Question

Derive condition for constructive and destructive interference

Answer 1

Answer:

constructive and destructive interferance

Answer 2

Answer:

Interference of light is the phenomena of multiple light waves interacting with one another under certain circumstances, causing the combined amplitudes of the waves to either increase or decrease.

Explanation:

Consider two light waves represented by

$\rm y_1 = a_1 \sin \omega t \: .......(1)$

$\rm y_2= a_2 \sin (\omega t + \theta\: ).......(2)$

Where y_{1} and y_{2}, and displacements, \alpha_{1} and \alpha_{2} are amplitudes, \omega is the angular frequency and \theta is the phase difference

Accordingto superposition principle the resulant displacement is given by

$\implies \rm \: y = y_{1} + y_2$

$\rm \implies \: y = a_1 \sin \omega t + a_2 \sin (\omega t + \theta\: )$

$\rm \implies \: y = a_1 \sin \omega t + a_2 \sin \omega t \cos \theta\: + a_2 \cos\omega t \sin \theta\:$

$\star\boxed{ \rm \sin(a + b) = \sin a \cos b + \cos a \sin b }$

$\rm \implies \: y = a_1 \sin \omega t + a_2 (\sin \omega t \cos \theta\: + \cos\omega t \sin \theta\:)$

$\rm \implies \: y = (a_1 + a_2 \cos \theta) (\sin \omega t + \cos\omega t (\ a_{2}sin \theta\:)) .....(3)$

$\rm Let : (a_1 + a_2 \cos \theta) = A \cos \Phi \: .....(4)$

$\rm \: and : \: a_{2}sin \theta\: \: = A \sin\Phi ....(5)$

$\rm \: Where \: A \: and \: \Phi \: are the \: constant$

Now using equation (3) , we get

$\rm \implies \: y = A \cos \Phi \sin \omega t +A \sin \Phi \cos\omega t...(6)$

This is the equation for resulant wave and A is the resulant amplitude.

Squaring equation (4) and (5) w

$\rm (a_1 + a_2 \cos \theta)^{2} = A ^{2} \cos ^{2} \Phi$

and

$\rm \: a_{2}^{2} sin ^{2} \theta\: \: = A ^{2} \sin ^{2} \Phi ....$

Now adding them

$\implies \rm (a_1 + a_2 \cos \theta)^{2} + a_{2}^{2} sin ^{2} \theta = A ^{2} \cos ^{2} \Phi + A ^{2} \sin ^{2} \Phi$

By solving we get,

$\boxed{ \rm \: A = \sqrt{ {a}^{2} _1 + {a}^{2} _2 + 2{a} _1{a} _2 \cos \theta } }$

For constructive interference the intensity is maximum and hense amplitudes is maximum

$\rm \: If \cos\theta = 1$

then

$\rm \: A = {a} _1 + {a} _2 = maximum$

Now

$\rm\cos\theta = \cos2n\pi$

when n = 0,1,2......

$\rm \theta = 2n\pi$

$\rm \theta = \dfrac{2\pi}{ \lambda} x$

Where \lambda is the wavelength and x is the path difference

$\rm \implies\dfrac{2\pi}{ \lambda} x = 2n\pi$

$\underline{ \boxed{\implies \rm x = 2n \dfrac{ \lambda}{2} }}$

This is the condition for constructive interference.

Again

For destructive interference the intensity is minimum and hense amplitudes is also minimum.

$\rm \: If \cos\theta = - 1$

then

$\rm \: A = {a} _1 - {a} _2 = minmum$

Now

$\rm\cos\theta = \cos(2n + 1)\pi$

when n = 0,1,2......

$\rm \theta = ( 2n + 1)\pi$

$\rm \implies\dfrac{2\pi}{ \lambda} x = (2n + 1)\pi$

$\underline{ \boxed{\implies \rm x =( 2n + 1) \dfrac{ \lambda}{2} }}$

This is the condition for destructive interference.