Question

The electric field of an electromagnetic wave in free space is given by \[E = 10\sin \left( {{{10}^7}t + kx} \right)\hat jV/m\], where \[t\] and \[x\] are in seconds and metres respectively. It can be inferred that
(a) The wavelength \[\lambda \] is \[188.4m\]
(b) The wave number \[k\] is \[0.033rad/m\]
(c) The wave amplitude is 10 V/m
(d) The wave is propagating along +x direction
Which one of the following pairs of statements is correct?
(A) (a) and (b)
(B) (b) and (c)
(C) (a) and (c)
(D) (c) and (d)

Answer 1

Hint: The amplitude is the maximum value of the electric field, and sine of an angle is either equal to or greater than one. The speed of an electromagnetic wave in one form can be given as the angular frequency divided by the wave number.
Formula used: In this solution we will be using the following formulae;
\[c = \dfrac{\omega }{k}\] where \[c\] is the speed of light (also EM wave), \[\omega \] is the angular frequency, \[k\] is the wave number of the wave. \[k = \dfrac{{2\pi }}{\lambda }\] where \[\lambda \] is the wavelength of the wave.
\[y = A\sin \left( {kx + \omega t} \right)\] where \[y\] is the instantaneous displacement at a particular time \[t\] and position \[x\], \[A\] is the amplitude of the wave. The equation is an equation of progressive waves moving in the negative x direction.

Complete Step-by-Step Solution:
To identify which of the statements is true, we need to identify each of the quantities stated in the options. To do so, we shall recall the general equation of a progressive wave given by
\[y = A\sin \left( {kx + \omega t} \right)\] where \[y\] is the instantaneous displacement at a particular time \[t\] and position \[x\], \[A\] is the amplitude of the wave. This, though, is for a wave moving in the negative x direction.
By comparing with the given equation \[E = 10\sin \left( {{{10}^7}t + kx} \right)\hat jV/m\], we see that:
The wave must be travelling in the negative x direction, hence statement (d) is false.
The amplitude \[A = 10V/m\], hence statement (c) is correct
The angular frequency is \[{10^7}{s^{ - 1}}\]
To find the wavenumber and wavelength, we note that
\[c = \dfrac{\omega }{k}\] where \[c\] is the speed of light (also EM wave), \[\omega \] is the angular frequency, \[k\] is the wave number of the wave.
Hence,
\[3 \times {10^8} = \dfrac{{{{10}^7}}}{k}\]
\[ \Rightarrow k = \dfrac{{{{10}^7}}}{{3 \times {{10}^8}}} = 0.033{m^{ - 1}}\], hence, statement (b) is false.
The wavelength is
\[k = \dfrac{{2\pi }}{\lambda }\]
Hence,
\[0.033 = \dfrac{{2\pi }}{\lambda }\]
\[ \Rightarrow \lambda = \dfrac{{2\pi }}{{0.033}} = 188.49m\]
Hence, roughly, statement (a) is true.

Hence, the correct option is C

Note: For exam purposes, since other options but option (a) have been proven false, it is unnecessary to calculate the wavelength, because the options are given in pairs and the only possible answer would be statement (a) and (c), which is option C.