Name: Waves and Their Phases
Uploaded: 2019-07-19
Duration: 6 min 51 s
Description: In this video, we will learn how two or more waves can be studied in terms of their phases. Equations of two waves and graphs of two waves. Learn more!

Question

In Young's double slit experiment, deduce the condition for destructive interference at a point on the screen. Draw graph showing variation of the resultant intensity in the interference pattern against position $x$ on the screen.

Accepted Answer

When two waves of same frequency and constant initial phase difference travel in the same direction along a straight line simultaneously, they superpose in such a way that the intensity of the resultant wave is maximum at certain points and minimum at certain other points. The phenomenon of redistribution of intensity due to superposition of two waves of same frequency and constant initial phase difference is called the interference. The waves of same frequency and constant initial phase difference are called coherent waves. At points of medium where the waves arrive in opposite phase, the resultant intensity is minimum and the interference at these points is said to be destructive.
Suppose two coherent waves travel in the same direction along a straight line, the frequency of each wave is $\frac{ω}{2 π}$ and amplitudes of electric field are a1 and a2 respectively. If at any time t, the electric fields of waves at a point are $y_{1}, y_{2}$ respectively and phase difference is $φ$ then equation of waves may be expressed as

$y_{1} = a_{1} \sin ω t y_{2} = a_{2} \sin (ω t + φ)$

According to Young's principle of superposition, the resultant displacement at that point will be

$y = y_{1} + y_{2}$

$= a_{1} \sin ω t + a_{2} \sin (ω t + φ)$

$\Rightarrow y = a_{1} \sin ω t + a_{2} (\sin ω t \cos φ + \cos ω t \sin φ)$

$= (a_{1} + a_{2} \cos φ) \sin ω t + (a_{2} \sin φ) \cos ω t L e t, a_{1} + a_{2} \cos φ = A \cos θ A n d a_{2} \sin φ = A \sin θ$ Where A and θ are new constants

$y = A \cos θ \sin ω t + A \sin θ \cos ω t = A \sin (ω t + θ)$

This is the equation of the resultant disturbance. By trigonometry, we get,

$A = \sqrt{{a_{1}}^{2} + {a_{2}}^{2} + 2 a_{1} a_{2} \cos φ} $

Intensity of resultant wave, $I = A^{2} = {a_{1}}^{2} + {a_{2}}^{2} + 2 a_{1} a_{2} \cos φ$

Destructive Interference: For minimum intensity at any point $\cos φ = - 1$

$p h a s e d i f f e r e n c e, φ = π, 3 π, 5 π, 7 π . . . . = (2 n - 1) π, n = 1, 2, 3 . . .$

In this case the minimum intensity, $I_{m i n} = {a_{1}}^{2} + {a_{2}}^{2} - 2 . a_{1} a_{2} = {(a_{1} - a_{2})}^{2}$

$Path difference, Δ = \frac{λ}{2 π} \times Phase difference$

$= (2 n - 1) \frac{λ}{2}$

In Young's double slit experiment, deduce the condition for destructive interference at a point on the screen. Draw graph showing variation of the resultant intensity in the interference pattern against position $x$ on the screen.

Important Questions on Wave Optics

Learn and Prepare for any exam you want !

In Young's double slit experiment, deduce the condition for destructive interference at a point on the screen. Draw graph showing variation of the resultant intensity in the interference pattern against position x on the screen.

Important Questions on Wave Optics

Learn and Prepare for any exam you want !

In Young's double slit experiment, deduce the condition for destructive interference at a point on the screen. Draw graph showing variation of the resultant intensity in the interference pattern against position $x$ on the screen.