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  • A long straight cable of length l is placed symmetrically along the z-axis and has radius a. The cable consists of a thin wire and a coaxial conducting tube.

A long straight cable of length l is placed symmetrically along the z-axis and has radius a. The cable consists of a thin wire and a coaxial conducting tube. An alternating current I(t)=I_{0}\; \sin \left ( 2\pi vt \right ) flows down the central thin wire and returns along the coaxial conducting tube. The induced electric field at a distance s from the wire inside the cable is E(s,t)=\mu _{0}I_{0}v\; coz\; \left ( 2\pi vt \right ). In \left ( \frac{s}{a} \right )\hat{k}

a) calculate the displacement current density inside the cable

b) integrate the displacement current density across the cross-section of the cable to find the total displacement current I

c) compare the conduction current I_{0} with the displacement current I_{0}d

Answers (1)

a) The displacement current density is given as

\overrightarrow{J}_{d}=\frac{2\pi I_{0}}{\lambda ^{2}}ln\frac{a}{s}\sin \; 2\pi vt\hat{k}

b) Total displacement current

I^{d}=\int J_{d}2\pi sds

I^{d}=\left ( \frac{\pi a}{\lambda } \right )2I_{0}\; \sin \; 2\pi vt

c) The displacement current is

\frac{I_{0}d}{I_{0}}=\left ( \frac{a\pi }{\lambda } \right )^{2}

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