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

The magnetic field of the dipole at O will be cylindrically symmetric. So, we will take only the component of magnetic field along the axial line of dipole as it will be perpendicular to area elements always. The magnetic field induction is given by:
`B = (mu_0)/(4pi) (2m cos theta)/(R^3)`

`dS = (2pi R sin theta) Rd theta = 2pi R^2 sin theta d theta `
Flux through the area ds will be:
`dphi = B .dS = (mu_0)/(4pi) (2m cos theta)/(R^3) xx 2pi R^2 sin theta d theta `
` dphi = (mu_0)/(2pi) m/R 2 cos theta sin theta d theta = (mu_0)/(2pi) m/R sin 2theta d theta `
`oint B.dS = oint dphi = int_(0)^(pi) (mu_0)/(2pi) m/R sin 2 theta d theta `
` = (mu_0 )/(2pi) m/R int_(0)^(pi) sin 2 theta d theta = (mu_0 )/(2pi ) m/R [- cos 2 theta]_(0)^(pi)`
` = (mu_0)/(2pi) m/R [ - cos 2pi - cos theta] = 0 `