Part I. A long cylindrical conductor of radius , conductivity and permeability…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Electromagnetic induction and alternating-current circuits

The Boundary-Value Problem, Ch Magnetic diffusion, shielding and flux exclusion

Problem

Part I. A long cylindrical conductor of radius aa, conductivity σ\sigma and permeability μ\mu carries a purely axial field Bz(s,t)B_z(s,t) (no dependence on zz or ϕ\phi), zero everywhere for t<0t<0, with the surface held at Bz(a,t)=B0B_z(a,t) = B_0 for t≥0t\ge0. (a) Write down the diffusion equation for Bz(s,t)B_z(s,t) in cylindrical coordinates and separate variables to find the equation obeyed by the radial function, showing it is Bessel's equation of order zero. (b) Apply the boundary condition to obtain the eigenvalue condition J0(kna)=0J_0(k_na) = 0, and give the mode times τn=μσa2/(kna)2\tau_n = \mu\sigma a^2/(k_na)^2, using the zeros of J0J_0 quoted earlier (k1a=2.405k_1a = 2.405, k2a=5.520k_2a=5.520, ...). (c) For a copper rod (σ=5.96×107 S m−1\sigma = 5.96\times10^{7}\,\mathrm{S\,m^{-1}}, μ=μ0\mu=\mu_0) of radius a=1.0 cma = 1.0\,\mathrm{cm}, evaluate the slowest mode time τ1\tau_1 and compare it with the slowest mode time of a copper slab of thickness d=1.0 cmd=1.0\,\mathrm{cm}, τ1=μ0σd2/π2\tau_1 = \mu_0\sigma d^2/\pi^2. Part II. A mu-metal spherical shell of mean radius a=3.0 cma = 3.0\,\mathrm{cm}, wall thickness d=0.80 mmd = 0.80\,\mathrm{mm} (inner radius ai=a−d/2a_i = a - d/2, outer radius ao=a+d/2a_o = a + d/2) and relative permeability μr=8.0×104\mu_r = 8.0\times10^{4} is placed in a steady uniform field B0B_0. The exact static shielding factor from matching the magnetic scalar potential across the three regions (outside, wall, cavity) is Ssh=(2μr+1)(μr+2)−2(ai/ao)3(μr−1)29μr.S_{\rm sh} = \frac{(2\mu_r+1)(\mu_r+2) - 2(a_i/a_o)^3(\mu_r-1)^2}{9\mu_r}. (a) Evaluate SshS_{\rm sh} exactly. (b) Evaluate the thin-wall approximation Ssh≈1+2μrd/3aS_{\rm sh}\approx1+2\mu_rd/3a and compare.

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