The relative permittivity of a dielectric enters

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

The relative permittivity of a dielectric enters Di=ε0εijEjD_i = \varepsilon_0\varepsilon_{ij}E_j. Someone examining a supposedly isotropic glass argues: "The glass is isotropic, so all nine components of εij\varepsilon_{ij} must equal the same number ε\varepsilon. Then for any field E=(E1,E2,E3)\mathbf E = (E_1, E_2, E_3), Di=ε0ε(E1+E2+E3)for every i=1,2,3,D_i = \varepsilon_0\varepsilon(E_1 + E_2 + E_3)\quad\text{for every } i = 1,2,3, so D\mathbf D points along (1,1,1)(1,1,1) no matter which way E\mathbf E points." ((a)) Explain why "isotropic ⇒\Rightarrow all nine components equal" is the wrong reading of isotropy for a rank-two tensor. ((b)) Give the correct isotropic form of εij\varepsilon_{ij} and the resulting relation between D\mathbf D and E\mathbf E, and say what it implies about the direction of D\mathbf D.

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