A block of copper ( , ) fills the region and is driven by a surface field with ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Partial differential equations and boundary-value problems

The Boundary-Value Problem, Ch The quasi-static approximation, eddy currents and the skin effect

Problem

A block of copper (σ=5.96×107 S m−1\sigma = 5.96\times10^7\,\mathrm{S\,m^{-1}}, μr=1\mu_r = 1) fills the region z>0z > 0 and is driven by a surface field B(0,t)=B0cos⁡(ωt) y^\mathbf{B}(0,t) = B_0\cos(\omega t)\,\hat{\mathbf{y}} with ω=2πf\omega = 2\pi f, f=200 Hzf = 200\,\mathrm{Hz}. Everywhere the displacement current is negligible. ((a)) Starting from Faraday's law, the quasi-static Ampere's law and Ohm's law, derive the magnetic diffusion equation ∇2B=μσ ∂B/∂t\nabla^2\mathbf{B} = \mu\sigma\,\partial\mathbf{B}/\partial t inside the copper. ((b)) Solve the diffusion equation for the driven half-space to obtain the skin depth δ=2/μσω\delta = \sqrt{2/\mu\sigma\omega}, and evaluate it numerically at f=200 Hzf = 200\,\mathrm{Hz}. ((c)) A separate thin sheet of the same copper, of thickness d=0.50 mmd = 0.50\,\mathrm{mm}, is placed in a spatially uniform field B0=0.010 TB_0 = 0.010\,\mathrm{T} oscillating at the same 200 Hz200\,\mathrm{Hz}. Check that d≪δd \ll \delta, then derive the eddy-current loss per unit volume in the thin-sheet limit and evaluate it numerically.

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