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 \((d \gg a)\). If the loop a 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. \(F\propto\left(\frac{a}{d}\right)^2\)

Hint:

By Newton's third law the force on the wire equals the force on the loop, which behaves as a magnetic dipole \(m=I'\pi a^2\) in the non-uniform field of the wire: \(F=m\,\frac{dB}{dx}\).

Answer 1

The Correct Option is D

Field of the wire at distance \(x\): \(B=\frac{\mu_0I}{2\pi x}\), perpendicular to the plane of the paper. The loop lies in the plane, so its magnetic moment \(m=I'\pi a^2\) is also perpendicular to the plane (parallel or antiparallel to \(B\)).
For a small loop, the force is \(F=m\left|\frac{dB}{dx}\right|=I'\pi a^2\cdot\frac{\mu_0I}{2\pi d^2}=\frac{\mu_0II'a^2}{2d^2}\).
By Newton's third law the loop exerts an equal and opposite force on the wire, so \(F\propto\frac{a^2}{d^2}=\left(\frac ad\right)^2\).
Answer: (4) \(F\propto\left(\frac{a}{d}\right)^2\)