Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude ${\mathbf{E}}_{\mathbf{0}}$, frequency $\mathbf{\omega }$, and phase angle zero that is (a) traveling in the negative $\mathbf{x}$direction and polarized in the direction; (b) traveling in the direction from the origin to the point$\left(1,1,1\right)$ , with polarization parallel to the$\mathbf{xy}$plane. In each case, sketch the wave, and give the explicit Cartesian components of $\widehat{\mathbf{k}}$and$\widehat{\mathbf{n}}$ .

The electric and magnetic fields for a monochromatic plane wave of amplitude is ${\mathrm{E}}_0$.

The frequency is $\mathrm{\omega }$.

The phase angle is zero.

The expression to calculate the electric field is given as follows,

$\mathbf{E}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{\mathbf{E}}_{\mathbf{o}}\mathbf{cos}\mathbf{(}\mathbf{k}\mathbf{\cdot }\mathbf{r}\mathbf{-}\mathbf{\omega t}\mathbf{+}\mathbf{\delta }\mathbf{)}\widehat{\mathbf{n}}$ …… (1)

Here,$\mathbf{\delta }$ is the phase angle.

The expression to calculate the magnetic field is given as follows,

The expression to calculate the magnetic field is given as follows,

$\mathbf{B}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\frac{{\mathbf{E}}_{\mathbf{0}}}{\mathbf{c}}\mathbf{cos}\mathbf{(}\mathbf{k}\mathbf{\cdot }\mathbf{r}\mathbf{-}\mathbf{\omega t}\mathbf{+}\mathbf{\delta }\mathbf{)}\mathbf{(}\widehat{\mathbf{k}}\mathbf{\times }\widehat{\mathbf{n}}\mathbf{)}$ …… (2)

The wave is travelling in the negative$x$ direction. Therefore, the wave vector is expressed as,

$k=-\frac{\omega }{c}\widehat{x}$

Calculate the dot product of$k$ and$r$ as,

$\begin{array}{l}k\cdot r=\left(-\frac{\omega }{c}\widehat{x}\right)\cdot (x\widehat{x}+y\widehat{y}+z\widehat{z})\\ k\cdot r=-\frac{\omega }{c}x\end{array}$

Calculate the expression for$\widehat{k}\times \widehat{n}$ as,

$\begin{array}{l}\widehat{k}\times \widehat{n}=-\widehat{x}\times \widehat{z}\\ \widehat{k}\times \widehat{n}=\widehat{y}\end{array}$

Calculate the expression for the electric field,

Substitute$-\frac{\omega }{c}x$ for$k.r$ ,$\hat{z}$ for $\hat{n}$and $0$for$\delta$ into equation (1).

$\begin{array}{l}\mathrm{E}(\mathrm{x},\mathrm{t})={\mathrm{E}}_0\cos \left(-\frac{\mathrm{\omega }}{\mathrm{c}}\mathrm{x}-\mathrm{\omega t}+0\right)\widehat{\mathrm{z}}\\ \mathrm{E}(\mathrm{x},\mathrm{t})={\mathrm{E}}_0\cos \left(\frac{\mathrm{\omega }}{\mathrm{c}}\mathrm{x}+\mathrm{\omega t}\right)\widehat{\mathrm{z}}\end{array}$

Calculate the expression for the magnetic field.

Substitute$-\frac{\omega }{c}x$ for $k\cdot r$,$\widehat{y}$ for$\widehat{k}\times \widehat{n}$ and$0$ for$\delta$ into equation (2).

$\begin{array}{c}\mathrm{B}(\mathrm{x},\mathrm{t})=\frac{{\mathrm{E}}_0}{\mathrm{c}}\cos \left(-\frac{\mathrm{\omega }}{\mathrm{c}}\mathrm{x}-\mathrm{\omega t}+0\right)\widehat{\mathrm{y}}\\ \mathrm{B}(\mathrm{x},\mathrm{t})=\frac{{\mathrm{E}}_0}{\mathrm{c}}\cos \left(\frac{\mathrm{\omega }}{\mathrm{c}}\mathrm{x}+\mathrm{\omega t}\right)\widehat{\mathrm{y}}\end{array}$

Sketch of wave for the electric and magnetic field is shown below.

