Question
An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The redius 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:
An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The redius 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:
A.
$$F = 0$$
B.
$$F \propto \left( {\frac{a}{d}} \right)$$
C.
$$F \propto \left( {\frac{{{a^2}}}{{{d^3}}}} \right)$$
D.
$$F \propto {\left( {\frac{a}{d}} \right)^2}$$
Answer :
$$F \propto {\left( {\frac{a}{d}} \right)^2}$$
Solution :
We know that $$F = - \frac{{dv}}{{dr}}$$ where $$r$$ = distance of the loop from straight current carrying wire
$$\eqalign{ & {\text{Here }}U = - \overrightarrow m .\overrightarrow B = - {I_2}\pi {a^2} \times \frac{{{\mu _0}}}{{4\pi }}\frac{{{I_1}}}{r} \times 2 \times \cos 0 \cr & = - \frac{{{\mu _0}{I_1}{I_2}{a^2}}}{{2r}} \cr & \therefore F = - \frac{d}{{d\left( r \right)}}\left[ { - \frac{{{\mu _0}{I_1}{I_2}{a^2}}}{{2r}}} \right] = - \frac{{{\mu _0}{I_1}{I_2}{a^2}}}{{{r^2}}} \cr & {\text{Here }}r = d \cr & \therefore F \propto \frac{{{a^2}}}{{{d^2}}}\left( {{\text{attractive}}} \right) \cr} $$
We know that $$F = - \frac{{dv}}{{dr}}$$ where $$r$$ = distance of the loop from straight current carrying wire
$$\eqalign{ & {\text{Here }}U = - \overrightarrow m .\overrightarrow B = - {I_2}\pi {a^2} \times \frac{{{\mu _0}}}{{4\pi }}\frac{{{I_1}}}{r} \times 2 \times \cos 0 \cr & = - \frac{{{\mu _0}{I_1}{I_2}{a^2}}}{{2r}} \cr & \therefore F = - \frac{d}{{d\left( r \right)}}\left[ { - \frac{{{\mu _0}{I_1}{I_2}{a^2}}}{{2r}}} \right] = - \frac{{{\mu _0}{I_1}{I_2}{a^2}}}{{{r^2}}} \cr & {\text{Here }}r = d \cr & \therefore F \propto \frac{{{a^2}}}{{{d^2}}}\left( {{\text{attractive}}} \right) \cr} $$