Question

A long straight cable of lenght $l$ is placed symmetrically along z - axis has radius $a(<<l)$. The cable consists of a thin wire and a coaxial conducting tube. An alternating current $I\left(t\right)=I_{0}sin\left(2\pi \upsilon t\right)$flows down the central thin wire and return along the coaxial conducting tube. The induced electric field at a distance s from the wire inside the cable is $E(s,t)={\mu }_o{I}_o\upsilon cos\left(2\pi \upsilon t\right)ln\left(\frac{s}{a}\right)\hat{k}$. The displacement current density inside the cable is

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

Answer 1

Answer: (A)

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

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

Displacement current density is given by:

${\overrightarrow{J}}_d={ϵ}_0\frac{d\overline{E}}{dt}$

$\Rightarrow {ϵ}_0{\mu }_0{I}_{0}v\frac{\partial }{\partial \left(t\right)}\left(cos2\pi vt\right)\left(\frac{s}{a}\right)\hat{k}$

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

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

$\Rightarrow \left(\frac{v}{c}{)}^{2}2\pi I_{0}sin\right(2\pi vt\left)ln\right(\frac{a}{s})\hat{k}$

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

Option $\left(A\right)$ is correct.