91Ó°ÊÓ

Consider thegiven electrical field is$\text{E}\left(\text{r,t}\right)=−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\mathrm{θ}(\mathrm{Ï\dots }t−r)\widehat{r}$.

Consider thegiven magnetic field is$\text{B}\left(\text{r,t}\right)=0$.

Consider the Maxwell’s equations are given by

$\begin{array}{l}∇â‹\dots \stackrel{→}{E}=\frac{\mathrm{ÒÏ}}{{\mathrm{Î\mu }}_0}\\ ∇â‹\dots \stackrel{→}{B}=0\\ ∇×\stackrel{→}{E}=−\frac{∂\stackrel{→}{B}}{∂t}\\ ∇×\stackrel{→}{B}={\mathrm{μ}}_0\stackrel{→}{J}+{\mathrm{μ}}_0{\mathrm{Î\mu }}_0\frac{∂\stackrel{→}{E}}{∂t}\end{array}$

Write the formula of first Maxwell’s equation for the given functions of $B$and $E$.

$∇â‹\dots \stackrel{\mathbf{→}}{\mathbf{E}}=\frac{\mathrm{ÒÏ}}{{\mathbf{Î\mu }}_0}$…… (1)

Here, $\mathrm{ÒÏ}$ is a charge and ${\mathrm{Î\mu }}_0$ is absolute permittivity.

Write the formula of second Maxwell’s equation for the given functions of $B$and$E$.

$∇â‹\dots \stackrel{\mathbf{→}}{\mathbf{B}}$

Here, $∇$ is derivative and $\stackrel{\mathbf{→}}{\mathbf{B}}$ is magnetic field.

Write the formula of third Maxwell’s equation for the given functions of $B$and$E$.

$∇×\stackrel{\mathbf{→}}{\mathbf{E}}$…… (2)

Here, $∇$ is derivative, $\stackrel{\mathbf{→}}{\mathbf{B}}$ is electrical field.

Write the formula of fourth Maxwell’s equation for the given functions of and$E$.

$∇×\stackrel{\mathbf{→}}{\mathbf{B}}={\mathrm{μ}}_0\stackrel{\mathbf{→}}{\mathbf{J}}+{\mathrm{μ}}_0{\mathbf{Î\mu }}_0\frac{∂\mathbf{Ε}}{∂\mathbf{t}}$…… (3)

Here,${\mathrm{μ}}_0$ is permeability and $\stackrel{\mathbf{→}}{\mathbf{J}}$ is current density, $\mathbf{E}$ is permittivity and is electric field.

Determine the value of first Maxwell’s equation for the given functions of $B$and.$E$

Substitute $\mathrm{θ}(\mathrm{Ï\dots }t−r)∇$ for and $(−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\widehat{r})−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\widehat{r}â‹\dots ∇\left|\mathrm{θ}(\mathrm{Ï\dots }t−r)\right|$ into equation (1).

$\begin{array}{c}∇â‹\dots E=\mathrm{θ}(\mathrm{Ï\dots }t−r)∇â‹\dots (−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\widehat{r})−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\widehat{r}â‹\dots ∇\left|\mathrm{θ}(\mathrm{Ï\dots }t−r)\right|\\ =−\frac{q}{{\mathrm{Î\mu }}_0}{\mathrm{δ}}^3\left(r\right)\mathrm{θ}(\mathrm{Ï\dots }t−r)−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}(\widehat{r}â‹\dots \widehat{r})\frac{∂}{∂r}\mathrm{θ}(\mathrm{Ï\dots }t−r)\end{array}$

From problem 1.45, we are given the next ${\mathrm{δ}}^3\left(r\right)\mathrm{θ}(\mathrm{Ï\dots }t−r)−{\mathrm{δ}}^3\left(r\right)\mathrm{θ}\left(t\right)$and $\frac{∂}{∂r}\mathrm{θ}(\mathrm{Ï\dots }t−r)−−\mathrm{δ}(\mathrm{Ï\dots }t−r)$. So, plug these expressions into equation (1) to get.

$\begin{array}{c}∇â‹\dots \text{E}=−\frac{q}{{\mathrm{Î\mu }}_0}{\mathrm{δ}}^3\left(r\right)\mathrm{θ}(\mathrm{Ï\dots }t−r)−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}(\widehat{r}â‹\dots \widehat{r})\frac{∂}{∂r}\mathrm{θ}(\mathrm{Ï\dots }t−r)\\ \mathrm{ÒÏ}={\mathrm{Î\mu }}_0∇â‹\dots \text{E}−−q{\mathrm{δ}}^3\left(r\right)\mathrm{θ}\left(t\right)+\frac{q}{4\mathrm{Ï€}{r}^2}\mathrm{δ}(\mathrm{Ï\dots }t−r)\end{array}$

Determine the value of second Maxwell’s equation for the given functions of $B$and.$E$

Plug the expression of$B$, so we get it by

Substitute $∇â‹\dots \left(0\right)$ for $∇$and $∇â‹\dots \left(0\right)=0$

Determine the value of third Maxwell’s equation for the given functions of $B$and $E$.

Given that the electric field is independent of $\mathrm{θ}$ and $\mathrm{ϕ}$ the vector product will be zero.

Substitute $\mathrm{ϕ}$ for $\frac{∂\stackrel{→}{B}}{∂t}$into equation (2).

$∇×\stackrel{→}{E}=0$

Determine the value of fourth Maxwell’s equation for the given functions of $B$and$E$.

To obtain the current density, plug in the formula for $B$ and $E$ as follows:

Substitute $(−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\mathrm{θ}(\mathrm{Ï\dots }t−r))\widehat{r}$ for $\stackrel{→}{E}$ into equation (3).

$\begin{array}{l}∇×0={\mathrm{μ}}_0\stackrel{→}{J}+{\mathrm{μ}}_0{\mathrm{Î\mu }}_0\frac{∂}{∂t}(−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\mathrm{θ}(\mathrm{Ï\dots }t−r))\widehat{r}\\ 0=\stackrel{→}{J}+{\mathrm{Î\mu }}_0\frac{∂}{∂t}(−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\mathrm{θ}(\mathrm{Ï\dots }t−r))\widehat{r}\\ \stackrel{→}{J}=−{\mathrm{Î\mu }}_0\frac{∂}{∂t}(−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r^2}\mathrm{θ}(\mathrm{Ï\dots }t−r))\widehat{r}\\ =−{\mathrm{Î\mu }}_0(−\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0})\frac{q}{r^2}\frac{∂}{∂t}\left(\mathrm{θ}(\mathrm{Ï\dots }t−r)\right)\widehat{r}\end{array}$

Solve further as

$\stackrel{→}{J}=\left(\frac{q}{4\mathrm{Ï€}{r}^2}\right)\left(\mathrm{δ}(\mathrm{Ï\dots }t−r)\right)\widehat{r}$.