Question

The magnetic field in a plane electromagnetic wave is given by ${B_y} = 2 \times {10^{ - 7}}\sin (0.5 \times {10^3}x + 1.5 \times {10^{11}}t)T$
A.What is the wavelength and frequency of wave?
B.Write an expression for the electric field.

Answer 1

Hint: this question is from the topic of electromagnetic waves. First we write the general form of an electromagnetic wave. Then we compare it with our wave given in the question. By that we get our required values of wavelength, frequency and expression for the electric field.

Complete answer:
The general equation of an electromagnetic wave is given as $B = {B_0}\sin (kx - \omega t)$
We have been given the magnetic field in a plane electromagnetic wave as
${B_y} = 2 \times {10^{ - 7}}\sin (0.5 \times {10^3}x + 1.5 \times {10^{11}}t)T$
Comparing both the equations with each other, we find that
${B_0} = 2 \times {10^{ - 7}}$
$k = 0.5 \times {10^3}$
$\omega = 1.5 \times {10^{12}}$
Solution for part A
We know, The number of complete cycles of a wave throughout its wavelength is theoretically described as the wavenumber equation given as
 $k = \dfrac{{2\pi }}{\lambda }$
Putting value of $k$ from above we get
 $ \Rightarrow 0.5 \times {10^3} = \dfrac{{2\pi }}{\lambda }$
Hence wavelength of the given wave is
$ \Rightarrow \lambda = 12.56 \times {10^{ - 3}}m$
As we know, The rate of change of the phase of the waveform or the angular displacement of any element of the wave per unit time is referred to as the angular frequency.
$\omega = 2\pi f$
Therefore frequency of the given wave will be
$ \Rightarrow f = \dfrac{{1.5 \times {{10}^{11}}}}{{2\pi }}$
$ \Rightarrow f = 2.39 \times {10^{10}}Hz$
Solution for part B
We know electric field is given by
${E_0} = c{B_0}$
$ \Rightarrow 3 \times {10^8} \times 2 \times {10^{ - 7}} = 60\dfrac{V}{m}$
Hence, expression for the electric field is
$E = 60\sin (0.5 \times {10^3}x + 1.5 \times {10^{11}}t)\dfrac{V}{m}$

Note:
Electromagnetic waves, or EM waves, are waves that are formed when an electric field and a magnetic field vibrate together. EM waves, in other words, are made up of oscillating magnetic and electric fields.