First slide

The bar magnet

Question

Points A and B are situated along the extended axis of a 2 cm long bar magnet at a distance x and 2 x cm respectively from the pole nearer to the points. The ratio of the magnetic field at A and B will be

Easy

A

4: 1 exactly
B

8: 1 exactly
C

4 : 1 approximately
D

8 : 1 approximately

Solution

$B = \frac{μ_{0}}{4 π} \frac{2 M}{r^{3}}$
where r is the distance of the point from the centre of the magnet. Hence
$\begin{array}{l} B_{A} = \frac{μ_{0}}{4 π} \times \frac{2 M}{(x + 1)^{3}} \\ and B_{B} = \frac{μ_{0}}{4 π} \times \frac{2 M}{(2 x + 1)^{3}} \\ ∴ \frac{B_{A}}{B_{B}} = \frac{(2 x + 1)^{3}}{(x + 1)^{3}} \approx \frac{(2 x)^{3}}{x^{3}} \approx 8 : 1 \end{array}$

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