Two slits S 1 and S 2 lies on the x-axis and symmetric with respect to y-axis are illuminated by a parallel monochromatic light beam of wavelength λ as shown. The distance between shits is d (>> λ ). Point O is the midpoint of the line S 1 S 2 and this point is considered as origin. The slits are in horizontal plane. The interference pattern is observed on a horizontal plate (acting as screen) of mass m which is connected to one end of a vertical massless spring of constant K. The other end of spring is fixed to the ground. At t = 0, plate is at C distance D (>>d) below the plane of slits and spring is at its natural length. The plate is released from rest from its initial position.
(i) The rate by which fringe width will increase when the acceleration of the plate is zero is -
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Ans.
(i)
Sol.
The equation governing motion of plates is
Let x' be instantaneous position of plate from O
x' = x 0 – x
x' = x 0 – x 0 cos ω t
x 1 = x 0 (1 – cos ω t)
Instantaneous fringe width is
W = 
= 
= 
(x 0 ω ) sin ω t
= 
. sin ω t
acceleration of plates is zero at equilibrium
sin ω t = 1
Rate at which fringe width will increase
= 
g 
(ii)
Sol. Plate is rest momentarily when
x' = 0, when x = x 0 = x 0 cos ω t
cos ω t = 1
fringe width is W = 
Plate is also at momentary rest when x' = 2x 0
fringe width W = (D + 2x 0 ) λ / d
Difference = 
– 
= 
= 
(iii)
Sol. Displacement of n th bright fringe is given as
Δ x = 
( μ – 1) t
Since first maxima shift to the position of central maxima hence Δ x = 
D λ / d = 
( μ – 1) t
t = 
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