Question

A long straight cable of length l is placed symmetrically along z-axis and has radius $a(<<l)$. the cable consists of a thin wire and co-axial conducting tube. An alternating current I(t) = $I_0\sin \left(2\pi \nu t\right)$flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance s from the wire inside the cable is E(s, t) = ${\mu }_0{I}_0\cos \left(2\pi \nu t\right)ln\left(\frac{s}{a}\right)\hat{k}$. The displacement current density inside the cable is

• A. $\frac{2\pi }{{\lambda }^2}{I}_{0}ln\left(\frac{a}{s}\right)\sin \left(2\pi \nu t\right)\hat{k}$
• B. $\frac{1}{{\lambda }^2}{I}_{0}ln\left(\frac{a}{s}\right)\sin \left(2\pi \nu t\right)\hat{k}$
• C. $\frac{\pi }{{\lambda }^2}{I}_{0}ln\left(\frac{a}{s}\right)\sin \left(2\pi \nu t\right)\hat{l}$
• D. Zero

Answer 1

Answer: (A)

$\frac{2\pi }{{\lambda }^2}{I}_{0}ln\left(\frac{a}{s}\right)\sin \left(2\pi \nu t\right)\hat{k}$

Given : The induced electric field at a distance from wire is $E(s,t)={\mu }_0{I}_{0}v\cos \left(2\pi vt\right)ln\left(\frac{s}{a}\right)\hat{k}$

Displacement current density is given by :

${\overrightarrow{J}}_d={ϵ}_0\frac{dE}{dt}$

$\Rightarrow {ϵ}_0{\mu }_0{I}_{0}v\begin{array}{c}\partial \\ \partial \left(t\right)\end{array}\left(\cos 2\pi vt\right)\left(\begin{array}{c}\frac{s}{a}\end{array}\right)\hat{k}$

Substitute for $\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$

$\Rightarrow \frac{1}{c^2}{I}_{0}2\pi v^2(-\sin (2\pi vt\left)\right)ln\left(\begin{array}{c}\frac{s}{a}\end{array}\right)\hat{k}$

$\Rightarrow \left(\begin{array}{c}\frac{v}{c}\end{array}{)}^{2}2\pi I_0\sin \right(2\pi vt\left)\right)ln\left(\begin{array}{c}\frac{a}{s}\end{array}\right)\hat{k}$

$\Rightarrow \frac{2\pi }{{\lambda }^2}{I}_{0}ln\left(\begin{array}{c}\frac{a}{s}\end{array}\right)\sin \left(2\pi vt\right)\hat{k}$

Option (A) is correct.