First slide

The bar magnet

Question

Two short magnetic dipoles ${\vec{N}}_{1} = M \hat{i} and \overset{⇀}{N_{2}} = M \hat{j}$ are placed at the origin. Then magnetic induction at the point $(r, 0)$ is

Moderate

A

$\frac{μ_{0}}{4 π} \frac{\sqrt{3} M}{r^{3}}$
B

$\frac{μ_{0}}{4 π} \frac{\sqrt{2} M}{r^{3}}$
C

$\frac{μ_{0}}{4 π} \frac{\sqrt{5} M}{r^{3}}$
D

$\frac{μ_{0}}{4 π} \frac{N}{r^{3}}$

Solution

${\vec{B}}_{1} = \frac{μ_{0}}{4 π} \frac{2 M}{r^{3}} \hat{i}$
${\vec{B}}_{2} = - \frac{μ_{0}}{4 π} \frac{M}{r^{3}} \hat{j}$
$∴ B = \sqrt{B_{1}^{2} + B_{2}^{2}} = \frac{μ_{0}}{4 π} \frac{\sqrt{5} M}{r^{3}}$

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