A capacitor has two circular parallel plates of radius separated by a vacuum gap that…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Maxwell's equations and conservation laws · Maxwell's equations and conservation laws

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

Problem

A capacitor has two circular parallel plates of radius

R = 6.0\,\mathrm{cm}

separated by a vacuum gap that is very small compared with

R

, so the electric field in the gap may be taken as uniform, perpendicular to the plates and zero outside the gap. The charge on the positive plate grows as

Q(t) = b\,t^{2}, \qquad b = 0.25\,\mathrm{C\,s^{-2}},

with

t

in seconds. Consider the instant

t = 1.2\,\mathrm{s}

. Take the flat circular surfaces below to be centred on the axis of the plates and to lie inside the gap, parallel to the plates. ((a)) Find the current

I

in the leads at this instant. ((b)) Find the rate of change of the electric field in the gap,

dE/dt

. ((c)) Find the displacement current

\varepsilon_0\,d\Phi_E/dt

through a flat surface of radius

6.0\,\mathrm{cm}

in the gap. ((d)) Find the magnitude of the magnetic field at distance

s = 4.0\,\mathrm{cm}

from the axis inside the gap, and at

s = 12\,\mathrm{cm}

from the axis outside the plates.

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