Question

Suppose

$\mathbf{E}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{\theta }\mathbf{,}\mathbf{\varphi }\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{A}\frac{\mathbf{sin\theta }}{r}\mathbf{[}\mathbf{cos}\mathbf{(}\mathbf{kr}\mathbf{-}\mathbf{\omega t}\mathbf{)}\mathbf{-}\mathbf{(}\mathbf{1}\mathbf{/}\mathbf{kr}\mathbf{)}\mathbf{sin}\mathbf{(}\mathbf{kr}\mathbf{-}\mathbf{\omega t}\mathbf{)}\mathbf{]}\widehat{\mathbf{\varphi }}$

(This is, incidentally, the simplest possible spherical wave. For notational convenience, let role="math" localid="1658817164296" $\mathbf{(}\mathbf{kr}\mathbf{-}\mathbf{\omega t}\mathbf{)}\mathbf{\equiv }\mathbf{u}$ in your calculations.)

(a) Show that $\mathbf{E}$ obeys all four of Maxwell's equations, in vacuum, and find the associated magnetic field.

(b) Calculate the Poynting vector. Average $\mathbf{S}$ over a full cycle to get the intensity vector $\mathbf{I}$. (Does it point in the expected direction? Does it fall off like${\mathbf{r}}^{\mathbf{-}\mathbf{2}}$, as it should?)

(c) Integrate role="math" localid="1658817283737" $\mathbf{I}\mathbf{\cdot }\mathbf{da}$ over a spherical surface to determine the total power radiated. [Answer: $\mathbf{4}{\mathbf{\pi A}}^2\mathbf{/}\mathbf{3}{\mathbf{\mu }}_0\mathbf{c}$ ]

Short answer

(a)

The value of divergence of electric field of Maxwell’s equation is $\nabla \cdot \text{E}=0$.

The value of curl of electric field is $\nabla \times \text{E}=\frac{1}{\mathrm{rsin\theta }}\frac{\partial }{\partial \mathrm{\theta }}\left({\mathrm{sin\theta E}}_{\mathrm{\varphi }}\right)\widehat{\mathrm{r}}-\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left({\mathrm{rE}}_{\mathrm{\varphi }}\right)\widehat{\mathrm{\theta }}$

The value of magnetic field is $\text{B}=\frac{2\mathrm{Acos\theta }}{{\mathrm{\omega r}}^2}\left(\mathrm{sinu}+\frac{1}{\mathrm{kr}}\mathrm{cosu}\right)\widehat{\mathrm{r}}+\frac{\mathrm{Asin\theta }}{\mathrm{\omega r}}\left(-\mathrm{kcosu}\frac{1}{{\mathrm{kr}}^2}\mathrm{cosu}+\frac{1}{\mathrm{r}}\mathrm{sinu}\right)\widehat{\mathrm{\theta }}$.

The value of Gauss law of magnetism is $\nabla \cdot \text{B}=0$.

The value of Ampere’s law.is $\frac{1}{{\mathrm{c}}^2}\frac{\partial \mathrm{E}}{\partial \mathrm{t}}=\nabla \times \text{B}$.

(b)

The value of Intensity vector is $\mathrm{I}=\frac{{\mathrm{A}}^2{\sin }^2\mathrm{\theta }}{2{\mathrm{\mu }}_0{\mathrm{cr}}^2}\widehat{\mathrm{r}}$ and the pointing vector S over the full cycle is $\text{S}=\frac{{\mathrm{A}}^2\mathrm{sin\theta }}{{\mathrm{\mu }}_0{\mathrm{\omega r}}^2}\left\{\begin{array}{l}\frac{2\mathrm{cos\theta }}{\mathrm{r}}\left[\mathrm{sinucosu}+\frac{1}{\mathrm{kr}}({\cos }^2\mathrm{u}-{\sin }^2\mathrm{u})-\frac{1}{{\mathrm{k}}^2{\mathrm{r}}^2}\mathrm{sinucosu}\right]\widehat{\mathrm{\theta }}\\ -\mathrm{sin\theta }\left(\begin{array}{l}-{\mathrm{kcos}}^2\mathrm{u}+\frac{1}{{\mathrm{kr}}^2}{\cos }^2\mathrm{u}+\frac{1}{\mathrm{r}}\mathrm{sinucosu}+\frac{1}{\mathrm{r}}\mathrm{sinucosu}\\ -\frac{1}{{\mathrm{k}}^2{\mathrm{r}}^3}\mathrm{sinucosu}-\frac{1}{{\mathrm{kr}}^2}{\sin }^2\mathrm{u}\end{array}\right)\widehat{\mathrm{r}}\end{array}\right\}$.

