in this example we are given that a solid
sphere of radius r, which is uniformly charged
with a charge q is rotating. about its central
axis at angular speed omega. and we are required
to find the magnetic moment of this rotating
sphere. in the solution here we can see if
this is solid sphere which is uniformly charged
with a charge q. and it is rotating at an
angular speed omega with respect to the central
axis it is of radius r. then as being a symmetric
object uniformly dense and uniformly charged.
we can make use of the relation that, for
any object the ratio of magnetic moment to
the angular momentum is a constant which is
q by 2 m. if m is considered to be the mass
of this sphere. then in this situation, we
know well that. angular momentum. of this
sphere is, l which can be written as i omega
we can write the moment of inertia of sphere
is 2 by 5 m r square. multiplied by omega.
so if we substitute the value of l. here we
directly get the value of magnetic moment.
of sphere. which can be given as m is equal
to q by 2-m multiplied by the angular momentum
which is 2 by 5. m r square omega. here this,
2 m gets cancelled out and the magnetic moment
we are getting is. 1 by 5 q r square omega
that will be the answer to this problem. and
the same result we can also evaluate by. considering
elemental shells within this sphere and integrating
the magnetic moment of the shells from zero
to r limits. that, alternative solution i
leave for. you guys to. attempt as an exercise
and verify you are getting the same result
by the method of integration also.
