91Ó°ÊÓ

The electric field for a current carrying straight wire is,$E\left(s,t\right)=\frac{{\mathrm{μ}}_0{∈}_0\mathrm{Ó\neg }}{2\mathrm{Ï€}}\sin \left(\mathrm{Ó\neg }t\right)\mathrm{In}\left(\frac{a}{s}\right)\hat{z}$.

The diameter of the outer cylinder is, $\mathrm{d}=2\mathrm{mm}=2×{10}^{-3}\mathrm{m}$.

The ratio $I_d$of and Iis,$\frac{I_d}{I}=1\%=\frac{1}{100}$.

Consider the electric field inside the conducting material of a capacitor changes, then a certain amount of current is produced. The current produced in the conductor is described as the ‘displacement current’.

The formula for the displacement current $\left(I_d\right)$is given by,

$I_d={∈}_0\frac{d{\mathrm{Φ}}_E}{dt}$

Here, ${\mathrm{Φ}}_E$ represents the electric flux and${∈}_0$ is the permittivity of vacuum.\

The given expression for the electric field having current flowing down the straight wire of radius and length is given by,

$E\left(s,t\right)=\frac{{\mathrm{μ}}_0{I}_0\mathrm{Ó\neg }}{2\mathrm{Ï€}}\sin \left(\mathrm{Ó\neg }t\right)\mathrm{In}\left(\frac{a}{s}\right)\hat{z}$

Then the formula for the displacement current density is given by,

$$\begin{gathered} J_d={∈}_0\frac{dE}{dt} \\ {J}_d={∈}_0\frac{d}{dt}\left(\frac{{\mathrm{μ}}_0{I}_0\mathrm{Ó\neg }}{2\mathrm{Ï€}}\sin \left(\mathrm{Ó\neg }t\right)In\left(\frac{a}{s}\right)\hat{z}\right) \end{gathered}$$

Solving it,

$$\begin{gathered} J_d={∈}_0\frac{{\mathrm{μ}}_0{I}_0\mathrm{Ó\neg }}{2\mathrm{Ï€}}\cos \left(\mathrm{Ó\neg }t\right)\mathrm{In}\left(\frac{a}{s}\right)\hat{z} \\ {J}_d={∈}_0\frac{{\mathrm{μ}}_0{\mathrm{Ó\neg }}^2}{2\mathrm{Ï€}}\left(I_0\cos \left(\mathrm{Ó\neg }t\right)\right)\mathrm{In}\left(\frac{\mathrm{a}}{\mathrm{s}}\right)\hat{\mathrm{z}} \end{gathered}$$

Putting, in expression,

$J_d=\frac{{\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}l}{2\mathrm{Ï€}}\mathrm{In}\left(\frac{\mathrm{a}}{\mathrm{s}}\right)\hat{\mathrm{z}}$

Hence, the displacement current density is $\frac{{\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}l}{2\mathrm{Ï€}}\mathrm{In}\left(\frac{\mathrm{a}}{\mathrm{s}}\right)\hat{\mathrm{z}}$.

Integrating the formula for the displacement current density $\left(J_d\right)$over a small area $\left(d_a\right)$to get the total displacement current $\left(I_d\right)$,

$$\begin{gathered} I_d=∫J_d.da \\ {I}_d=∫\frac{{\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I}{2\mathrm{Ï€}}\mathrm{In}\left(\frac{\mathrm{a}}{\mathrm{s}}\right)\hat{\mathrm{z}}.\mathrm{da} \end{gathered}$$

Putting, $\hat{z}.da=2\mathrm{Ï€}s.ds$

Taking constant terms out of integral and integrating between 0 to a,

$I_d=\frac{{\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I}{2\mathrm{Ï€}}{∫}_0^a\mathrm{In}\left(\frac{\mathrm{a}}{\mathrm{s}}\right)\left(2\mathrm{Ï€}s.ds\right)$

$$\begin{gathered} I_d={\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I{∫}_0^a\left(\mathrm{In}a-\mathrm{In}s\right)\left(s.ds\right) \\ {I}_d={\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I{∫}_0^a\left(s.\mathrm{In}a-s.\mathrm{In}s\right)ds \end{gathered}$$

Solving the integral,

$$\begin{gathered} I_d={\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I{\left[\frac{s^2}{2}\left(\mathrm{In}a\right)-\frac{s^2}{2}\left(\mathrm{In}s\right)+\frac{s^2}{4}\right]}_0^a \\ {I}_d={\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I\left[\frac{a^2}{2}\left(\mathrm{In}a\right)-\frac{a^2}{2}\left(\mathrm{In}a\right)+\frac{a^2}{4}\right] \\ {I}_d=\frac{{\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I{a}^2}{4} \end{gathered}$$

Hence, the total displacement current is $\frac{{\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I{a}^2}{4}$.

According to the question, the ratio of $I_d$ and I is given by,

$$\begin{gathered} \frac{I_d}{I}=\frac{\frac{{\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^{2}I{a}^2}{4}}{I} \\ \frac{I_d}{I}=\frac{{\mathrm{μ}}_0{∈}_0{\mathrm{Ó\neg }}^2{a}^2}{4} \end{gathered}$$

It is known that, ${\mathrm{μ}}_0{∈}_0=\frac{1}{c^2}$, here c is speed of light. So,

$$\begin{gathered} \frac{I_d}{I}=\frac{{\mathrm{Ó\neg }}^2{a}^2}{4c^2} \\ \frac{I_d}{I}={\left(\frac{\mathrm{Ó\neg }a}{2c}\right)}^2 \end{gathered}$$

It is given that $\frac{I_d}{I}=\frac{1}{100}$, so,

$$\begin{gathered} \frac{1}{100}={\left(\frac{\mathrm{Ó\neg }a}{2c}\right)}^2 \\ \frac{1}{10}=\frac{\mathrm{Ó\neg }a}{2c} \\ \frac{\mathrm{Ó\neg }a}{2c}=\frac{1}{10} \\ \mathrm{Ó\neg }=\frac{2c}{10a} \end{gathered}$$

Putting the value of radius $a=\frac{2×{10}^{-3}}{2}={10}^{-3}\mathrm{m}$, and speed of light

$$\begin{gathered} c=3×{10}^8\frac{m}{s}, \\ \mathrm{Ó\neg }=\frac{2×3×{10}^8{\displaystyle \frac{m}{s}}}{10×{10}^{-3}\mathrm{m}} \\ \mathrm{Ó\neg }=0.6×{10}^{11}{\mathrm{s}}^{-1} \\ \mathrm{Ó\neg }=6×{10}^{10}{\mathrm{s}}^{-1} \end{gathered}$$

Frequency in Hertz is given by,

$$\begin{gathered} v=\frac{\mathrm{Ó\neg }}{2\mathrm{Ï€}} \\ v=\frac{6×{10}^{10}{s}^{-1}}{2\mathrm{Ï€}} \\ v=0.955×{10}^{10}\mathrm{Hz} \end{gathered}$$

It can be written as,

$$\begin{gathered} v≈{10}^{10}\mathrm{Hz} \\ \mathrm{v}≈{10}^4\mathrm{MHz} \end{gathered}$$

Hence, the value of the frequency has to be ${10}^4\mathrm{MHz}$.