Since the wave is travelling in the direction from $(1,1,1)$with polarization to $xy$plane, therefore the wave vector is given by.

$k=\frac{\omega }{c}\left(\frac{\widehat{x}+\widehat{y}+\widehat{z}}{\sqrt{3}}\right)$

The normal vector is given by,

$\widehat{n}=\frac{\widehat{x}-\widehat{z}}{\sqrt{2}}$

Calculate the expression for $k\cdot r$as,

$\begin{array}{l}k\cdot r=\frac{\omega }{c}\left(\frac{\widehat{x}+\widehat{y}+\widehat{z}}{\sqrt{3}}\right)\cdot (x\widehat{x}+y\widehat{y}+z\widehat{z})\\ k\cdot r=\frac{\omega }{\sqrt{3}c}(\widehat{x}+\widehat{y}+\widehat{z})(x\widehat{x}+y\widehat{y}+z\widehat{z})\\ k\cdot r=\frac{\omega }{\sqrt{3}c}(x+y+z)\end{array}$

Calculate the expression for $\widehat{k}\times \widehat{n}$as,

$\begin{array}{l}\widehat{k}\times \widehat{n}=\frac{1}{\sqrt{6}}\left|\begin{array}{ccc}\widehat{x}& \widehat{y}& \widehat{z}\\ 1& 1& 1\\ 1& 0& -1\end{array}\right|\\ \widehat{k}\times \widehat{n}=\frac{1}{\sqrt{6}}(-\widehat{x}+2\widehat{y}-\widehat{z})\end{array}$

Calculate the expression for the electric field,

Substitute $\frac{\omega }{\sqrt{3}c}(x+y+z)$for $k\cdot r$, $\frac{\widehat{x}-\widehat{z}}{\sqrt{2}}$for $\widehat{n}$ and $0$for $\delta$into equation (1).

$\begin{array}{c}\mathrm{E}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})={\mathrm{E}}_0\cos \left(\frac{\mathrm{\omega }}{\sqrt{3}\mathrm{c}}(\mathrm{x}+\mathrm{y}+\mathrm{z})-\mathrm{\omega t}+0\right)\frac{\widehat{\mathrm{x}}-\widehat{\mathrm{z}}}{\sqrt{2}}\\ \mathrm{E}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})={\mathrm{E}}_0\cos \left(\frac{\mathrm{\omega }}{\sqrt{3}\mathrm{c}}(\mathrm{x}+\mathrm{y}+\mathrm{z})-\mathrm{\omega t}\right)\frac{\widehat{\mathrm{x}}-\widehat{\mathrm{z}}}{\sqrt{2}}\end{array}$

Calculate the expression for the magnetic field.

Substitute$\frac{\omega }{\sqrt{3}c}(x+y+z)$for $k.r$,$\frac{1}{\sqrt{6}}(-\widehat{x}+2\widehat{y}-\widehat{z})$for$\widehat{k}\times \widehat{n}$and$0$for$\delta$into equation (2).

$\begin{array}{l}B(x,y,z,t)=\frac{E_0}{c}\cos \left(\frac{\omega }{\sqrt{3}c}(x+y+z)-\omega t+0\right)\frac{1}{\sqrt{6}}(-\widehat{x}+2\widehat{y}-\widehat{z})\\ B(x,y,z,t)=\frac{E_0}{c}\cos \left(\frac{\omega }{\sqrt{3}c}(x+y+z)-\omega t\right)\frac{1}{\sqrt{6}}(-\widehat{x}+2\widehat{y}-\widehat{z})\end{array}$

Sketch of wave for the electric and magnetic field is shown below.