Question

If the plane difference between two interfering waves of equal wavelength is $13\pi $, then _________ interference or _________ order will occur.
A. Constructive, ${13^{th}}$
B. Constructive, ${7^{th}}$
C. Destructive, ${13^{th}}$
D. Destructive, ${7^{th}}$

Answer 1

Hint: We are given the phase difference between two waves of equal wavelength. We first need to find out whether the interference is constructive or destructive and then find the path difference and then use the path difference to find the order of the interference.

Complete step by step answer:
Given the phase difference between two interfering waves of equal wavelength is $13\pi $.Therefore the phase difference is an odd integral multiple of half wavelength, hence the interference is destructive. Now we will find the path difference of the two wavelengths.We know;
$\dfrac{{{\text{phase difference}}}}{{2\pi }} = \dfrac{{\Delta x}}{\lambda }$
Substituting the values we get
$ \Rightarrow \dfrac{{13\pi }}{2} = \dfrac{{\Delta x}}{\lambda }$
$ \Rightarrow \Delta x = \dfrac{{13}}{2} \times \lambda $

Now we will find the order of the destructive interference
We know that for destructive interference
$\Delta x = (2n - 1) \times \dfrac{\lambda }{2}$
Substituting the value we got from the above equation here we get
$\dfrac{{13}}{2} \times \lambda = (2n - 1) \times \dfrac{\lambda }{2}$
$ \Rightarrow 2n - 1 = 13$
Further solving we get,
$ \Rightarrow 2n = 14$
$ \therefore n = 7$
Therefore it is destructive interference with an order of ${7^{th}}$.

Hence, option D is correct.

Additional information: When monochromatic light passing through two narrow slits illuminates a distant screen, a characteristic pattern of bright and dark fringes is observed.This interference pattern is caused by the superposition of overlapping light waves originating from two slits. The bright fringes are formed by constructive interference while the dark fringes are formed by destructive interference.

Note: The difference between two waves is an integral multiple of the wavelength, which satisfies conditions for constructive interference. Whereas, if the path difference between two waves is an odd integral multiple of half-wavelength, it satisfies the condition of destructive interference. In the given question the wavelengths of the two interfering waves are equal.