A homogeneous

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter

Invariance, Ch Index notation and Cartesian tensors

Problem

A homogeneous, isotropic, linear, non-conducting dielectric, whose relative permittivity εr\varepsilon_r is the same at every point, carries no free charge or current. Maxwell's equations for the medium (held from the electromagnetism course) are ∇⋅D=0,∇×E=−∂B∂t,∇⋅B=0,∇×H=∂D∂t,\nabla\cdot\mathbf D=0,\qquad \nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t},\qquad \nabla\cdot\mathbf B=0,\qquad \nabla\times\mathbf H=\frac{\partial\mathbf D}{\partial t},with Di=ε0εijEjD_i=\varepsilon_0\varepsilon_{ij}E_j and B=μ0H\mathbf B=\mu_0\mathbf H. ((a)) Explain, in one or two sentences and citing the isotropic tensor result for rank two, why an isotropic dielectric must have εij=εrδij\varepsilon_{ij}=\varepsilon_r\delta_{ij} rather than a general tensor. ((b)) Using the contracted Levi-Civita identity to evaluate [∇×(∇×E)]i=ϵijkϵklm∂j∂lEm[\nabla\times(\nabla\times\mathbf E)]_i=\epsilon_{ijk}\epsilon_{klm}\partial_j\partial_l E_m, derive the wave equation ∇2E=μ0ε0εr ∂2E∂t2\nabla^2\mathbf E=\mu_0\varepsilon_0\varepsilon_r\,\frac{\partial^2\mathbf E}{\partial t^2} and give the wave speed in the medium in terms of c=1/μ0ε0c=1/\sqrt{\mu_0\varepsilon_0}.

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