Verify the Gauss's law for magnetic field of a point dipole of dipole moment `overset(to)( M)` at the origin for the surface which is a sphere of radius R.

Explanation

Gauss.s law of magnetism `oint overset(to) (B).d overset(to)(S) = 0`.
Now, magnetic moment of dipole at origin "O" is along z-axis `overset(to)(M) = M hat(k)`
Let P be a point at distance r from O and OP makes an angle `theta` with z-axis component of `overset(to)(M)` along `OP= M cos theta`
Now, the magnetic field induction at P due to dipole of moment `overset(to)(M) cos theta` is `overset(to)(B)= (mu_0)/( 4 pi) (2M cos theta)/( r^3) hat(r)`

From the diagram, r is the radius of sphere with centre at O lying in yz-plane. Take an elementary area `d overset(to)(S)` of the surface at P, then
`therefore doverset(to)(S) = r (r sin theta) hat(r) = r^(2) sin theta d theta hat(r)`
`therefore oint overset(to)(B). d overset(to)(S) = oint (mu_0)/( 4 pi) (2M cos theta) /( r^3) hat(r) ( r^(2) sin theta d theta) hat(r)`
`= (mu_0)/( 4pi) (M)/( r) int_(0)^(2pi) 2 sin theta cos theta d theta`
`= (mu_0)/( 4 pi) (M)/( r) int_(0)^(2pi) sin 2 theta d theta`
`= (mu_0)/( 4 pi ) (M)/(r ) (- ( cos 2 theta)/( 2))_(0)^(2pi)`
`= (mu_0)/( 4 pi) (M)/( r) [ ( cos 2 theta)/( 2)]_(0)^(2pi)`
`= -(mu_0)/( 4 pi) (M)/( 2r) [ cos 4 pi - cos 0]`
`= (mu_(0) M)/(4 pi (2r) ) [ 1-1]`
`=0`