Introduction to Electrodynamics · Problem 7.36

Conservation Laws — David J. Griffiths

Physics · displacement-current

Problem

Refer to Prob. 7.16, to which the correct answer was E(s,t)=μ0I0ω2πsin⁡(ωt)ln⁡(as)z^.\mathbf{E}(s,t) = \frac{\mu_0 I_0 \omega}{2\pi} \sin(\omega t) \ln\left(\frac{a}{s}\right) \hat{z}. (a) Find the displacement current density Jd\mathbf{J}_d. (b) Integrate it to get the total displacement current, Id=∫Jd⋅da.I_d = \int \mathbf{J}_d \cdot d\mathbf{a}. (c) Compare IdI_d and II. (What's their ratio?) If the outer cylinder were, say, 2 mm in diameter, how high would the frequency have to be, for IdI_d to be 1\% of II? [This problem is designed to indicate why Faraday never discovered displacement currents, and why it is ordinarily safe to ignore them unless the frequency is extremely high.]

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