(c) The value of total power radiated is $\frac{4\mathrm{\pi }}{3}\frac{{\mathrm{A}}^2}{{\mathrm{\mu }}_0\mathrm{c}}$.

Step-by-step solution

Write the given data from the question.

Consider this, incidentally, the simplest possible spherical wave. For notational convenience, let $(\text{kr}-\omega \text{t})\equiv \text{u}$in your calculations.

Determine the formula of divergence of electric field of Maxwell’s equation, curl of electric field, magnetic field, Gauss law of magnetism, Ampere’s law, Intensity vector and total power radiated.

Write the formula of electric field is,

$\nabla \cdot \mathbf{E}=\frac{1}{{\mathbf{r}}^2}\frac{\partial }{\partial \mathbf{r}}\left({\mathbf{r}}^2{\mathbf{E}}_r\right)+\frac{1}{\mathbf{rsin}\mathrm{\theta }}\frac{\partial }{\partial \mathrm{\theta }}\left(\mathbf{sin}\mathrm{\theta }{\mathbf{{\rm E}}}_{\mathrm{\theta }}\right)+\frac{1}{\mathbf{rsin}\mathrm{\theta }}\frac{\partial {\mathbf{{\rm E}}}_{\mathrm{\varphi }}}{\partial \mathrm{\varphi }}$ …… (1)

Here, $\mathbf{E}$ is the electric field component of a spherical wave, ${\mathbf{E}}_{\mathbf{r}}$ is electric field component of a spherical wave, $\mathbf{r}$ is radius, ${\mathbf{E}}_{\mathbf{\theta }}$electric field component of a spherical wave and ${\mathbf{E}}_{\mathbf{\varphi }}$is electric field component of a spherical wave.

Write the formula of curl of electric field.

localid="1658817884404" $\nabla \times \mathbf{E}=\frac{-\partial \mathbf{B}}{\partial \mathbf{r}}$ …… (2)

Here, localid="1658817920962" $\mathbf{B}$ is the magnetic field strength and $\mathbf{r}$is radius.

Write the formula of magnetic field.

$\begin{array}{l}\mathbf{B}\mathbf{=}\frac{1}{\mathbf{rsin\theta }}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{\theta }}\left[\frac{{\mathrm{Asin}}^2\theta }{}\left(\int \mathrm{cosu}-\frac{}{}\int \mathrm{sinu}\right)\right]\widehat{\mathbf{r}}\\ \mathbf{-}\frac{1}{r}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{r}}\left[\mathrm{Asin\theta }\left(\int \mathrm{cosu}-\frac{}{}\int \mathrm{sinu}\right)\right]\widehat{\mathbf{\theta }}\end{array}$ …… (3)

Here, $\mathbf{r}$ is radius, $\mathbf{k}$represent the wave number and $\mathbf{A}$ is constant.

Write the formula of Gauss law of magnetism.

localid="1658818098642" $\nabla \cdot \mathbf{B}=\mathbf{0}$ …… (4)

Here, $\mathbf{B}$isthe magnetic field strength.

Write the formula of Ampere’s law.

