Question

The distance of two points on the axis of a magnet from its center are $10$cm . The ratio of magnetic field at these points is $12.5:1$. The length of the magnet is :

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

Answer 1

The correct option is C $10$cm

magnetic intensity on the axis of a magnet from its centre at a distance r

$B=\frac{{\mu }_{0}2rM}{4\pi (r^2-L^2{)}^2}$

The ratio of magnetic intensity on the axis at a distance $r_1$ and $r_2$ is $B_1$ and $B_2$

$\Rightarrow \frac{B_1}{B_2}=\frac{12.5}{1}$

$\Rightarrow r_1=10cm,r_2=20cm$

$\Rightarrow \frac{B_1}{B_2}=\frac{\frac{{\mu }_{0}2r_{1}M}{4\pi (r_1^2-L^2{)}^2}}{\frac{{\mu }_{0}2r_{2}M}{4\pi (r_2^2-L^2{)}^2}}=\frac{r_1(r_2^2-L^2{)}^2}{r_2(r_1^2-L^2{)}^2}$

$\Rightarrow \frac{12.5}{1}=\frac{10({20}^2-L^2{)}^2}{20({10}^2-L^2{)}^2}$

$\Rightarrow \frac{5}{1}=\frac{{20}^2-L^2}{{10}^2-L^2}$

$\Rightarrow 500-5L^2=400-L^2$

$\Rightarrow 4L^2=100$

$\Rightarrow L=5cm$

Length of the magnet is 2L

$\Rightarrow 2L=2\times 5cm=10cm$

Hence correct option is C.