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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)

Explanation:-

a)The displacement current density (Jd?) is related to the time rate of change of the electric field. According to Maxwell’s equations, 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) The total displacement current III is obtained by integrating the displacement current density Jd? over the cross-sectional area of the cable. For a coaxial cable, the cross-sectional area of interest is the region between the inner wire 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 conduction current depends on the total current supplied to the wire. The displacement current depends on the geometry of the cable (i.e., the area between the inner wire and the outer tube) and the frequency of the alternating current.

The displacement current is

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

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