Question

Three short bar magnets of different magnetic moments are as shown. Find resultant magnetic induction at the point '$P$'.

• A. $\frac{{\mu }_{0}M}{4\pi d^3}\sqrt{3}$
• B. $\frac{{\mu }_{0}M}{4\pi d^3}\sqrt{5}$
• C. $\frac{{\mu }_{0}M}{4\pi d^3}\sqrt{7}$
• D. $\frac{{\mu }_{0}M}{4\pi d^3}\sqrt{6}$

Answer 1

The correct option is B $\frac{{\mu }_{0}M}{4\pi d^3}\sqrt{5}$

In case of short Bar magnet magnetic field on axial line

$B=\frac{{\mu }_0}{4\pi }\ast \frac{2M}{r^3}$

magnetic field on equitorial line

$B=\frac{{\mu }_0}{4\pi }\ast \frac{M}{r^3}$

${\overrightarrow{B}}_{net}=\overrightarrow{B_1}+\overrightarrow{B_2}+\overrightarrow{B_3}$

$\frac{{\mu }_0}{4\pi }\left(\frac{2M}{d^3}\right)\hat{j}+\frac{{\mu }_0}{4\pi }\left(\frac{2M}{d^3}\right)\widehat{-j}+\frac{{\mu }_0}{4\pi }\left(\frac{2M}{d^3}\right)\widehat{-i}$

$=\frac{{\mu }_{0}M}{4\pi d^3}[\hat{j}-2\hat{j}-2\hat{i}]$

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

$\therefore \left|\overrightarrow{B_{next}}\right|=\frac{{\mu }_{0}M}{4\pi d^3}\sqrt{2^2+1^2}$

$=\frac{{\mu }_{0}M}{4\pi d^3}\sqrt{5}$