Question

The distance of two points on the axis of a magnet from its centre is $10 \, cm$ and $20 \, cm$ respectively. The ratio of magnetic intensity at these points is $12.5:1$ . The length of the magnet will be

• A. $5cm$
• B. $25cm$
• C. $20cm$
• D. $10 \, cm$

Answer 1

Answer: (D)

The magnetic field on the axis of a bar magnet of length $2l$ , having a pole strength $m$ , at a distance $d$ is
$B=\frac{\mu _{0}}{4 \pi }\frac{m}{\left[d - l\right]^{2}}-\frac{\mu _{0}}{4 \pi }\frac{m}{\left[d + l\right]^{2}}$
$B=\frac{\mu _{0} m}{4 \pi }\frac{\left[d + l\right]^{2} - \left[d - l\right]^{2}}{\left[d^{2} - l^{2}\right]^{2}}=\frac{\mu _{0} mld}{\pi \left[d^{2} - l^{2}\right]^{2}}$
$\Rightarrow \frac{B_{1}}{B_{2}}= \, \frac{d_{1}}{d_{2}} \, \left[\frac{d_{2}^{2} - l^{2}}{d_{1}^{2} - l^{2}}\right]^{2}$
$\Rightarrow \frac{12 .5}{1}=\frac{10}{20} \, \left[\frac{400 - l^{2}}{100 - l^{2}}\right]^{2}$
$\Rightarrow l= 5cm$
Hence, length of magnet $=2l= 10cm$