Question

A monochromatic beam of light has a frequency $v = \dfrac{3}{{2\pi }} \times {10^{12}}\,{\text{Hz}}$ and is propagating along the direction $\dfrac{{{\hat i} + {\hat j}}}{{\sqrt 2 }}$. It is polarized along the ${\hat k}$ direction. The acceptable form for magnetic field is:
A. $\dfrac{{{E_0}}}{c}\left( {\dfrac{{{\hat i} - {\hat j}}}{{\sqrt 2 }}} \right)\cos \left[ {{{10}^4}\left( {\dfrac{{{\hat i} - {\hat j}}}{{\sqrt 2 }}} \right) \cdot \hat r - \left( {3 \times {{10}^{12}}} \right)t} \right]$
B. $\dfrac{{{E_0}}}{c}\left( {\dfrac{{{\hat i} - {\hat j}}}{{\sqrt 2 }}} \right)\cos \left[ {{{10}^4}\left( {\dfrac{{{\hat i} + {\hat j}}}{{\sqrt 2 }}} \right) \cdot \hat r - \left( {3 \times {{10}^{12}}} \right)t} \right]$
C. $\dfrac{{{E_0}}}{c}{\hat k}\cos \left[ {{{10}^4}\left( {\dfrac{{{\hat i} - {\hat j}}}{{\sqrt 2 }}} \right) \cdot \hat r + \left( {3 \times {{10}^{12}}} \right)t} \right]$
D. $\dfrac{{{E_0}}}{c}\left( {\dfrac{{{\hat i} + {\hat j} + {\hat k}}}{{\sqrt 3 }}} \right)\cos \left[ {{{10}^4}\left( {\dfrac{{{\hat i} + {\hat j}}}{{\sqrt 2 }}} \right) \cdot \hat r + \left( {3 \times {{10}^{12}}} \right)t} \right]$

Answer 1

Using the given direction of propagation of the monochromatic beam of light and direction of polarization of the monochromatic beam of light, determine the direction of the magnetic field in the monochromatic beam of light and substitute these values of unit vectors in the formula for the magnetic field. Thus, check which of the option given represent an acceptable form of magnetic field.

Complete step by step answer:
We have given that the frequency of the monochromatic beam of light is $\dfrac{3}{{2\pi }} \times {10^{12}}\,{\text{Hz}}$.
$v = \dfrac{3}{{2\pi }} \times {10^{12}}\,{\text{Hz}}$
The direction of propagation of the monochromatic beam of light is $\dfrac{{{\hat i} + {\hat j}}}{{\sqrt 2 }}$.
$\hat n = \dfrac{{{\hat i} + {\hat j}}}{{\sqrt 2 }}$
The monochromatic beam of light is polarized along the direction ${\hat k}$.

We have asked to determine an acceptable form of magnetic field.Let ${\hat i}$, ${\hat j}$ and ${\hat k}$ be the unit vectors along X, Y and Z directions respectively. The direction ${\hat B}$ of the magnetic field in the monochromatic beam of light is given by the cross produce of the direction of propagation of the monochromatic beam of light and the direction of polarization of the monochromatic beam of light.
${\hat B} = \hat n \times {\hat k}$

Substitute for $\hat n$ in the above equation.
${\hat B} = \left( {\dfrac{{{\hat i} + {\hat j}}}{{\sqrt 2 }}} \right) \times {\hat k}$
$\Rightarrow {\hat B} = \left( {\dfrac{{{\hat i} \times {\hat k}}}{{\sqrt 2 }}} \right) + \left( {\dfrac{{{\hat j} \times {\hat k}}}{{\sqrt 2 }}} \right)$
$\Rightarrow {\hat B} = \left( {\dfrac{{ - {\hat j}}}{{\sqrt 2 }}} \right) + \left( {\dfrac{{{\hat i}}}{{\sqrt 2 }}} \right)$
$\Rightarrow {\hat B} = \dfrac{{{\hat i} - {\hat j}}}{{\sqrt 2 }}$
Hence, the above equation represents the direction of the magnetic field in the monochromatic beam of light.Hence, the options C and D are incorrect.

The acceptable form for the magnetic field $B = \dfrac{{{E_0}}}{c}{\hat B}\cos \left[ {K{\hat B} - \omega t} \right]$ is given by
$\vec B = \dfrac{{{E_0}}}{c}{\hat B}\cos \left[ {K{\hat B} - \omega t} \right]$
Here, ${E_0}$ is the electric field , $c$ is speed of light, $K$ is wave number, $\omega$ is angular frequency and $t$ is time.

Let us substitute the values of the direction vectors in the above equation. The above equation becomes
$\therefore\vec B = \dfrac{{{E_0}}}{c}\left( {\dfrac{{{\hat i} - {\hat j}}}{{\sqrt 2 }}} \right)\cos \left[ {K\left( {\dfrac{{{\hat i} - {\hat j}}}{{\sqrt 2 }}} \right) - \omega t} \right]$
The above equation for magnetic field resembles only with the option A if we compared the directions of the unit vectors in all the options.We can solve the above equation to obtain the same value as in option A by using the formula for the angular frequency and wave number.

Hence, the correct option is A.

Note: The students should be careful while performing the cross product of the direction of propagation of the monochromatic beam of light and direction of polarization of the beam of light. If one does not perform this cross product correctly then the final answer for the acceptable form of magnetic field will be incorrect.