Question

The midpoint of two small magnetic dipoles $d$ in end-on positions, are separated by a distance $x,\left(\right.x>>d\left.\right).$ The force between them is proportional to $x^{- n}$ where $n$ is:

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (D)

Magnetic field due to a magnetic dipole at the end-on position is $B=\frac{\mu _{0}}{4 \pi }\frac{2 M}{r^{3}}$ .
Potential energy, $U=-\overset{ \rightarrow }{\mu }\cdot \overset{ \rightarrow }{B}$ .
i.e., $U \propto \frac{1}{r^{3}}$
Force between the dipoles, $F=-\frac{d U}{d r}$ .
Therefore, on differentiating the expression of potential energy, the force between them, $F \propto \frac{1}{r^{4}}$ or $F \propto r^{- 4}$ .