Question

The condition for constructive interference in Lloyd’s single mirror experiment is the path difference which is equal to
A. $N\lambda$
B. $(2N-1){\displaystyle \frac{\lambda }{2}}$
C. $(N-1){\displaystyle \frac{\lambda }{2}}$
D. ${\displaystyle \frac{\lambda }{2(2N-1)}}$

Answer 1

Hint: When reflection occurs then there is phase change of $\pi$ radians between the incident ray and the reflected rays. The path difference between incident and the reflected ray is due to the phase change.

Complete step by step solution:
If $\theta$ is the phase difference between two waves (incident light wave and the reflected light ray) then the path difference between the two is calculated using formula,
$\mathrm{\Delta }x=\left({\displaystyle \frac{\theta \lambda }{2\pi }}\right)$
During reflection, $\theta =\pi$ radians

Hence, the path difference between the incident light and the reflected light can be calculated as,
$\mathrm{\Delta }{x}_1=\left({\displaystyle \frac{\pi \lambda }{2\pi }}\right)={\displaystyle \frac{\lambda }{2}}$
At any point on the screen which is at distance of $y$ from the central point of the screen the path difference is given as,
$\mathrm{\Delta }{x}_2={\displaystyle \frac{yd}{D}}$
Where,
$\begin{array}{rl}& d=\text{ slit width}\\ & D=\text{ distance of screen from the source}\end{array}$

Then,
For constructive interference at point on the screen,
Phase difference $=N\lambda$
Where, $N=0,1,2\dots \dots$
Total path difference $=$$\mathrm{\Delta }x=\mathrm{\Delta }{x}_1+\mathrm{\Delta }{x}_2$
$\begin{array}{rl}& \mathrm{\Delta }x=N\lambda \\ & {\displaystyle \frac{yd}{D}}+{\displaystyle \frac{\lambda }{2}}=N\lambda \\ & {\displaystyle \frac{yd}{D}}=N\lambda -{\displaystyle \frac{\lambda }{2}}\\ & =(2N-1){\displaystyle \frac{\lambda }{2}}\end{array}$

Therefore,
For the constructive interference at any point on the screen in Llyod’s single mirror experiment the path difference is equal to $(2N-1){\displaystyle \frac{\lambda }{2}}$.

Note: 1. Phase difference during reflection is $\pi$ radian
2. Phase difference during refraction depends on the path in which light is travelling. When light is moving from rarer medium to denser medium then the phase change is $\pi$