Question

Two short bar magnets of magnetic moments $M,4M$ are arranged as shown. Find magnetic induction at the point '$P$'. Also, find magnetic induction at the same point if poles of one magnet are reversed.

Answer 1

Magnetic field due to a bar magnet at a point on equatorial line is given by

$B=\frac{{\mu }_{0}M}{4\pi r^3}$

and its direction is opposite to the magnetic moment of dipole.

${\overrightarrow{B}}_{net}=\overrightarrow{B_1}+\overrightarrow{B_2}$

$=\frac{{\mu }_{0}M}{4\pi d^3}\hat{i}+\frac{{\mu }_{0}4M}{4\pi (2d{)}^3}\hat{i}$

$=\frac{{\mu }_{0}M}{4\pi d^3}(1+\frac{4}{8})\hat{i}=\frac{3{\mu }_{0}M}{8\pi d^3}\hat{i}$

if poles of one magnet are reserved,

${\overrightarrow{B}}_{net}=\frac{{\mu }_{0}M}{4\pi d^3}\widehat{-i}+\frac{{\mu }_{0}4M}{4\pi (2d{)}^3}\hat{i}$

$=\frac{{\mu }_{0}M}{4\pi d^3}(-1+\frac{4}{8})\hat{i}$

$\frac{-{\mu }_{0}M}{8\pi d^3}\hat{i}$