2016 Moving Charges and MagnetismForce on a Current-carrying Conductor in a Magnetic Field Medium Question 4 of 7 in Force on a Current-carrying Conductor in a Magnetic Field

A square loop ABCD carrying a current i, is placed near and coplanar with a long straight conductor XY carrying a current I, the net force on the loop will be

A $\frac{\mu_0Ii}{2\pi}$
B $\frac{2\mu_0IiL}{3\pi}$
C $\frac{\mu_0IiL}{2\pi}$
D $\frac{2\mu_0Ii}{3\pi}$

The answer is D.

Solution

Forces on the two parallel sides are unequal because they are at different distances.

Sides BC and DA experience equal and opposite forces; only AB and CD contribute.
Field at AB (distance L/2): $B_{AB} = \frac{\mu_0I}{2\pi(L/2)} = \frac{\mu_0}{2\pi}\frac{2I}{L}$
Field at CD (distance 3L/2): $B_{CD} = \frac{\mu_0I}{2\pi(3L/2)} = \frac{\mu_0}{2\pi}\frac{2I}{3L}$
Net force $F = iLB_{AB} - iLB_{CD} = \frac{2\mu_0IiL}{2\pi L} - \frac{2\mu_0IiL}{6\pi L} = \frac{\mu_0Ii}{\pi} - \frac{\mu_0Ii}{3\pi}$
$F = \frac{2\mu_0Ii}{3\pi}$ (towards the wire)

Concept square-loop-near-wire-force

Force on a Current-carrying Conductor in a Magnetic Field in past papers

7 questions from this chapter have appeared across 6 exam years.