Solution:
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We assume that the electron is a small sphere of radius R,
its charge and mass being spread uniformly throughout its volume. Under
this assumption the charge density inside the electron will be

In this model of the electron uniformly charged sphere of
radius R containing total charge e is rotating with angular speed
. We will
calculate the magnetic moment due to the rotating charged sphere.

We use spherical polar coordinate
.
The magnetic moment due to rotating charge contained in the ring as shown in the
diagram will be

where q the total charge contained in the ring shown
is
.
As the magnetic moments due to each element of rotating charges are parallel,
their contributions add. Therefore, the magnetic moment due to charge contained
in the rotating sphere of radius R will be

Note that the integral

Therefore,

Substituting the value of
, we find
for the magnetic moment of a sphere of radius R containing charge e
and rotating with angular speed
the
expression

We recall that the rotational inertia of a sphere of radius
R containing mass total m which is uniformly distributed is
.
Therefore, the angular momentum of the “spinning” sphere of
radius R rotating with angular speed
will be
.
We thus find that

The experimental value of magnetic moment of an electron is

where

The spin of an electron
.
Therefore,
.
Therefore, the result of our model is in disagreement with
experiment. Our model of electron is too mechanistic and is not in the spirit of
quantum mechanics.
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