Question

Monochromatic light of wavelength `5000 Å` is used in YDSE, with slit width, `d = 1 mm`, distance between screen and slits, `D = 1 M`. If intensites at the two slits are `I_1 = 4I_0 and I_2 = I_0`, find:
a. finge width `beta:`
b. distance of 5th minima from the central maxima on the screen,
c. intensity at `y = (1)/(3) mm,`
d. distance of the 1000th maxima, and
e. distance of the 5000th maxima.

Answer 1

(i) `beta=(lambdaD)/(d) =(5000xx10^(-10)xx1)/(1xx10^(-3))=0.5 mm`
(ii) `y=(2n-1) (lambdaD)/(2d),n=5 rArr y=2.25 mm`
(iii) At `y=(1)/(3) mm, y lt lt D` Hence `Deltap=(dy)/(D)`
`Delta phi=(2pi)/(lambda)Deltap=2pi (dy)/(lambdaD)=(4pi)/(3)`
Now resultant intensity
`I= I_(1)+I_(2) sqrt(I_(1)I_(2)) cos Delta phi =4I_(0)+I_(0)+2 sqrt(4I_(0)^(2)) cos Delta phi =5I_(0)+4I_(0) cos (4pi)/(3) =3I_(0)`
(iv) `(d)/(lambda)=(10^(-3))/(0.5xx10^(-6))=2000`
`n=1000` is not ` lt lt 2000`
Hence now `Deltap=d sin theta` must be used
Hence, `d sin theta =nlambda=1000lambda`
`rArr sin theta=1000(lambda)/(d)=(1)/(2) rArr theta =30^(@)`
`y=D tan theta=(1)/(sqrt3)` meter
(v) Higest order maxima
`n_(max)=[(d)/(lambda)]=2000`
Hence, `n=5000` is not possible.