Question

A short magnetic dipole is placed at the origin with its dipole moment directed along the $+vex$- axis . If magnetic induction at a point $\text{P}(r,0)$ is $B\hat{i}$, the magnetic induction at point $\text{Q}(0,2r)$ will be:

• A. $-\frac{B}{16}\hat{i}$
• B. $-\frac{B}{8}\hat{j}$
• C. $\frac{B}{16}\hat{j}$
• D. $-\frac{B}{16}\hat{j}$

Answer 1

The correct option is A $-\frac{B}{16}\hat{i}$


$\text{P}$ is an axial point at a distance $r$ from the centre of dipole

So, $\overrightarrow{B_P}=K\frac{2M}{r^3}\hat{i}(hereK=\frac{{\mu }_0}{4\pi })$

$\Rightarrow B=\frac{K\left(2M\right)}{r^3}$

Now $\text{Q}(0,2r)$ is an equatorial point, at distance $2r$ from centre of the dipole.

$\overrightarrow{B_Q}=\frac{K.M}{(2r{)}^3}(-\hat{i})$

$\overrightarrow{B_Q}=\frac{K}{8}.\left(\frac{M}{r^3}\right).(-\hat{i})$

$\overrightarrow{B_Q}=\frac{K}{16}.\left(\frac{2M}{r^3}\right).(-\hat{i})$

$\therefore \overrightarrow{B_Q}=\frac{B}{16}(-\hat{i})$

Hence, option $\left(A\right)$ is the correct answer.