At a point inside a charging capacitor plate

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Vector calculus and field geometry

Electricity, Magnetism and Electromagnetism, Ch Displacement current, Maxwell's equations and boundary conditions

Problem

At a point PP inside a charging capacitor plate, in vacuum apart from the conductor itself, charge is piling up: the charge density there is increasing at ∂ρ/∂t=2.5×10−4 C m−3 s−1\partial\rho/\partial t = 2.5\times10^{-4}\,\mathrm{C\,m^{-3}\,s^{-1}}. All fields and their derivatives are well behaved at PP. Recall from Chapter 9 that the divergence of any curl vanishes identically. ((a)) Use the continuity equation to find ∇⋅J\nabla\cdot\mathbf{J} at PP. ((b)) Show that the uncorrected local law ∇×B=μ0J\nabla\times\mathbf{B} = \mu_0\mathbf{J} cannot hold at PP, by taking the divergence of both sides. ((c)) Starting from Gauss's law in local form, show that ∇⋅(ε0 ∂E/∂t)=∂ρ/∂t\nabla\cdot\left(\varepsilon_0\,\partial\mathbf{E}/\partial t\right) = \partial\rho/\partial t, and evaluate it at PP. ((d)) Hence find the divergence of J+ε0 ∂E/∂t\mathbf{J} + \varepsilon_0\,\partial\mathbf{E}/\partial t at PP, and say in one sentence why this settles the two-surface trouble of the charging capacitor.

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