Ncert Exercises & Solutions

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

Hint: Use Gauss' law.

Step 1: Let us draw the figure for the given situation.

We have to prove that

\oint B \cdot dS = 0

. This is called Gauss's law in magnetization.

According to the question, the magnetic moment of the dipole at origin O is;

M = M \hat{k}

Let P be a point at distance r from O and OP makes an angle

θ

with the z-axis.

Component of M along OP= M cos

θ

Now, the magnetic field induction at P due to the dipole of moment Mcos

θ

is:

B = \frac{μ_{0}}{4 π} \frac{2 M \cos θ}{r^{3}} \hat{r}

From the diagram, r is the radius of the sphere with center at O lying in the yz-plane. Take an elementary area dS of the surface at P. Then,

\begin{array}{l} d S = r (r \sin θ d θ) \hat{r} = r^{2} \sin θ d θ \hat{r} \\ \oint B . d S = \oint \frac{μ_{0}}{4 π} \frac{2 M \cos θ}{r^{3}} \hat{r} (r^{2} \sin θ d θ \hat{r}) \\ = \frac{μ_{0}}{4 π} \frac{M}{r} \int_{0}^{2 π} 2 \sin θ . \cos θ d θ \\ = \frac{μ_{0}}{4 π} \frac{M}{r} \int_{0}^{2 π} \sin 2 θ d θ \\ = \frac{μ_{0}}{4 π} \frac{M}{r} (\frac{- \cos 2 θ}{2})_{0}^{2 π} \\ = - \frac{μ_{0}}{4 π} \frac{M}{2 r} [\cos 4 π - \cos 0] \\ = \frac{μ_{0}}{4 π} \frac{M}{2 r} [1 - 1] = 0 \end{array}