Question

Consider two coherent sources S1 and S2 producing monochromatic waves to produce interference pattern. let, the displacement of the Waves produced by S1 given by y1= acosõt and displacement by s2 be y2= a cos(õt+ø). find out the expression for amplitude of the resultant displacement.

Answer 1

Answer:

Explanation:

(a)

Resultant displacement at the point will be

Y = Y_1+Y_2Y=Y  

1

​  

+Y  

2

​  

 

   = a \cos(\omega t) + a \cos(\omega t +\phi)=acos(ωt)+acos(ωt+ϕ)

   = 2 a \cos (\omega t + \phi/2) \cos (\phi/2)=2acos(ωt+ϕ/2)cos(ϕ/2)

Intensity is square of amplitude of displacement

I = 4a^2 \cos^2(\phi/2)I=4a  

2

cos  

2

(ϕ/2)

For constructive interference, resultant intensity is maximum.  

Hence, \phi/2 = n\piϕ/2=nπ where n is an integer

            \phi = 2n \piϕ=2nπ

For destructive interference, resultant intensity is minimum.

Hence, \phi /2 = n\pi \pm \pi/2ϕ/2=nπ±π/2  

             \phi = 2n\pi \pm \piϕ=2nπ±π

(b)

(i) When the width of the slit is increased,  intensity of the light waves incident increases and hence, the intensity of the fringes formed increases. However, the fringes formed are not very sharp and increasing the slit width too much, spreads the fringes in a greater area.

(ii)

When a monochromatic light source is replaced by a white light source, fringes formed at a point consists of only a single colour (wavelength). Multiple fringes of various colours are formed. Central fringe remains white as all colours constructuvely interfere there.

Answer 2

$A=\sqrt{2a^2(1+cos\phi)}$

Explanation:

The resultant amplitude of waves is given by the sum of the displacement of the waves:

Given:

$y_1=acos\omega t\\y_2=acos(\omega t+\phi)$

Expression of the amplitude:

$y=y_1+y_2\\y=acos \omega t+a cos(\omega t+\phi)\\y=a(cos\omega t+cos (\omega t+\phi))$

$A=\sqrt{a_1^2+a_2^2+2a_1a_2cos\phi}\\A=\sqrt{a^2+a^2+2a^2cos \phi}\\A=\sqrt{2a^2(1+cos\phi)}$

The phase difference Ф depends on whether there is constructive interference or destructive interference

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