Question

The distance of two points on the axis of a magnet from its centre is $10cm$ and $20cm$ 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. $10cm$

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 $=2l=10cm$