An anisotropic conductor has principal resistivities

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Index notation and Cartesian tensors

Problem

An anisotropic conductor has principal resistivities ρ1=2 Ω m\rho_1=2\,\Omega\,\mathrm m, ρ2=3 Ω m\rho_2=3\,\Omega\,\mathrm m and ρ3=5 Ω m\rho_3=5\,\Omega\,\mathrm m, so that in its principal axes Ei=ρijJjE_i=\rho_{ij}J_j with ρij=diag(ρ1,ρ2,ρ3)\rho_{ij}=\mathrm{diag}(\rho_1,\rho_2,\rho_3), the inverse of the conductivity relation. It carries a current density J=(1,2,1) A m−2\mathbf J=(1,2,1)\,\mathrm{A\,m^{-2}}, and the power dissipated per unit volume is p=EiJip=E_iJ_i. Someone writes p=ρiJiJi,p = \rho_iJ_iJ_i, arguing that summing over the repeated index ii gives p=ρ1J12+ρ2J22+ρ3J32p=\rho_1J_1^2+\rho_2J_2^2+\rho_3J_3^2, so the expression is a legitimate use of the summation convention. ((a)) Find the error in this reasoning. ((b)) Write a correct index expression for pp using only properly paired indices, and evaluate it for the numbers given.

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