$\mathbf{\nabla }\mathbf{\times }\mathbf{B}\mathbf{=}\mathbf{\mu \sigma E}\mathbf{+}\frac{1}{{\mathbf{c}}^2}\frac{\mathbf{\partial }\mathbf{E}}{\mathbf{\partial }\mathbf{t}}$ …… (5)

Here, $\mathbf{E}$ is the electric field component of a spherical wave, $\mathbf{\mu }$ is permeability, $\mathbf{c}$denotes the speed of light.

Write the formula of intensity vector.

$\mathbf{I}\mathbf{=}\mathbf{⟨}\mathbf{S}\mathbf{⟩}$ …… (6)

Here, localid="1658818241979" $\mathbf{S}$ is Poynting vector.

Write the formula of total power radiated.

$\mathbf{P}\mathbf{=}\int I\cdot \mathrm{da}$ …… (7)

Here, $\mathbf{I}$ is intensity vector.

(a) Determine the electric field of Maxwell’s equation.

From Gauss’s law,

$\nabla \cdot \text{E}=\frac{{\rho }_f}{{\epsilon }_0}$

Here, ${\rho }_f$is the free charge density.

Determine the divergence of electric field is,

Substitute $0$ for $E_r,E_{\theta }$and $\frac{A\sin \theta }{r}\left[\cos (kr-\omega t)-\frac{1}{kr}\sin (kr-\omega t)\right]$ for $E_{\varphi }$.

$\begin{array}{c}\nabla \cdot \text{E}=\frac{1}{{\mathrm{r}}^2}\frac{\partial }{\partial \mathrm{r}}\left(\begin{array}{l}{\mathrm{r}}^2\\ \left(0\right)\end{array}\right)+\frac{1}{\mathrm{rsin\theta }}\frac{\partial }{\partial \mathrm{\theta }}\left(\begin{array}{l}\mathrm{sin\theta }\\ \left(0\right)\end{array}\right)+\frac{1}{\mathrm{rsin\theta }}\frac{\partial \left(\begin{array}{l}\frac{\mathrm{Asin\theta }}{\mathrm{r}}\\ \left[\cos (\mathrm{kr}-\mathrm{\omega t})-\frac{1}{\mathrm{kr}}\sin (\mathrm{kr}-\mathrm{\omega t})\right]\end{array}\right)}{\partial \mathrm{\varphi }}\\ =\frac{1}{\mathrm{rsin\theta }}\frac{\partial {\mathrm{E}}_{\mathrm{\varphi }}}{\partial \mathrm{\varphi }}\\ =0\end{array}$

As there is no free charge density here, therefore$\nabla \cdot \text{E}=0$.

Hence, Gauss’s law is obeyed.

According to Faraday’s Law.

Determine the curl of electric field is,

$\nabla \times \text{E}=\frac{1}{\mathrm{rsin\theta }}\left[\frac{\partial }{\partial \mathrm{\theta }}\left({\mathrm{sin\theta E}}_{\mathrm{\varphi }}\right)-\frac{\partial {\mathrm{E}}_{\mathrm{\theta }}}{\partial \mathrm{\varphi }}\right]\widehat{\mathrm{r}}+\frac{1}{\mathrm{r}}\left[\frac{1}{\mathrm{sin\theta }}\frac{\partial {\mathrm{E}}_{\mathrm{r}}}{\partial \mathrm{r}}\left({\mathrm{rE}}_{\mathrm{\varphi }}\right)\right]\widehat{\mathrm{\theta }}+\frac{1}{\mathrm{r}}\left[\frac{\partial }{\partial \mathrm{r}}\left({\mathrm{rE}}_{\mathrm{\theta }}\right)-\frac{\partial {\mathrm{E}}_{\mathrm{r}}}{\partial \mathrm{\theta }}\right]\widehat{\mathrm{\varphi }}$

Therefore, the value of curl of electric field is

$\nabla \times \text{E}=\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta E_{\varphi }\right)\widehat{r}-\frac{1}{r}\frac{\partial }{\partial r}\left(r{E}_{\varphi }\right)\widehat{\theta }$

Substitute $\frac{-\partial B}{\partial t}$ for $\nabla \times \text{E}$.

