Question

Use Bohr’s model of hydrogen atom to obtain the relationship between the angular momentum and the magnetic moment of the revolving electron.

Answer 1

In the Bohr model of the hydrogen atom, the electron is modeled as a point negative charge rotating in a circular orbit about a fixed axis about a nucleus.

Let us consider,

r = radius of the orbit

v = velocity 

e = charge of electron

m = mass of electron

Time period (T) = `"circumference"/"velocity"`

= `(2 pi r)/v`

Current (I) = `(-e)/T`

= `(-e)/((2 pi r)/v)`

= `(-e v)/(2 pi r)`

The magnetic moment due to a current loop enclosing an area A is given by:

M_L = IA

= `(-e V)/(2 pi r) xx A`

= `(-e v)/(2 pi r) xx pi r^2`

= `(-e r v)/2`

= `(-m e r v)/(2 m)`

L = Angular momentum = mvr

So, M_L = `(-e)/(2 m) L`