91影视

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

The frequency is $\mathrm{蝇}$.

The phase angle is zero.

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

$\mathbf{E}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{\mathbf{E}}_{\mathbf{o}}\mathbf{cos}\mathbf{(}\mathbf{k}\mathbf{鈰�}\mathbf{r}\mathbf{-}\mathbf{蝇t}\mathbf{+}\mathbf{未}\mathbf{)}\widehat{\mathbf{n}}$ 鈥︹�� (1)

Here,$\mathbf{未}$ 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,

$\mathbf{B}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\frac{{\mathbf{E}}_{\mathbf{0}}}{\mathbf{c}}\mathbf{cos}\mathbf{(}\mathbf{k}\mathbf{鈰�}\mathbf{r}\mathbf{-}\mathbf{蝇t}\mathbf{+}\mathbf{未}\mathbf{)}\mathbf{(}\widehat{\mathbf{k}}\mathbf{脳}\widehat{\mathbf{n}}\mathbf{)}$ 鈥︹�� (2)

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

$k=鈭�\frac{蝇}{c}\widehat{x}$

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

$\begin{array}{l}k鈰�r=\left(鈭�\frac{蝇}{c}\widehat{x}\right)鈰�(x\widehat{x}+y\widehat{y}+z\widehat{z})\\ k鈰�r=鈭�\frac{蝇}{c}x\end{array}$

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

$\begin{array}{l}\widehat{k}脳\widehat{n}=鈭�\widehat{x}脳\widehat{z}\\ \widehat{k}脳\widehat{n}=\widehat{y}\end{array}$

Calculate the expression for the electric field,

Substitute$鈭�\frac{蝇}{c}x$ for$k.r$ ,$\hat{z}$ for $\hat{n}$and $0$for$未$ into equation (1).

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

Calculate the expression for the magnetic field.

Substitute$鈭�\frac{蝇}{c}x$ for $k鈰�r$,$\widehat{y}$ for$\widehat{k}脳\widehat{n}$ and$0$ for$未$ into equation (2).

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

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.

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

The normal vector is given by,

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

Calculate the expression for $k鈰�r$as,

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

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

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

Calculate the expression for the electric field,

Substitute $\frac{蝇}{\sqrt{3}c}(x+y+z)$for $k鈰�r$, $\frac{\widehat{x}鈭�\widehat{z}}{\sqrt{2}}$for $\widehat{n}$ and $0$for $未$into equation (1).

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

Calculate the expression for the magnetic field.

Substitute$\frac{蝇}{\sqrt{3}c}(x+y+z)$for $k.r$,$\frac{1}{\sqrt{6}}(鈭�\widehat{x}+2\widehat{y}鈭�\widehat{z})$for$\widehat{k}脳\widehat{n}$and$0$for$未$into equation (2).

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

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