Classical Electrodynamics · Problem 5.12

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · concentric-loops-torque

Problem

Two concentric circular loops of radii aa, bb and currents II, I′I', respectively (b<ab < a), have an angle α\alpha between their planes. Show that the torque on one of the loops is about the line of intersection of the two planes containing the loops and has the magnitude N=μ0πII′b22a∑n=0∞(n+1)(2n+1)[Γ(n+32)Γ(n+2)Γ(32)]2(ba)2nP2n+11(cos⁡α)N = \frac{\mu_0 \pi I I' b^2}{2a}\sum_{n=0}^\infty \frac{(n+1)}{(2n+1)}\left[\frac{\Gamma(n+\frac{3}{2})}{\Gamma(n+2)\Gamma(\frac{3}{2})}\right]^2 \left(\frac{b}{a}\right)^{2n} P^1_{2n+1}(\cos\alpha) where Pl1(cos⁡α)P_l^1(\cos\alpha) is an associated Legendre polynomial. Determine the sense of the torque for α\alpha an acute angle and the currents in the same (opposite) directions.

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