Question

The electric field part of an electromagnetic wave in a medium is represented by
$
  {E_x} = 0 \\
  {E_y} = 2.5\dfrac{N}{C}\cos \left[ {(2\pi \times {{10}^6}\dfrac{{rad}}{s})t - (\pi \times {{10}^{ - 2}}\dfrac{{rad}}{m}x)} \right]; \\
  {E_x} = 0. \\
 $
The wave is:
(A) Moving along x direction with frequency $2\pi \times {10^6}Hz$ and the wavelength 200 m.
(B) Moving along y direction with frequency $2\pi \times {10^6}Hz$ and wavelength 200 m.
(C) Moving along x direction with frequency ${10^6}Hz$ and wavelength 100 m
(D) Moving along x direction with frequency ${10^6}Hz$ and wavelength 200 m.

Answer 1

Hint: To answer this question, we need to know the standard equation of an electromagnetic wave. The equation of the electromagnetic wave that is then given in the question should be compared with the standard equation. From the comparison we can obtain the quantities required for the answer. From the sign of the answer that we will obtain, we can find the coordinate to which it is belonging.

Complete step by step answer:
We know that the standard equation of electromagnetic wave is given by:
${E_y} = {E_0}\cos (wt - kx)$
The equation that is given is:
$
    \\
  {E_y} = 2.5\dfrac{N}{C}\cos \left[ {(2\pi \times {{10}^6}\dfrac{{rad}}{s})t - (\pi \times {{10}^{ - 2}}\dfrac{{rad}}{m}x)} \right] \\
    \\
 $
Now we have to compare the given equation with the standard equation.
On the comparison we get that:
$
  w = 2\pi f \\
   \Rightarrow w = 2\pi \times {10^6} \\
   \Rightarrow \lambda f = {10^6}Hz \\
 $
It is also known to us that:
$
  \dfrac{{2\pi }}{\lambda } = k = \pi \times {10^{ - 2}}{m^{ - 1}} \\
   \Rightarrow f = \lambda = 200m \\
 $
Hence we can say that the wave is moving along the positive x direction with a frequency of ${10^6}Hz $and a wavelength of 200 m.

So option D is the correct answer.

Note: In this question we have come across the term electromagnetic wave. For better understanding we should be defining the electromagnetic waves. The waves that are created due to the occurrence of the vibrations between the electric and the magnetic field are known as electromagnetic waves.
The standard equation of the electromagnetic wave is described as a second order partially differentiated equation. The main aim of the equation is to describe the propagation of the electromagnetic waves through a medium or even through a vacuum.
The standard equation of the electromagnetic waves represents a three dimensional form of the wave equation.