Question

An electromagnetic wave Here is a particular electromagnetic field in free space: (9.64) Er Eo sin at), E B2 (Eo/c) sin (kx -t cat).

(a) Show that this field can satisfy Maxwell's equations if w and k are related in a certain way.

(b) Suppose w=10¹⁰s^-1 and E₀=1kV/m. What is the wavelength? What is the energy density in joules per cubic meter, averaged over a large region? From this calculate the power density, the energy flow in joules per square meter per second.

(c) Show also that the electric field of associated with a spherically symmetric wave may have the dependence E_i = {Acos[k(r − ct)]}/r , where A is a constant.

Answer 1

Maxwell's equations are :

$\begin{aligned} \nabla \cdot \mathbf{E} &=\frac{\rho}{\varepsilon_{0}} \\ \nabla \cdot \mathbf{B} &=0 \\ \nabla \times \mathbf{E} &=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &=\mu_{0} \mathbf{j}+\frac{1}{c^{2}} \frac{\partial \mathbf{E}}{\partial t} \end{aligned}$

a) Now let's check maxwell's equations:

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

$\left(\frac{\partial E_{z}}{\partial y}-\frac{\partial E_{y}}{\partial z}\right) \mathbf{i}+\left(\frac{\partial E_{x}}{\partial z}-\frac{\partial E_{z}}{\partial x}\right) \mathbf{j}+\left(\frac{\partial E_{y}}{\partial x}-\frac{\partial E_{x}}{\partial y}\right) \mathbf{k} = -\frac{\partial \mathbf{B}}{\partial t}$

$\Rightarrow \frac{\partial E_{y}}{\partial x} = -\frac{\partial B_z}{\partial t}$

$\Rightarrow E_0cos(kx+\omega t)k = -\frac{-E_0}{c} cos(kx+\omega t) \omega$

$\Rightarrow k = \frac{\omega}{c}$

b) when $\omega = 10^{10}s^{-1}\Rightarrow \frac{2\pi}{\lambda} = \frac{10^{10}}{3 \times 10^8}\Rightarrow \lambda = \frac{6\pi}{100} = 0.19m$

$\text{Energy density} = \frac{1}{2} \varepsilon_{0} E^{2} + \frac{1}{2} \frac{B^{2}}{\mu_{0}} = \varepsilon_{0} E^{2} = \varepsilon_{0} E_0^{2}\sin^2 (k x+\omega t)$

  $\text{Space average} = \frac{1}{2} \varepsilon_{0} E_0^2$ .   $\text{Since Average of } \sin^2 \theta\text{ over large space is }\frac{1}{2}$

$\text{Power density} = \frac{\frac{1}{2} \varepsilon_{0} E^{2} + \frac{1}{2} \frac{B^{2}}{\mu_{0}}}{dt} = \frac{d(\varepsilon_{0} E^{2})}{dt} = \frac{d (\varepsilon_{0} E_0^{2}\sin^2 (k x+\omega t))}{dt}=\varepsilon_{0} E_0^{2}\omega\sin(2(k x+\omega t))$

c)

$\mathrm{E_i}=\mathrm{A} \cos [\mathrm{k}(\mathrm{r}-\mathrm{ct})] / \mathrm{r}$

The Wave equation is given by:

$\nabla^{2} u=\frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}}$

$\nabla^{2} E=\frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}}$

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