A capacitor with circular plates of radius and a narrow vacuum gap is charged through…

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 with circular plates of radius

R = 5.0\,\mathrm{cm}

and a narrow vacuum gap is charged through a straight lead carrying a steady current

I = 0.40\,\mathrm{A}

. Draw a circular loop of radius

s = 3.0\,\mathrm{cm}

around the lead, at right angles to it, traversed in the sense whose right-hand normal points along the current. Two open surfaces have this loop as their boundary:

S_1

, the flat disc, pierced once by the lead;

S_2

, a bag-shaped surface that widens past the edge of the near plate and then closes across the whole of the gap between the plates, so that it is pierced by no conduction current at all. The field in the gap is uniform, perpendicular to the plates and zero beyond the plate edge (edge effects ignored). ((a)) Using Ampère's law in the form

\oint\mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\text{enc}}

with

S_1

, find the value predicted for the circulation of

\mathbf{B}

round the loop. ((b)) Repeat with

S_2

and state the value that same law predicts. ((c)) Show from

E = Q/\varepsilon_0\pi R^2

that the displacement current

\varepsilon_0\,d\Phi_E/dt

through

S_2

equals

I

, and give its value. Hence state the circulation that the Ampère-Maxwell law gives for either surface. ((d)) Find the magnitude of

\mathbf{B}

on the loop.

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