Question

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$\widehat{\mathbf{n}}$ .

Short answer

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

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

Step-by-step solution

Write the given data from the question

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.

Sketch the electric and magnetic field wave.

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)

Sketch the electric and magnetic field wave for travelling in the negative x direction and polarized in z direction.

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.

Sketch the electric and magnetic field wave for travelling in the direction from the origin to the point (1,1,1), with polarization parallel to the plane xy.

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.