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.

Explanation

Let us draw the figure for given situation.

We have to prove that `intBdS=0`. This is called Gauss's law in magnetisation.
According to question.
Magnetic moment of dipole at origin O is
`M=Mhatk`
Let P be a point at distance r from O and OP makes an angle `theta` with z-axis . Component of M along OP=M cos `theta`
Now. the magnetic field induction at P due to dipole of moment M `cos theta` is
`B=(mu_(0)2Mcostehta)/(4pi r^(3))hatr`.
From the diagram r is the radius of sphere with centre at O lying in yz-plane . take an elementary area d S of the surface at P Then,
`dS=r( sin theta dtheta)hatr=r^(2)sin theta dthetahatr`
`ointB.dS=oint(mu_(0))/(4 pi)(2M cos theta)/(r^(3))hatr(r^(2)sin thetad theta hatr)`
`=(mu_(0))/(4pi)(M)/(l)int_(0)^(2)2sintheta.costhetad theta`
`=(mu_(0))/(4pi)(M)/(r)int_(0)^(2x)2sintheta,costhetad theta`
`=(mu_(0))/(4pi)(M)/(r)( (-cos2theta)/(2))_(0)^(2x)`
`=-(mu_(0))/(4pi)(M)/(2r)[cos 4pi-cos 0]`
`=(mu_(0))/(4pi)(M)/(2r)(1-1)=0`