The plates of a parallel-plate capacitor are being charged

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

The plates of a parallel-plate capacitor are being charged. In the gap between the plates the four-current is Jμ=(cρq,ȷ⃗)J^\mu = (c\rho_{\rm q}, \vec\jmath) and the field-strength tensor FμνF_{\mu\nu} is assembled from E⃗\vec E and B⃗\vec B with the convention F0i=−Ei/cF_{0i} = -E_i/c, Fij=ϵijkBkF_{ij} = \epsilon_{ijk}B_k, obeying ∂μFνμ=μ0Jν\partial_\mu F^{\nu\mu} = \mu_0 J^\nu. Take JμJ^\mu to be a known type (1,0)(1,0) tensor. ((a)) Suppose only that the combination FμνJνF_{\mu\nu}J^\nu is known to transform as a covector, a type (0,1)(0,1) tensor, for every four-current JνJ^\nu that can occur in nature. Use the quotient rule to show that FμνF_{\mu\nu} itself must be a type (0,2)(0,2) tensor. ((b)) Apply ∂ν\partial_\nu to ∂μFνμ=μ0Jν\partial_\mu F^{\nu\mu} = \mu_0 J^\nu and use the antisymmetry Fνμ=−FμνF^{\nu\mu} = -F^{\mu\nu} to show that ∂νJν=0\partial_\nu J^\nu = 0, and identify this with the continuity equation ∂ρq/∂t+∇⋅ȷ⃗=0\partial\rho_{\rm q}/\partial t + \nabla\cdot\vec\jmath = 0. ((c)) Set ν=0\nu = 0 in ∂μFνμ=μ0Jν\partial_\mu F^{\nu\mu} = \mu_0 J^\nu to recover Gauss's law ∇⋅E⃗=ρq/ϵ0\nabla\cdot\vec E = \rho_{\rm q}/\epsilon_0 (using μ0ϵ0=1/c2\mu_0\epsilon_0 = 1/c^2). At some instant during the charging the free charge density between the plates is ρq=8.85×10−10 C m−3\rho_{\rm q} = 8.85\times10^{-10}\ \mathrm{C\,m^{-3}}. Find ∇⋅E⃗\nabla\cdot\vec E at that instant.

Related theory

Browse the problem library