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

• A

  $3$
• Correct answer
  B

  $4$
• C

  $2$
• D

  $1$

Solution

 

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

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

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

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

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

Hence, the correct answer is option: 'B'.