Question

In an interference pattern the ${\left( {n + 4} \right)^{th}}$ blue bright fringe and ${n^{th}}$ red bright fringe are formed at the spot. If ref and blue light have the wavelength of $7800\;\dot A$. and $5200\;\dot A$, the value of n should be
A. 2
B. 4
C. 6
D 8

Answer 1

This question is based on the interference pattern. We have to know the term interference. Interference is a phenomenon that occurs in waves. It happens when two waves coincide at the time of traveling along with the same medium. First, we have to find the expression of the centers of the bright fringes. Then equate the expression of center distance of ${\left( {n + 4} \right)^{th}}$ the blue bright fringe and ${n^{th}}$ red bright fringe to get the value of number of fringes.

Complete step by step answer:
Given: The wavelength of red light is $7800\;\dot A$ and the wavelength of blue light is $5200\;\dot A$.
The ${\left( {n + 4} \right)^{th}}$ blue bright fringe is overlapped on ${n^{th}}$ red bright fringe. So, we have to find the center distance of both the fringes.
The expression for the center distance of ${\left( {n + 4} \right)^{th}}$ the blue bright fringe is given as,
$C.D = \dfrac{{\left( {n + 4} \right)D{\lambda _B}}}{d}$
Here, $n$ is the number of fringes, $D$ is the separation distance between the slits and the screen, and $d$ is the distance between the slits and ${\lambda _B}$ is the wavelength of blue light.
The expression for the center distance of ${\left( {n + 4} \right)^{th}}$ the blue bright fringe is given as,
$C.D = \dfrac{{\left( n \right)D{\lambda _R}}}{d}$
Here, ${\lambda _R}$ is the wavelength of blue light.
Now, we have to equate both the expressions to get the number of fringes.
$\dfrac{{\left( {n + 4} \right)D{\lambda _B}}}{d} = \dfrac{{\left( n \right)D{\lambda _R}}}{d}$
Now, substitute the value in the above equation we get,
$\begin{array}{l} \dfrac{{\left( {n + 4} \right)D \times 5200\;\dot A}}{d} = \dfrac{{\left( n \right)D \times 7800\;\dot A}}{d}\\\ n = 8 \end{array}$
Thus, the value of $n$ is 8.

So, the correct answer is “Option D”.

Note:
In this question, students have the knowledge of the term interference and fringes. Fringes are groups that are bright or dark formed by diffraction or interference of radiation The expression of ${\left( {n + 4} \right)^{th}}$ term and ${n^{th}}$ term should be known.