Question

(a) A monochromatic source of light of wavelength λ illuminates a narrow slit of width d to produce a diffraction pattern on the screen. Obtain the conditions when secondary wavelets originating from the slit interfere to produce maxima and minima on the screen.

(b) How would the diffraction pattern be affected when 

(i) the width of the slit is decreased? 

(ii) the monochromatic source of light is replaced by white light?

Answer 1

(a) When plane wavefront coming from distant source illuminate the slit of size (=d), each point within the slit becomes the source of secondary wavelets, and these wavelets superpose on each other to generate the maxima and minima on the screen; path difference between the rays, directing to the point P on the screen can be given as

Condition of Minima: If set AB is divided into the equal halves (or in even parts) each of size d/2, for every point in part AM, there is a point in part MB that contribute the secondary wavelets out of phase (i.e., 180º). So net contribution from two halves becomes zero and hence intensity falls to zero for path difference Δ = nλ.

∴ d sinθ = nλ
d.θ = nλ where n is integer except n = 0.

Condition of maxima:
If slit AB is divided into three equal parts (or in odd parts). First two thirds of the slit having a path difference λ/2 between them cancel each other, and only the remaining one third of the set contributes to the intensity at the point between two minima, so for path difference Δ =(n + 1/2)λ.
We have dsinθ =(n + 1/2)λ.
where n is integer except n = 0.

(b) (i) Effect of the width of the slit —
For given monochromatic waves, if slit width is decreased, the fringe pattern becomes broader.

(ii) If monochromatic source of light is replaced by white light, instead of white fringes we have few coloured fringes on either side of central white fringe, and then uniform illumination on the screen.

y_n α λ (λ for VIBGYOR)