Classical Electrodynamics · Problem 5.34

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

Physics — E&M · coaxial-loops-mutual-inductance-expansion

Problem

Two identical circular loops of radius aa are initially located a distance RR apart on a common axis perpendicular to their planes. (a) From the expression W12=∫d3x J1⋅A2W_{12} = \int d^3x\,\mathbf{J}_1 \cdot \mathbf{A}_2 and the result for AϕA_\phi from Problem 5.10b, show that the mutual inductance of the loops is M12=μ0πa2∫0∞dk e−kRJ12(ka)M_{12} = \mu_0 \pi a^2 \int_0^\infty dk\, e^{-kR} J_1^2(ka) (b) Show that for R>2aR > 2a, M12M_{12} has the expansion, M12=μ0πa2[(aR)3−3(aR)5+758(aR)7+⋯ ]M_{12} = \frac{\mu_0\pi a}{2}\left[\left(\frac{a}{R}\right)^3 - 3\left(\frac{a}{R}\right)^5 + \frac{75}{8}\left(\frac{a}{R}\right)^7 + \cdots\right] (c) Use the techniques of Section 3.3 for solutions of the Laplace equation to show that the mutual inductance for two coplanar identical circular loops of radius aa whose centers are separated by a distance R>2aR > 2a is M12=−μ0πa4[(aR)3+94(aR)5+37564(aR)7+⋯ ]M_{12} = -\frac{\mu_0\pi a}{4}\left[\left(\frac{a}{R}\right)^3 + \frac{9}{4}\left(\frac{a}{R}\right)^5 + \frac{375}{64}\left(\frac{a}{R}\right)^7 + \cdots\right] (d) Calculate the forces between the loops in the common axis and coplanar configurations. Relate the answers to those of Problem 5.18.

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