Question

Verify the Gauss’s law for magnetic field of a point dipole of dipole moment m at the origin for the surface which is a sphere of radius R.

Answer 1

Let us draw the figure for given situation,

We have by Gauss's law in magnetism `oint_s vecB.vec(ds) = 0`

Magnetic moment (m) of dipole at origin O is `vecm = mhatk`

Let P be a point at a distance r from O and OP makes an angle θ with z-axis. Component of `vecm` along OP is equal to `vecm` cos θ

Now the magnetic field of induction at P  due to dipole of moment `vecm` cos θ is

`B = mu_0/(4pi) (2vecm cos theta)/r^3 hatr`

Here r is the radius of sphere with centre 'O' lying in X-Y-Z plane. Centre at origin.

Take an elementary area ds at P then

`vec(ds) = r(r sin theta)d theta hatr = r^2 sin theta. d theta hatr`  .....`[because d theta = (ds)/r^2  "or"  ds = r^2 d theta]`

`oint_s vecB.vec(ds) = oint mu_0/(4pi) (2vecm cos theta)/r^3 hatr * (r^2 sin theta * d thetahatr)`