question_answer

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 \[d\left( d>>a \right)\]. If the loop applies a force F on the wire then: [JEE Main Online Paper (Held On 09-Jan-2019 Morning]

A) \[F\propto \left( \frac{{{a}^{2}}}{{{d}^{3}}} \right)\]

B) \[F\propto \left( \frac{a}{d} \right)\]

C) \[F\,\,=\,\,0\]               

D) \[F\propto {{\left( \frac{a}{d} \right)}^{2}}\]

Correct Answer: D

Solution :

work done by magnetic force = fdx \[fdx=-\,dU\] \[f\,\,=\,\,-\frac{dU}{dx}\] dx \[U=-\,\overrightarrow{M}.\text{ }\overrightarrow{B}\] \[U\,\,=\,\,-\frac{{{I}_{2}}\times \pi {{a}^{2}}\,\times \,{{\mu }_{0}}{{I}_{1}}}{2\pi x}\,\,\cos \,0{}^\circ \] \[\frac{dU}{dx}\,\,=\,\,\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}\pi {{a}^{2}}\,\,}{2\pi }\,\,\left( \frac{1}{{{x}^{2}}} \right)\] \[f\,\,=\,\,\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}\pi {{a}^{2}}}{2\pi {{d}^{2}}}\,\,(put\,\,x\,\,=\,\,d)\] \[f\,\,\propto \,\,\frac{{{a}^{2}}}{{{d}^{2}}}\]