#### 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 cableb) integrate the displacement current density across the cross-section of the cable to find the total displacement current Ic) compare the conduction current $I_{0}$ with the displacement current $I_{0}d$

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

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