Question

Screen Problem 3: The figure shows a so called Lloyd's mirror. It is used to generate an interference pattern from the light emitted by a point source onto a screen. Assume the reflection coefficient of the mirror plate is R = 1. (a) What is the condition for the difference of lengths r2 - r of the two rays for an interference hi maximum and minimum to appear? (b) At what position y on the screen do the first Mirror plate minima and maxima of the interference pattern appear? (Hint: Justify that the difference of lengths could be written as 72 - 71 = 2hsin() in case we assume d >>h and d >>y to simplify the result.) (c) Calculate the values of y for the first two minima and the first two maxima when h = 1 cm and d = 10 cm, and the wavelength of the source radiation is 1 = 532 nm.

Answer 1

(a)
r₂ - r₁ =  
(2n – 1))/2
for constructive interference or maximum
( odd integer multiples of
(orX/2)
difference as there is an additional  $\pi$ phase difference due to reflection from the mirror )

r₂ - r₁ = $n \lambda$ for destructive interference or minimum
( integer multiples of )

27 (or)

(b)

For , small $\theta$ , $\theta$ = Sin$\theta$ = Tan$\theta$

r₂ - r₁ = 2h Sin $\theta$ from the figure below

90 , -- Ꮎ Ꮎ _h . Ax = 2hSine s2{ .

2h Sin $\theta$ = 2h Tan $\theta$ = 2h y/d

r₂ - r₁ = $\lambda$ = 2h y /d [ for first minimum ( as n= 1 in n$\lambda$ ) ]

$\lambda$ = 2h y /d
y = d $\lambda$ / 2h ....For First Minima

Similarly for n=2
y = d $\lambda$ / h ... For Second Minima and so on

r₂ - r₁ = $\lambda /2$   = 2h y /d [ for first maximum ( as n= 1 in ) ]

(2n - 1)/2

$\lambda /2$ = 2h y /d
y = d $\lambda$ / 4h ... For First Maxima
Similarly , for n=2
y = 3 d $\lambda$ / 4h ...  For Second Maxima and so on

(c)

y = d $\lambda$ / 2h ....For First Minima
y = 10 cm * 532nm / 2 *1cm
= 2660 nm

y = d $\lambda$ / h .. For 2nd minima
y =Twice of 1st minima = 2* 2660 nm = 5320 nm

y = d $\lambda$ / 4h ... For First Maxima
y =10cm * 532nm / 4*1cm = 1330 nm

y = 3 d $\lambda$ / 4h ..  For 2nd maxima
y = 3 * 1330 nm = 3990 nm