Classical Electrodynamics · Problem 5.35

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

Physics — E&M · spherical-coil-magnetic-diffusion

Problem

An insulated coil is wound on the surface of a sphere of radius aa in such a way as to produce a uniform magnetic induction B0B_0 in the zz direction inside the sphere and dipole field outside the sphere. The medium inside and outside the sphere has a uniform conductivity σ\sigma and permeability μ\mu. (a) Find the necessary surface current density K\mathbf{K} and show that the vector potential describing the magnetic field has only an azimuthal component, given by Aϕ=B0a22r<r>2sin⁡θA_\phi = \frac{B_0 a^2}{2}\frac{r_<}{r_>^2}\sin\theta where r<(r>)r_< (r_>) is the smaller (larger) of rr and aa. (b) At t=0t = 0 the current in the coil is cut off. [The coil's presence may be ignored from now on.] With the neglect of Maxwell's displacement current, the decay of the magnetic field is described by the diffusion equation, (5.160). Using a Laplace transform and a spherical Bessel function expansion (3.113), show that the vector potential at times t>0t > 0 is given by Aϕ=3B0aπsin⁡θ∫0∞e−νtk2j1(k)j1(kra)dkA_\phi = \frac{3B_0 a}{\pi}\sin\theta\int_0^\infty e^{-\nu tk^2} j_1(k) j_1\left(\frac{kr}{a}\right)dk where ν=1/μσa2\nu = 1/\mu\sigma a^2 is a characteristic decay rate and j1(x)j_1(x) is the spherical Bessel function of order one. Show that the magnetic field at the center of the sphere can be written explicitly in terms of the error function Φ(x)\Phi(x) as Bz(0,t)=B0[Φ(14νt)−1πνtexp⁡(−14νt)]B_z(0, t) = B_0\left[\Phi\left(\frac{1}{\sqrt{4\nu t}}\right) - \frac{1}{\sqrt{\pi\nu t}}\exp\left(-\frac{1}{4\nu t}\right)\right] (c) Show that the total magnetic energy at time t>0t > 0 can be written Wm=6B02a3μ∫0∞e−2νtk2[j1(k)]2 dkW_m = \frac{6B_0^2 a^3}{\mu}\int_0^\infty e^{-2\nu t k^2}[j_1(k)]^2\,dk Show that at long times (νt≫1\nu t \gg 1) the magnetic energy decays asymptotically as Wm→2πB02a324μ(νt)3/2W_m \to \frac{\sqrt{2\pi}B_0^2 a^3}{24\mu(\nu t)^{3/2}} (d) Find a corresponding expression for the asymptotic form of the vector potential (at fixed rr, θ\theta and νt→∞\nu t \to \infty) and show that it decays as (νt)−3/2(\nu t)^{-3/2} as well. Since the energy is quadratic in the field strength, there seems to be a puzzle here. Show by numerical or analytic means that the behavior of the magnetic field at time tt is such that, for distances small compared to R=a(νt)1/2≫aR = a(\nu t)^{1/2} \gg a, the field is uniform with strength (B0/6π1/2)(νt)−3/2(B_0/6\pi^{1/2})(\nu t)^{-3/2}, and for distances large compared to RR, the field is essentially the original dipole field. Explain physically.

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