Name: Wavelength Measurement: Fresnel's Biprism
Uploaded: 2019-07-19
Duration: 5 min 11 s
Description: A Fresnel biprism is a thin double prism placed base to base, which has a tiny refracting angle used to determine the wavelength of monochromatic light. Let’...

Question

Derive the expression for the intensity at a point where interference of light occurs. Arrive at the conditions for maximum and zero intensity.

Accepted Answer

Let the two waves S₁and S₂from the two coherent sources interfere at a point P. Let their amplitudes be $a$ and the initial phase difference between them be $ϕ$ . Then the displacements produced by S₁ and S₂will be given as:

$y_{1} = a \cos ω t$ ....(i)

$y_{2} = a \cos (ωt) (+ ϕ)$ ....(ii)

The intensity produced by each wave, $I_{0} \propto a^{2}$ ....(iii)

The resultant displacement at point P will be:

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

$y = a [\cos ω t] [+ \cos (ω t) (+ ϕ)]$

$y = 2 a \cos (\frac{ϕ}{2}) \cos (ω t + \frac{ϕ}{2})$ ......(iv)

From (iii), it is clear that the amplitude of the resultant displacement is $2 a \cos (\frac{ϕ}{2})$ and therefore the intensity at that point will be:

$I \propto 4 a^{2} \cos^{2} (\frac{ϕ}{2})$

On using (iii), we get:

$I = 4 I_{0} \cos^{2} (\frac{ϕ}{2})$ .....(v)

Where $I_{0}$ is the intensity of each of the individual sources.

Using (v) , For $ϕ = 0, \pm 2 π, \pm 4 π . . . . .$ , the intensity will be maximum and given by:

$I = 4 I_{0}$

For $ϕ = \pm π, \pm 3 π, \pm 5 π . . . .$ , the intensity will be zero:

$I = 0$

Derive the expression for the intensity at a point where interference of light occurs. Arrive at the conditions for maximum and zero intensity.

Important Questions on Wave Optics

Learn and Prepare for any exam you want !