Question

The wave function (in S.I. units) for an electromagnetic wave is mentioned to be as
$\psi \left( x,t \right)={{10}^{3}}\times \sin \pi \left( 3\times {{10}^{6}}x-9\times {{10}^{14}}t \right)$
Therefore, the speed of the wave can be given as,
$\begin{align}
  & A.9\times {{10}^{14}}m{{s}^{-1}} \\
 & B.3\times {{10}^{8}}m{{s}^{-1}} \\
 & C.3\times {{10}^{6}}m{{s}^{-1}} \\
 & D.3\times {{10}^{7}}m{{s}^{-1}} \\
\end{align}$

Answer 1

Hint: The equation given in the question should be compared with the one which is the general equation of the wave function for an electromagnetic wave. By comparing, find out the angular velocity and the value of the propagation vector of the wave. The velocity of a wave will be obtained by taking the ratio of the angular velocity to the propagation vector of the wave. This will help you in solving this question.

Complete step by step answer:
The general equation of an electromagnetic wave can be written as,
$E={{E}_{0}}\cos \left( kz-\omega t \right)$
Where ${{E}_{0}}$ be the initial amount of electric field, $k$ be the propagation vector, $\omega $ be the angular velocity of the wave and $t$ be the time taken.
The wave function mentioned in the question can be written as,
$\psi \left( x,t \right)={{10}^{3}}\times \sin \pi \left( 3\times {{10}^{6}}x-9\times {{10}^{14}}t \right)$
We have to compare both these equations together.
That is,
The angular velocity will be equivalent to,
$\omega =9\times {{10}^{14}}$
And the propagation vector can be expressed as,
$k=3\times {{10}^{6}}$
The velocity of a wave can be found by taking the ratio of the angular velocity to the propagation vector of the wave. That is we can write that,
$v=\dfrac{\omega }{k}$
Substituting the values in it will give,
$v=\dfrac{9\times {{10}^{14}}}{3\times {{10}^{6}}}=3\times {{10}^{8}}m{{s}^{-1}}$
This will be the velocity of the wave.
Hence the answer has been obtained as the option B.

Note:
Angular velocity is defined as the quantity which measures how the angular position of a body or the orientation of a body varies with the variation of time. There are two types of angular velocity. One is orbital angular velocity and the other is spin angular velocity. The propagation vector shows the direction at which the wave travels.