(a) Derive the relation a `sin theta=lambda` for the first minimum of the diffraction pattern produced due to a single slit of width ‘a’ using light of wavelength`lambda`.
(b) State with reason, how the linear width of central maximum will be affected if
(i) monochromatic yellow light is replaced with red light, and
(ii) distance between the slit and the screen is increased.
(c) Using the monochromatic light of same wavelength in the experimental set-up of the diffraction pattern as well as in the interference pattern where the slit separation is 1 mm, 10 interference fringes are found to be within the central maximum of the diffraction pattern. Determine the width of the single slit, if the screen is kept at the same distance from the slit in the two cases.

Explanation

From diagram path difference between the waves from L and N `= a sin theta`
When first minimum is obtained at P, then path difference `= lambda`
Wavelets from first half of the slit has a corresponding wavelet from second half of the slit differing by a path of `(lambda)/(2)` and cancel each other ]
Condition for first minimum -
`lambda= a sin theta`
(b) We know that `beta_( cm) = ( 2 lambda D)/(d)`
(i) increases , (ii) increases
(c) `10(lambda)/(d)= 2 (lambda)/(a)`
Or `a = (d)/(5)= 0.2 mm`