Question

Two sinusoidal waves of intensity I having the same frequency and same amplitude interfere constructively at a point. The resultant intensity at a point will be -
A) I
B) 2I
C) 4I
D) 8I

Answer 1

Hint: We are given that two coherent waves, with same amplitude and same frequency interfere at a point. Also, it is given that interferes constructively. So, we can use the superposition of waves method very easily to find the resultant amplitude and hence the intensity.

Complete answer:
We are given two sinusoidal waves which have the same amplitude and frequency. From the additional data given, we know that they interfere constructively. This gives us a conclusion that the two waves are coherent to each other and have constant phase, i.e., they do not have a phase difference between them at the time of interference.
So, we can find the equation of the two waves to be –
\[\begin{align}
  & {{y}_{1}}=a\sin (\omega t) \\
 & {{y}_{2}}=a\sin (\omega t+\phi ) \\
\end{align}\]
We can find the resultant amplitude at the time of constructive interference as –
\[\begin{align}
  & A=\sqrt{{{a}_{1}}^{2}+{{a}_{2}}^{2}+2{{a}_{1}}{{a}_{2}}\cos \phi } \\
 & \text{Here,} \\
 & {{a}_{1}}={{a}_{2}}=a \\
 & \text{and,} \\
 & \phi ={{0}^{0}} \\
 & \Rightarrow \text{ }A=\sqrt{{{a}^{2}}+{{a}^{2}}+2{{a}^{2}}} \\
 & \Rightarrow \text{ }A=2a \\
\end{align}\]
Now, we can relate the amplitude to the intensity of the wave at the point of constructive interference as –
\[\begin{align}
  & I\propto {{A}^{2}} \\
 & \Rightarrow \text{ }{{I}_{2}}\propto 4{{a}^{2}} \\
\end{align}\]
The initial intensity is given when the amplitude is a as –
\[{{I}_{1}}=I\propto {{a}^{2}}\]
From this, we can conclude the intensity at the point of interference is –
\[\begin{align}
  & \dfrac{{{I}_{2}}}{I}=4 \\
 & \Rightarrow \text{ }{{I}_{2}}=4I \\
\end{align}\]
The intensity at the point of constructive interference is 4I.

So, the correct answer is “Option C”.

Note:
We can directly substitute the amplitudes in the equation of intensity as we did for the amplitude. We can use any method as we wish. After all, the square relation between the amplitude and the intensity is the major point to be considered in this interference.