$\begin{array}{l}\frac{-\partial \mathrm{B}}{\partial \mathrm{t}}=\frac{1}{\mathrm{rsin\theta }}\frac{\partial }{\partial \mathrm{\theta }}\left[\frac{{\mathrm{Asin}}^2\mathrm{\theta }}{\mathrm{r}}\left(\mathrm{cosu}-\frac{1}{\mathrm{kr}}\mathrm{sinu}\right)\right]\widehat{\mathrm{r}}\\ -\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left[\mathrm{Asin\theta }\left(\mathrm{cosu}-\frac{1}{\mathrm{kr}}\mathrm{sinu}\right)\right]\widehat{\mathrm{\theta }}\end{array}$ …… (8)

Here, $\mathrm{u}=(\mathrm{kr}-\mathrm{\omega t})$.

Integrate equation (8),

$\begin{array}{l}\text{B}=\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }\left[\frac{A{\sin }^2\theta }{r}\left(\int \cos u-\frac{1}{kr}\int \sin u\right)\right]\widehat{r}\\ -\frac{1}{r}\frac{\partial }{\partial r}\left[A\sin \theta \left(\int \cos u-\frac{1}{kr}\int \sin u\right)\right]\widehat{\theta }\end{array}$ …… (9)

Substitute $-\frac{1}{\omega }\sin u$ for $\int \cos udt$and $\frac{1}{\omega }\cos u$ for $\int \sin udt$into equation (9).

$\text{B}=\frac{2A\cos \theta }{\omega r^2}\left(\sin u+\frac{1}{kr}\cos u\right)\widehat{r}+\frac{A\sin \theta }{\omega r}\left(-k\cos u\frac{1}{k{r}^2}\cos u+\frac{1}{r}\sin u\right)\widehat{\theta }$

Therefore, the value of magnetic field is .

$\text{B}=\frac{2\mathrm{Acos\theta }}{{\mathrm{\omega r}}^2}\left(\mathrm{sinu}+\frac{1}{\mathrm{kr}}\mathrm{cosu}\right)\widehat{\mathrm{r}}+\frac{\mathrm{Asin\theta }}{\mathrm{\omega r}}\left(-\mathrm{kcosu}\frac{1}{{\mathrm{kr}}^2}\mathrm{cosu}+\frac{1}{\mathrm{r}}\mathrm{sinu}\right)\widehat{\mathrm{\theta }}$

Determine the Gauss’s law of magnetism,

Substitute $\frac{2A\cos \theta }{\omega r^2}\left(\sin u+\frac{1}{kr}\cos u\right)\widehat{r}+\frac{A\sin \theta }{\omega r}\left(-k\cos u\frac{1}{k{r}^2}\cos u+\frac{1}{r}\sin u\right)\widehat{\theta }$ for $B$ into equation (4).

$\begin{array}{c}\nabla \cdot \text{B}=\frac{1}{{\mathrm{r}}^2}\frac{\partial }{\partial \mathrm{r}}\left({\mathrm{r}}^2\mathrm{B}\right)+\frac{1}{\mathrm{rsin\theta }}\frac{\partial }{\partial \mathrm{\theta }}\left({\mathrm{sin\theta B}}_{\mathrm{\theta }}\right)\\ =\frac{1}{{\mathrm{r}}^2}\frac{\partial }{\partial \mathrm{r}}\left[\frac{2\mathrm{Acos\theta }}{\mathrm{\omega }}\left(\mathrm{sinu}+\frac{1}{\mathrm{kr}}\mathrm{cosu}\right)\right]\\ +\frac{1}{\mathrm{rsin\theta }}\frac{\partial }{\partial \mathrm{\theta }}\left[\frac{{\mathrm{Asin}}^2\mathrm{\theta }}{\mathrm{\omega r}}\left(-\mathrm{kcosu}+\frac{1}{{\mathrm{kr}}^2}\mathrm{cosu}+\frac{1}{\mathrm{r}}\mathrm{sinu}\right)\right]\\ =\frac{1}{{\mathrm{r}}^2}\frac{2\mathrm{Acos\theta }}{\mathrm{\omega }}\left(\mathrm{kcosu}-\frac{1}{{\mathrm{kr}}^2}\mathrm{cosu}-\frac{1}{\mathrm{r}}\mathrm{sinu}\right)\\ +\frac{1}{\mathrm{rsin\theta }}\frac{2\mathrm{Asin\theta cos\theta }}{\mathrm{\omega r}}\left(-\mathrm{kcosu}+\frac{1}{{\mathrm{kr}}^2}\mathrm{cosu}+\frac{1}{\mathrm{r}}\mathrm{sinu}\right)\end{array}$

