Discuss diffraction at single slit and obtain the consition for `n^(th)` minimum. Diffraction at single slit:

Explanation

Let a parallel beam of light fall normally on a single slit AB of width. The diffracted beam falls on a screen kept at a distance. The center of the slit is C. A straight line through C A straight line through C perpendicular to the plane of slit meets the center of the screen at O. We would like to find the intensity at any point P on the screen. The lines joining P to the different points on the slit can be treated as parallel lines, making an angle `theta` with the normal CO.
All the waves start parallel to each other from different points of the slit and interface at point P and other to give the resultant intensities. The point P is the geometrically shadowed region, up to which the central maximum is spread due to diffraction. We need to give the condition for the point P to be of various manima.
The basic idea is to divide the slit into much smaller even number of parts. Then, add their contributions at P with the proper path difference to show that destructive interference takes place at that point to make it minimum. To explain maximum, the slit is diveded into add number of parts.

Condition for P to be first minimum: ,br> Let us divide the slit AB into two half.s AC and CB. Now width of AC is (a/2). We have difference points on the slit which are separated by the same width (here a/2) called corresponding points.
The path difference of ligth waves from different correponding points.
The path difference of light waves from different corresponding points meeting at point P and interfere destructively to make it first minimum. The path difference `delta` between waves from these corresponding points is `delta = (a)/(2) sin theta`
The condition for P to be first minimum, `(a)/(2)sintheta=(lambda)/(2)`
`a sin theta = lambda` (first minimum)
Condition for P to be second minimum:
Let us divide the slit AB into four equal parts. Now, the points is, a/4. The path difference `delta` between waves from these corresponding points is, `delta = (a)/(4) sin theta`.
The condition for P to be second minimum, `(a)/(4) sin theta = (lambda)/(2)`
`a sin theta = 2 lambda` (second minimum)
Condition for P to be third order minimum:
The same way the slit is divided into parts to explain the condition for P to be third minimum is, `(a)/(6) sin theta = (lambda)/(2)`
`a sin theta = 3 lambda` (third minimum)
Condition for P to be `n^(th)` order minimum:
Dividing the slit into 2n number of (even number of) equal parts makes the light produced by one of the corresponding points to be counterpart. Thus, the condition for ` n^(th)` order minimum is, `(a)/(2n) sin theta = (lambda)/(2)`.
`a sin theta = n lambda (n^(th)"minimum")`