Question

Two short magnets placed along the same axis with their like poles facing each other repel each other with a force which varies inversely as

• A. Square of the distance
• B. Cube of the distance
• C. Distance
• D. Fourth power of the distance

Answer 1

Answer: (D)

Both the magnets are placed in the field of one another, hence potential energy of dipole $(2)$ is
$U_2=-M_2{B}_1\cos 0^{\circ }=-M_2{B}_1$
$=M_2\times \frac{{\mu }_0}{4\pi }\cdot \frac{2M_1}{r^3}$
By using $F=-\frac{dU}{dr}$, force on magnet $(2)$ is
$F_2=-\frac{d{U}_2}{dr}=-\frac{d}{dr}\left(\frac{{\mu }_0}{4\pi }\cdot \frac{2M_1{M}_2}{r^3}\right)$
$=-\frac{{\mu }_0}{4\pi }\cdot 6\frac{M_1{M}_2}{r^4}$
It can be proved $\left|F_1\right|=\left|F_2\right|=F$
$=\frac{{\mu }_0}{4\pi }\cdot \frac{6M_1{M}_2}{r^4}$
$\Rightarrow F\propto \frac{1}{r^4}$