Solve further as

$\begin{array}{c}\nabla \cdot \text{B}=\frac{2A\cos \theta }{\omega r^2}\left(k\cos u-\frac{1}{k{r}^2}\cos u-\frac{1}{r}\sin u-k\cos u+\frac{1}{r}\sin u\right)\\ =0\end{array}$

Hence, the Gauss law of magnetism is obeyed.

Determine the Ampere’s law,

As $\mathrm{\sigma }=0$, therefore,

$\begin{array}{c}\nabla \times \text{B}=\frac{1}{c^2}\frac{\partial \text{E}}{\partial t}\\ =\frac{1}{r}\left[\frac{\partial }{\partial r}\left(r{B}_{\theta }\right)-\frac{\partial B}{\partial \theta }\right]\varphi \end{array}$

Substitute $\frac{2\mathrm{Acos\theta }}{{\mathrm{\omega r}}^2}\left(\mathrm{sinu}+\frac{1}{\mathrm{kr}}\mathrm{cosu}\right)\widehat{\mathrm{r}}+\frac{\mathrm{Asin\theta }}{\mathrm{\omega r}}\left(-\mathrm{kcosu}\frac{1}{{\mathrm{kr}}^2}\mathrm{cosu}+\frac{1}{\mathrm{r}}\mathrm{sinu}\right)\widehat{\mathrm{\theta }}$ for $B$.

$\begin{array}{c}\nabla \times \text{B}=\frac{1}{r}\left\{\begin{array}{c}\frac{\partial }{\partial r}\left[\frac{A\sin \theta }{\omega }\left(-k\cos u+\frac{1}{k{r}^2}\cos u+\frac{1}{r}\sin u\right)\right]\\ -\frac{\partial }{\partial \theta }\left[\left(\frac{2A\cos \theta }{\omega r^2}\left(\sin u+\frac{1}{kr}\cos u\right)\right)\right]\widehat{\varphi }\end{array}\right\}\\ =\frac{k}{\omega }\frac{A\sin \theta }{r}\left(k\sin u+\frac{1}{r}\cos u\right)\widehat{\varphi }\\ =\frac{A\sin \theta }{cr}\left(k\sin u+\frac{1}{r}\cos u\right)\widehat{\varphi }\end{array}$

Solve the term $\frac{1}{c^2}\frac{\partial \text{E}}{\partial t}$,

$\begin{array}{c}\frac{1}{c^2}\frac{\partial \text{E}}{\partial t}=\frac{1}{c^2}\frac{A\sin \theta }{r}\left(\omega \sin u+\frac{\omega }{kr}\cos u\right)\widehat{\varphi }\\ =\frac{1}{c^2}\frac{\omega }{k}\frac{A\sin \theta }{r}\left(k\sin u+\frac{1}{r}\cos u\right)\varphi \\ =\frac{1}{c}\frac{A\sin \theta }{r}\left(k\sin u+\frac{1}{r}\cos u\right)\varphi \\ =\nabla \times \text{B}\end{array}$

Hence, Ampere’s law is obeyed.

(b) Determine the Poynting vector and energy per unit time.

