n.
