Question 25 of 57

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

A$3$B$4$C$2$D$1$

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Answer: (B) $4$

$\text{B}=\dfrac{{\mu }_{\text{o}}}{4\pi }\dfrac{2\text{M}}{{\text{x}}^{3}}$ for end-on position

$U=-\vec{M}.\vec{B}$

$=\dfrac{\mu o}{4\pi }\dfrac{2{M}^{2}}{{x}^{3}}$

$F=-\dfrac{dU}{dx}=\dfrac{-3\mu o}{4\pi }\dfrac{2{M}^{2}}{{x}^{4}}$

therefore force between them $\propto {\text{x}}^{-4}$