Question

An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The radius of the loop is $a$ and the distance of its centre from the wire is $d(d>>a)$. If the loop applies a force $F$ on the wire, then:

• A. $F\propto \left(\frac{a^2}{d^3}\right)$
• B. $F=0$
• C. $F\propto \left(\frac{a}{d}\right)$
• D. $F\propto {\left(\frac{a}{d}\right)}^2$

Answer 1

The correct option is D $F\propto {\left(\frac{a}{d}\right)}^2$

The circular loop can be considered as a magnetic dipole.

Therefore, Force on dipole,

$F=m.\frac{dB}{dr}$

$m\to \text{magnetic moment}$

The magnetic field due to the long current carrying wire is given by,

$B=\frac{{\mu }_{0}i}{2\pi r}$

$\Rightarrow \frac{dB}{dr}=-\frac{{\mu }_{0}i}{2\pi r^2}$

$\text{Magnetic moment,}m=I\pi a^2$

$\Rightarrow |F|=I\pi a^2\times \frac{{\mu }_{0}i}{2\pi r^2}$

$\Rightarrow F\propto \frac{a^2}{r^2}$

$\text{Here},r=d$

$\Rightarrow F\propto \frac{a^2}{d^2}$

Note: By Newton's third law, the same force is exerted by the loop on the wire.

Hence, option $\left(D\right)$ is correct.