Question

(a) (i)'Two independent monochromatic sources of light cannot produce a sustained interference pattern'. Give reason.
(ii) Light wave each of amplitude "a" and frequency "$\omega$", emanating from two coherent light sources superpose at a point. If the displacements due to these waves is given by $y_1=acos\omega t$ and $y_2=acos(\omega t+\varphi )$ where $\varphi$ is the phase difference between the two, obtain the expression for the resultant intensity at the point.
(b) In Young's double slit experiment, using monochromatic light of wavelength $\lambda$, the intensity of light at a point on the screen where path difference is $\lambda$, is K units. Find out the intensity of light at a point where path difference is $\lambda /3$.

Answer 1

(a)(i) Two independent monochromatic sources of light cannot produce a sustained interference because :

(1) If the sources are not coherent , they cannot emit waves continuously .

(2) Independent sources , emit the waves , which don't have same phase or a constant phase difference .

(ii) given $y_1=a\cos \omega t$ ,

$y_2=a\cos (\omega t+\varphi )$ ,

by superposition principle ,

resultant displacement , $y=y_1+y_2$ ,

or $y=a\cos \omega t+a\cos (\omega t+\varphi )$ ,

or $y=2a\cos \left(\varphi /2\right).\cos (\omega t+\varphi /2)$ ,

or $y=A\cos (\omega t+\varphi /2)$ ,

it is an equation of simple harmonic plane progressive wave , whose amplitude is $A$ ,

here $A=2a\cos \left(\varphi /2\right)$ ,

now intensity is proportional to square of amplitude , therefore

$I=K{A}^2=4K{a}^2{\cos }^2\left(\varphi /2\right)$ ,

where $K$ is proportionality constant .

(b) In interference the intensity $I$ at a point is given by ,

$I=I_o{\cos }^2\left(\pi /\lambda \right)x$ ,

where $x=$ path difference ,

$\lambda =$ wavelength ,

$I_o=$ intensity of central maximum ,

when $x=\lambda$ , $I=K$ ,

$K=I_o{\cos }^2\left(\pi /\lambda \right)\lambda$ ,

or $K=I_o{\cos }^2\pi =I_o$ ,

when $x=\lambda /3$ , $I=I^{\prime }$ ,

$I=I_o{\cos }^2\left(\pi /\lambda \right)\lambda /3$ ,

or $I^{\prime }=I_o{\cos }^2\left(\pi /3\right)=I_0(1/2{)}^2=K/4$ ,