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

What is interference? Describe Young's experiment for demonstrating the phenomenon of interference of light and explain it by the wave theory. Derive the formula used.

Accepted Answer

Interference is a phenomenon in which two waves superpose to form the resultant wave. At the point of interference we get minimum to maximum brightness of light. Young's double slit experiment is an example of interference of light in which a pattern of bright and fringes are obtained on the screen.

In Young's double slit experiment, waves coming out from two coherent sources undergo interference on the screen as shown below:

The path difference between these two waves is $∆ x$ .

From the the figure we can write that,

$d \sin θ = ∆ x$ and $\tan θ = \frac{y}{D}$

Where, $d$ is the distance between slits, $y$ is the distance of point from centre and $D$ is the distance between slits and the screen.

Since $θ$ is very small it means $\sin θ ~ \tan θ$

$\Rightarrow$ $\frac{∆ x}{d} = \frac{y}{D}$

$∆ x = \frac{y d}{D}$

Now intensity at any point $P$ will be given by,

$I = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos ϕ$

Where, $I_{1}$ is the intensity of 1st wave source $S_{1}$ , $I_{2}$ is the intensity of 2nd wave source $S_{2}$ and $ϕ$ is the phase difference between these two waves.

For maxima, $\cos ϕ = 1$

$\Rightarrow ϕ = 2 n π$ where $n$ is integer

As we know that $ϕ = \frac{2 π}{λ} ∆ x$ where $λ$ is wavelength of each wave.

$\Rightarrow \frac{2 π}{λ} ∆ x = 2 n π$

$\Rightarrow ∆ x = n λ$ for bright fringes.

Similarly, for minima $\cos ϕ = 0$

$\Rightarrow ϕ = (2 n + 1) π$

$\Rightarrow \frac{2 π}{λ} ∆ x = (2 n + 1) π$

$\Rightarrow ∆ x = (n + \frac{1}{2}) λ$ for dark fringes.

Also we have calculated that, $∆ x = \frac{y d}{D}$

Hence, position of bright fringes will be given by,

$\frac{y d}{D} = n λ$

$\Rightarrow y = \frac{n λ D}{d}$

Similarly, position of dark fringes on the screen will be given by,

$\frac{y d}{D} = (n + \frac{1}{2}) λ$

$\Rightarrow y = (n + \frac{1}{2}) \frac{λ D}{d}$

What is interference? Describe Young's experiment for demonstrating the phenomenon of interference of light and explain it by the wave theory. Derive the formula used.

Important Questions on Wave Optics and Interference

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