Determine the Poynting vector is given by the following equation.

$S=\frac{1}{{\mu }_0}(\text{E}\times \text{B})$ …… (10)

Substitute $\frac{2A\cos \theta }{\omega r^2}\left(\sin u+\frac{1}{kr}\cos u\right)\widehat{r}+\frac{A\sin \theta }{\omega r}\left(-k\cos u\frac{1}{k{r}^2}\cos u+\frac{1}{r}\sin u\right)\widehat{\theta }$ for $B$ and $\frac{A\sin \theta }{r}\left[\cos u-\frac{1}{kr}\sin u\right]\widehat{\varphi }$ for $E$ into above equation (10).

$\begin{array}{c}\text{S}=\frac{1}{{\mu }_0}\left[\frac{A\sin \theta }{r}\left[\cos u-\frac{1}{kr}\sin u\right]\widehat{\varphi }\right]\times \\ \left[\begin{array}{l}\frac{2A\cos \theta }{\omega r^2}\left(\sin u+\frac{1}{kr}\cos u\right)\widehat{r}\\ +\frac{A\sin \theta }{\omega r}\left(-k\cos u+\frac{1}{k{r}^2}\cos u+\frac{1}{r}\sin u\right)\widehat{\theta }\end{array}\right]\\ =\frac{A^2\sin \theta }{{\mu }_0\omega r^2}\left\{\begin{array}{l}\frac{2\cos \theta }{r}\left[\sin u\cos u+\frac{1}{kr}({\cos }^{2}u-{\sin }^{2}u)-\frac{1}{k^2{r}^2}\sin u\cos u\right]\widehat{\theta }\\ -\sin \theta \left(\begin{array}{l}-k{\cos }^{2}u+\frac{1}{k{r}^2}{\cos }^{2}u+\frac{1}{r}\sin u\cos u+\frac{1}{r}\sin u\cos u\\ -\frac{1}{k^2{r}^3}\sin u\cos u-\frac{1}{k{r}^2}{\sin }^{2}u\end{array}\right)\widehat{r}\end{array}\right\}\end{array}$

Average over a full cycle is,

$\begin{array}{c}⟨\sin u\cos u⟩=0\\ ⟨{\sin }^{2}u⟩=⟨{\cos }^{2}u⟩=\frac{1}{2}\end{array}$

Determine theIntensity vector.

Substitute $\frac{A^2\sin }{{\mu }_0\omega r^2}\left(\frac{k}{2}\sin \theta \right)\widehat{r}$ for $⟨\text{S}⟩$into equation (6).

$\begin{array}{c}\text{I}=\frac{A^2\sin }{{\mu }_0\omega r^2}\left(\frac{k}{2}\sin \theta \right)\widehat{r}\\ =\frac{A^2{\sin }^2\theta }{2{\mu }_{0}c{r}^2}\widehat{r}\end{array}$

The intensity fluctuates as $\frac{1}{r^2}$ and faces in the direction of $\widehat{r}$. A spherical wave is predicted to behave in this way.

Therefore, the intensity vector is $\frac{A^2{\sin }^2\theta }{2{\mu }_{0}c{r}^2}\widehat{r}$.

(c) Determine the total power radiated.

Determine the total power radiated is,

Substitute $\frac{A^2{\sin }^2\theta }{2{\mu }_{0}c{r}^2}\widehat{r}$ for $I$.

$\begin{array}{c}P=\frac{A^2}{2{\mu }_{0}c}\int \frac{{\sin }^2\theta }{r^2}{r}^2\sin \theta d\theta d\varphi \\ =\frac{A^2}{2{\mu }_{0}c}2\pi {\int }_0^{\pi }{\sin }^2\theta d\theta \\ =\frac{4\pi }{3}\frac{A^2}{{\mu }_{0}c}\end{array}$

Therefore, the value of total power radiated is $\frac{4\pi }{3}\frac{A^2}{{\mu }_{0}c}$.