Question

It is found that when waves of same intensity from two coherent sources superpose at a certain point, then the resultant intensity is equal to intensity one wave only.This means that the phase difference between two waves at that point is
A. Zero
B. $\dfrac{\pi}{3}$
C. $\dfrac{2 \pi}{3}$
D. $\pi$

Answer 1

Coherent sources are that source of light which travels with constant phase difference along with it they have the same frequency. When these sources superpose on each other either forms a constructive or destructive interference depending on their phase difference.

Formula used:
The amplitude of resultant wave is given $\dfrac{\pi }{3}$ by:
$A = {\left( {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos \theta } \right)^{\dfrac{1}{2}}}$
where ${A_1}$ and ${A_2}$ are amplitudes of waves.

Complete step by step answer:
The intensity of the resultant wave varies with amplitude of resultant wave as proportionally to square of amplitude. Intensity of resultant wave is defined as: $I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \delta$ (where $\delta = \dfrac{{2\pi }}{\lambda }\left( {\Delta x} \right)$)
$\delta$ is the phase difference and $\Delta x$ is the path difference between waves.

As given in the question the intensity of the waves are equal then ${I_1} = {I_2}$ $= {I_0}$.Substitute the value in the equation of intensity.
$I = {I_0} + {I_0} + 2\sqrt {{I_0}^2} \cos \delta$
$\Rightarrow I = 4{I_0}{\cos ^2}\dfrac{\delta }{2} \\\$
Now it is given the resultant intensity is equal to the intensity of one wave only then $I = {I_0}$, substituted in the above equation.
${I_0} = 4{I_0}{\cos ^2}\dfrac{\delta }{2}$
Solve the equation
\dfrac{1}{4} = {\cos ^2}\dfrac{\delta }{2} \\\ \Rightarrow \dfrac{1}{2} = \cos \dfrac{\delta }{2} \\\
$\Rightarrow \cos \dfrac{\pi }{3} = \cos \dfrac{\delta }{2}$
$\therefore \delta = \dfrac{{2\pi }}{3}$
The phase difference between the waves is $\dfrac{{2\pi }}{3}$.

Hence, option C is the correct answer.

Note: When we calculate the average of the resultant intensity of two waves we find that it is equal to the sum of the given intensities which tell us that the interference of waves also follows the law of conservation of energies. As light is also a form of energy ‘ It can never be created and destroyed by its own’.