Question

The dipole moment of a circular loop carrying a current I, is m and the magnetic field at the centre of the loop is $B_1$. When the dipole moment is doubled by keeping the current constant, the magnetic field at the centre of the loop is $B_2$. The ratio $\frac{B_1}{B_2}$ is?

• A. $\sqrt{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $2$
• D. $\sqrt{3}$

Answer 1

The correct option is A $\sqrt{2}$

Dipole moment $M=IA=I\left(\pi R^2\right)$
$⟹R=\sqrt{\frac{M}{\pi I}}$
Magnetic field at the center $B_c=\frac{{\mu }_{o}I}{2R}$
We get $B_c=\frac{{\mu }_{o}I}{2}.\sqrt{\frac{\pi I}{M}}$
$⟹B\propto \frac{1}{\sqrt{M}}$
$⟹\frac{B_1}{B_2}=\sqrt{\frac{M_2}{M_1}}=\sqrt{\frac{2}{1}}=\sqrt{2}$