Aluminium has conductivity

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Condensed Matter, Materials & Soft Matter · Electronic transport in solids

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

Problem

Aluminium has conductivity σ=3.5×107 S m−1\sigma = 3.5\times10^7\,\mathrm{S\,m^{-1}}, mass density ρm=2700 kg m−3\rho_m = 2700\,\mathrm{kg\,m^{-3}} and atomic mass 27 u27\,\mathrm{u} (take 1 u=1/NA g1\,\mathrm{u} = 1/N_A\,\mathrm{g} with NA=6.022×1023 mol−1N_A = 6.022\times10^{23}\,\mathrm{mol^{-1}}). Assume three conduction electrons per aluminium atom. ((a)) Estimate the conduction-electron number density NvN_v. ((b)) The mean time between electron collisions is τc≈meσ/(Nve2)\tau_c \approx m_e\sigma/(N_ve^2), with me=9.109×10−31 kgm_e = 9.109\times10^{-31}\,\mathrm{kg} and e=1.602×10−19 Ce = 1.602\times10^{-19}\,\mathrm{C}. Estimate τc\tau_c and the frequency f≈1/(2πτc)f \approx 1/(2\pi\tau_c) above which a field reverses faster than the electrons collide, so that σ\sigma can no longer be treated as a frequency-independent constant. ((c)) Evaluate Λ=ε0ω/σ\Lambda = \varepsilon_0\omega/\sigma at that frequency and use it to explain why aluminium stops being amenable to the quasi-static diffusion treatment there even though Λ\Lambda is still far below one.

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