Physics help please!?

Question

Show that the magnetic dipole moment (M) of an electron orbiting the proton nucleus of a hydrogen atom is related to the orbital angular momentum (L) of the electron by:

M= (e/(2m))(L)

thank you so much in advance!

Answer 1

In case of electric dipole moment [p = q *2l] the pair of opposite charge constitute a dipole moment.

I magnetic dipole moment, the moving charge - around a nucleus - here also conjugate pair of charges - produces a magnetic dipole moment by virtue of equivalent current produced due to orbital motion in each revolution around.

If there is I current in the circular ring of A - area of cross section then the magnetic dipole moment (derived from torque in mag field) is given by

magnetic moment M = current * Area = I A

The orbiting electron (- e charge, m mass, T time period) is equivalent to the rate of change of charge with time - which is current by definition

I = -e / T

if v is its orbital velocity in circle of radius r then

v = r w = r * 2pi / T OR T = 2 pi r /v

I = - e *v / 2pi r multiply+divide with r because orbital area (A= pi r^2) has to be retained in denominator

I = - e *v r / 2pi r^2

I = - e *v r / 2 A

I *A = - e *v r / 2
so magnetic moment M is
M = - e *v r / 2 --- (1)
we know the orbital angular momentum
L = (m v) cross (r) = m v r ---(2)
divide (1) by (2)

M / L = {- e *v r / 2}{1/m v r}

M / L = {- e / 2}{1/m}

M = - e / 2m { L}
this is it

Answer 2

From definitions of M and L
M=IA=-erv/2 where v=frequency
L=mrv so rv=L/m