Bohr's postulate of quantisation of angular momentum

Question

The Bohr's postulate of quantisation of angular momentum can be written in a formula as $mvr = nh/(2π)$

where $m$ is mass of electron

$v$ is its velocity

$r$ is the radius of that shell

$h$ is Plank's constant

So, I was wondering that if, I want to find where approximately the thing with $n=2$ exists, or in other words, what is the velocity of electron and radius of its circular path in the second shell, I could rewrite it as

$vr = nh/(2mπ)$

or

$vr = constant$

So if I was to just increase the radius by a factor of say $z$ and velocity by a factor of $1/z$ it would give me the exact same value for $vr$ every time. Doesn't that mean I could indefinitely increase the radius and keep reducing the velocity and still stay in second shell forever?

Answer 1

What you're missing is that the electron (or rather, Bohr's version thereof) moves in a curved path because of the centripetal force provided by electrostatic attraction to the proton. You therefore have

$mvr=nh/2\pi$, quantized angular momentum

and

$mv^2/r=\epsilon_0e^2/r$ centripetal force from electrostatic attraction

For each quantum number $n$ we have just one shell radius $r$ and one speed $v$ meeting both requirements.