Question

A plane electromagnetic wave of wavelength $\lambda$ has an intensity $I$. It is propagating along the positive y – direction. The allowed expressions for electric and magnetic fields are given by:
$\text{A.}\quad \vec E = \sqrt{\dfrac{I}{\epsilon_{\circ}C}}cos \left(\dfrac{2\pi}{\lambda}(y-ct) \right) \hat i ; \vec B = \dfrac 1c E \hat k$
$\text{B.}\quad \vec E = \sqrt{\dfrac{I}{\epsilon_{\circ}C}}cos \left(\dfrac{2\pi}{\lambda}(y-ct) \right) \hat k ; \vec B = -\dfrac 1c E \hat i$
$\text{C.}\quad \vec E = \sqrt{2\dfrac{I}{\epsilon_{\circ}C}}cos \left(\dfrac{2\pi}{\lambda}(y-ct) \right) \hat k ; \vec B = \dfrac 1c E \hat i$
$\text{D.}\quad \vec E = \sqrt{2\dfrac{I}{\epsilon_{\circ}C}}cos \left(\dfrac{2\pi}{\lambda}(y+ct) \right) \hat k ; \vec B = \dfrac 1c E \hat i$

Answer 1

Here, we need to realize that only that equation, which will obey the relation $\vec E = \vec B \times \vec c$ can be the correct option. Further-more, we have to check for the values of the intensity of the electromagnetic wave if needed. The direction of the wave can also be checked from the presence of the term $y-ct$. If the coefficients of the term ‘t’ and ‘y’ are of the same sign, the wave is travelling in –y direction and if the coefficients are of opposite sign, then the wave is travelling in +y direction.

Formula used:
$I = \dfrac 12 \epsilon_{\circ}E_{\circ}^2, \ I = \dfrac{B_{\circ}^2}{2\mu_{\circ}}, \vec E = \vec B \times \vec c$

Complete step by step answer:
Let’s understand the standard wave equation.
$Y=asin\left( \omega t+\phi \right)$ is called the standard wave equation.
Here ‘Y’ can represent the value of displacement of wave particles or the electric or magnetic field at time ‘t’. Coefficient of trigonometric function ‘a’ is called the amplitude of the wave.’$\omega$’ is the angular frequency of the wave, which is the measure of angular displacement. ‘$\phi$’ is the initial phase difference of the wave. It is also called ‘epoch’.
We will now see option one by one:
A. Given the electric field is in x-direction and magnetic field in z-direction. Hence using $\vec E = \vec B \times \vec c$, c must be in –y direction. But in the question it is given that the wave is propagating in positive y direction. Hence this option is incorrect.
B. For this option too, the direction of the wave is towards negative ‘y’ ($\vec E = \vec B \times \vec c$). Hence this option is also incorrect.
For option D. we can directly say that the direction of the wave is towards –y as the coefficients of term ‘y’ and ‘t’ are of the same sign.
For option C, the direction of the wave is towards +y direction.
Therefore, answer is option C.

Note:
By chance in this question, we found the answer by just getting the direction of the wave. But incase two or more option have correct direction of wave, we shall proceed by calculating the intensity of the wave by using any one of the formula $I = \dfrac 12 \epsilon_{\circ}E_{\circ}^2, \ I = \dfrac{B_{\circ}^2}{2\mu_{\circ}}$. Here $E_{\circ} \ and \ B_{\circ}$ are the amplitude of respective waves.