Equation for path difference:
(i) Let d be the distance between the double slits `S_1` and `S_2` which act as coherent sources of wavelength `lamda`. A screen is placed parallel to the double slit at a distance D from it. (ii) The mid-point of `S_1` and `S_2` is C and the i mid-point of the screen O is equidistant from `S_1` and `S_2` P is any point at a distance y from O. (iii) The waves from `S_1` and `S_2` meet at P either n-phase or out-of-phase depending upon the path difference between the two waves. (iv) The path difference `delta` between the light waves from `S_1` and `S_2` to the point P is,
` delta = S_2 P - S_1 P`
(v) A perpendicular is dropped from the point `S_1` to the line `S_2P ` at M to find the path difference more precisely.
` delta = S_2 P - MP = S_2M`
The angular position of the point P from C is `theta. angle OCP = theta`
From the geometry, the angles `angleOCP` and `angleS_2S_1M` are equal.
` angle OCP = angle S_2 S_1 M = theta`
In right angle triangle `Delta S_1 S_2 M` , the path difference, ` S_2 M = d sin theta`
` delta = d sin theta`
If the angle `theta` is small, `sin theta ~~ tan theta ~~ theta` .From the right angle triangle `DeltaOCP, tan theta = y/D`
The path difference, ` delta = (dy)/(D)`
Based on the condition on the path difference, the point P may have a bright or dark fringe.
Condition for bright fringe (or) maxima :
(i) The condition for the constructive interference or the point P to be have a bright fringe is,
Path difference, `delta = nlamda`
where, n =1,2,3...
` therefore (dy)/(D) = nlamda`
` y = n (D lamda)/(d) " or " y_n = n (D lamda)/(d)`
(ii) This is the condition for the point P to be a bright fringe. The distance is the distance of the nth bright fringe from the point O.
Condition for dark fringe (or) minima :
(i) The condition for the destructive interference or the point P to be have a dark fringe is,
Path difference, ` delta = (2n - 1) lamda/2`
where, n = 1, 2, 3, .. ..
` therefore (dy)/(D) = (2n -1) lamda/2`
` y = ((2n - 1))/(2) (lamda D)/(d) " or " y_n = ((2n -1) )/(2) (lamda D)/(d)`
(ii) This is the condition for the point P to be a dark fringe. The distance `y_n` is the distance of the nth dark fringe from the point O.
(iii) The formation of bright and dark fringes is, shown in Figure.
(iv) This shows that on the screen, alternate bright and dark bands are seen on either side of the central bright fringe. (v) The central bright is referred as `0^(th)` bright followed by 1st dark and 1st bright and then 2nd dark and 2nd bright and so on, on either side of O successively as shown in Figure.
Equation for bandwidth :
(i) The bandwidth (`beta`) is defined as the distance between any two consecutive bright or dark fringes. (ii) The distance between `(n+1)^(th)` and `n^(th)` consecutive bright fringes from O is given by
`beta = y_((n+1)) - y_n = ((n+1) (lamdaD)/(d)) - (n (lamdaD)/(d))`
`beta = (D lamda)/(d)`
(iii) Similarly, the distance between `(n+1)^(th)` and nth consecutive dark fringes from O is given by,
` beta = y_((n+1)) - y_n`
` = (((2(n+1) - 1))/(2) (lamdaD)/(d) ) - (((2n -1))/(2) (lamdaD)/(d))`
`beta = (Dlamda)/(d)`
