Q.A long straight cable of length $l$ is placed symmetrically along the $z$-axis and has radius $a\ (\ll l)$. The cable consists of a thin wire and a co-axial conducting tube. An alternating current $I(t) = I_0\sin(2\pi\nu t)$ 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 $E(s,t) = \mu_0 I_0\nu\cos(2\pi\nu t)\ln\left(\dfrac{s}{a}\right)\hat{k}$.

The displacement current density is $\mathbf{J}_d=\varepsilon_0\dfrac{\partial \mathbf{E}}{\partial t}=-\dfrac{2\pi\nu^2 I_0}{c^2}\sin(2\pi\nu t)\ln\!\left(\dfrac{s}{a}\right)\hat{k}$ (using $\varepsilon_0\mu_0=1/c^2$). Integrating the axial $\mathbf{J}_d$ over the circular cross-section, with $\int_0^a s\ln(s/a)\,ds=-a^2/4$, gives $I_d=\dfrac{\pi^2\nu^2 a^2 I_0}{c^2}\sin(2\pi\nu t)$. So the displacement-current amplitude is $I_0^d=\left(\dfrac{\pi\nu a}{c}\right)^2 I_0$, tiny at power frequencies. …

Concept understanding

A time-varying electric field is itself a source of magnetic field through Maxwell's displacement-current term

Here $\mathbf{E}$ is given explicitly, so we only differentiate and integrate.

(i) Displacement current density

Given

Differentiating, $\dfrac{\partial}{\partial t}\cos(2\pi\nu t)=-2\pi\nu\sin(2\pi\nu t)$, so

Using $\varepsilon_0\mu_0=1/c^2$,

(ii) Total displacement current

$\mathbf{J}_d$ points along the axis $\hat{k}$, so its flux through the cross-section (rings of area $2\pi s\,ds$, from $0$ to $a$) is

With $u=s/a$, $\displaystyle\int_0^a s\ln(s/a)\,ds=a^2\int_0^1 u\ln u\,du=a^2\left(-\tfrac14\right)=-\frac{a^2}{4}$ (the boundary term $u^2\ln u\to0$ as $u\to0$). Hence

(iii) Comparison with the conduction current …

Method: Finding Displacement Current from a Given Time-Varying Electric Field

Use this method whenever the electric field inside a region is given explicitly as a function of position and time, and you need the displacement current density, then the total displacement current through a cross-section.

Steps

Step 1: Differentiate the given E-field with respect to time

Displacement current density is defined as $\mathbf{J}_d = \varepsilon_0\dfrac{\partial \mathbf{E}}{\partial t}$ (in vacuum/air-filled space). If $\mathbf{E}(s,t)$ is given as a product of a spatial function and a time-oscillating factor, only the time factor changes under $\partial/\partial t$ — differentiate that factor and keep the spatial dependence and direction unchanged.

Step 2: Integrate the current density over the relevant cross-section to get total current

Since $\mathbf{J}_d$ is a current density, the total displacement current through a surface is $I_d = \int \mathbf{J}_d\cdot d\mathbf{s}$. For a field that only depends on the radial/transverse coordinate $s$ inside a circular cross-section, use ring-shaped area elements $dA = 2\pi s\,ds$ and integrate from the axis out to the boundary.

$I_d = \int_0^{a} J_d(s)\,(2\pi s)\,ds$ …