31.

State the importance of coherent sources in the phenomenon of interference.

In Young’s double slit experiment to produce interference pattern, obtain the conditions for constructive and destructive interference. Hence, deduce the expression for the fringe width.

How does the fringe width get affected, if the entire experimental apparatus of Young is immersed in water?

If coherent sources are not taken, the phase difference between two interfering waves, will change continuously and a sustained interference pattern will not be obtained. Thus, coherent sources provide sustained interference pattern.

Let S₁ and S₂ be two coherent sources separated by a distance d.

Let the distance of the screen from the coherent sources be D. Let point M be the foot of the perpendicular drawn from O, which is the midpoint of S₁ and S₂ on the screen. Obviously point M is equidistant from S₁ and S₂. Therefore the path difference between the two waves at point M is zero. Thus the point M has the maximum intensity.

Now, Consider a point P on the screen at a distance y from M. Draw S₁N perpendicular from S₁ on S₂ P. 

 

The path difference between two waves reaching at P from S₁ and S₂ is given by, 

 

Expression for Fringe Width:

Distance between any two consecutive bright fringes or any two consecutive dark fringes are called the fringe width. It is denoted by  . 

Let,  be the distance of two consecutive fringes. Then, we have

So, fringe width = 

Fringe width is same for both bright and dark fringe.

When the entire apparatus is immersed in water, the fringe width decreases.