Question

The oscillating magnetic field in a plane electromagnetic wave is given by: ${B_y} = \left( {8 \times {{10}^{ - 6}}} \right)\sin \left[ {2 \times {{10}^{11}}t + 300\pi x} \right]T$
(i) Calculate the wavelength of the electromagnetic wave.
(ii) Write down the expression for the oscillating electric field.

Answer 1

Electromagnetic waves are produced when an electric field comes in contact with a magnetic field. Electromagnetic waves are the composition of oscillating electric and magnetic fields.
Formula Used:
${B_y} = {B_ \circ }\sin \left( {\omega t + kx} \right)$
$k = \dfrac{{2\pi }}{\lambda }$
$\dfrac{{{E_ \circ }}}{{{B_ \circ }}} = c$
${E_z} = {E_ \circ }\sin \left( {kx + \omega t} \right)V/m$

Complete answer:
General equation of a magnetic field is given by,
${B_y} = {B_ \circ }\sin \left( {\omega t + kx} \right)$ --(2)
Now, comparing equation (1) and equation (2) we will get,
\eqalign{ & {B_ \circ } = 8 \times {10^{ - 6}} \cr & \omega = 2 \times {10^{11}} \cr & k = 300\pi \cr}
Propagation constant is known as the measure of change in its amplitude and phase per unit time, it is denoted as $k$. It can also be written as
$k = \dfrac{{2\pi }}{\lambda }$
Substituting the value of propagation constant $k$ from above we will get the value of wavelength of the electromagnetic field,
$\Rightarrow 300\pi = \dfrac{{2\pi }}{\lambda }$
$\lambda = \dfrac{2}{{300}} = \dfrac{1}{{150}} = 0.0067m$
In an electromagnetic wave the ratio of amplitudes of electric field and magnetic field is equal to the velocity of the electromagnetic waves in free space or the speed of light.
Mathematically it can be written as,
$\dfrac{{{E_ \circ }}}{{{B_ \circ }}} = c$
Where ${E_ \circ }$ denotes the electric field, ${B_ \circ }$denotes the magnetic field and $c$ is the speed of light whose value is taken as $3 \times {10^8}m/s$.
${E_ \circ } = c{B_ \circ }$
$\Rightarrow {E_ \circ } = 3 \times {10^8} \times 8 \times {10^{ - 6}} = 24 \times {10^2} = 2400V/m$
The oscillating magnetic field in a plane electromagnetic wave is given along the y- axis. So the electric field in a plane electromagnetic wave will be along the z- axis, and the direction of propagation will be along x- axis.
Thus, electric field along z- axis is given by,
${E_z} = {E_ \circ }\sin \left( {kx + \omega t} \right)V/m$
Now, substituting the values we will get the expression for the oscillating electric field,
$\Rightarrow {E_z} = 2400\sin \left( {30\pi x + 2 \times {{10}^{11}}t} \right)V/m$
Hence, (i) the wavelength of the electromagnetic wave is $0.0067m$ and (ii) the expression for the oscillating electric field is given as ${E_z} = 2400\sin \left( {30\pi x + 2 \times {{10}^{11}}t} \right)V/m$

Note:
Electromagnetic waves are transverse in nature because they propagate by varying the electric and magnetic fields such that the two fields are perpendicular to each other. Electromagnetic waves are produced by accelerated charges. It does not require any material medium to travel for its propagation.