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

Answer 1

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

The circular loop will behave as a magnetic dipole with upper face as north pole .
Equivalent diagram of the question from top view



Force on the dipole placed in a non-uniform magnetic field is given by

$F=M\frac{dB}{dx}$

$(M=\text{magnetic moment of the coil})$

$F=i\pi a^2$ $\left(\frac{d}{dx}\left(\frac{{\mu }_{0}i}{2\pi x}\right)\right)$

$[\text{magnetic moment = i}\times area]$

field $B$ due to wire at distance $x$ from wire $=\frac{{\mu }_{0}i}{2\pi x}$

$F=i\pi a^2$ $\left(\frac{{\mu }_{0}i}{2\pi }\right)\left(\frac{-1}{x^2}\right)$

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

(here $x=d=$ distance of loop from wire )

Force by loop on wire will be equal and opposite to the force on loop by wire (Newton's $3$rd law action reaction pair )

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