The magnetic component of a polarized wave of light is

${\mathit{B}}_{\mathbf{x}}\mathbf{=}\left(4.0\times {10}^{-6}\text{T}\right)\mathit{s}\mathit{i}\mathit{n}\left[\left(1.57\times {10}^7{\text{m}}^{\text{- 1}}\right)y+\omega t\right]\mathbf{.}$

(a) Parallel to which axis is the light polarized? What are the (b) frequency and (c) intensity of the light?

The magnetic component of the polarized light is

$B_x=(4.0\times {10}^{-6}T)\sin \left[(1.5\times {10}^7{m}^{-1})y+\omega t\right]$

Here, we need to use the concept of polarization of light. Also, using the equation of frequency related to wave number, we can calculate the frequency. We can use the equation of amplitude ratio to find the magnitude of the electric field. Then, to calculate the intensity of light, we need to use the equation of intensity related to the RMS value of the electric field.

Formula,

The electric field of a propagating wave,$E_m=B_m\times c······\left(1\right)$

The vector identity of the directional vectors,$\hat{k}\times \hat{i}=\hat{j}······\left(2\right)$

The wavelength of the propagating wave due to wave vector,$\lambda =\frac{2\pi }{k}······\left(3\right)$

The frequency of a light wave,$f=\frac{c}{\lambda }······\left(4\right)$

The RMS value of the electric field,$E_{rms}=\frac{E_m}{\sqrt{2}}······\left(5\right)$

The intensity of a wave,$I=\frac{1}{c{\mu }_0}\times E_{rms}^2······\left(6\right)$

According to the given equation, the magnetic field is along the x direction and the wave is along the negative y direction.

It means using equation (2), $\hat{E}\times \hat{i}=-\hat{j}$

This shows that the electric field direction is $\left(-\hat{k}\right)$, that is, along the negative zdirection.

As the polarization direction is along the electric field, we can say that the light is polarized in the direction parallel to z-axis.

Hence, the direction of the polarizing light is parallel to the z-axis.

From the given equation of magnetic field, we have the wave vector as:$k=1.57\times {10}^7{m}^{-1}T$

Now, substituting equation (3) in equation (4), we get the frequency of the wave using the given values as:$\begin{array}{rcl}f& =& \frac{kc}{2\pi }\\ & =& \frac{1.57\times {10}^7\times 3\times {10}^8}{2\pi }\\ & =& 7.5\times {10}^{14}\text{Hz}\end{array}$

Hence, the frequency is $7.5\times {10}^{14}\text{Hz}$.

From the given equation, the magnitude of magnetic field is,$B_m=4.0\times {10}^{-6}\text{T}$

The magnitude of the electric field is given by the RMs value, thus it is given using equation (1) in equation (5) as follows:

$E_m^2=\frac{B_m^2\times c^2}{2}$

Now, substituting the above value in equation (6), we can get the intensity of the wave as follows:

localid="1663004881536" $\begin{array}{rcl}I& =& \frac{1}{c{\mu }_0}\times \frac{B^{2}m\times c^2}{2}\\ & =& \frac{B^{2}m\times c}{2{\mu }_0}\\ & =& \frac{{(4.0\times {10}^{-6})}^2\times (3.0\times {10}^{-8})}{2\times 4\pi \times {10}^{-7}}\\ & =& 1.9kW/m^2\end{array}$

Hence, the value of the intensity is $1.9kW/m^2$.