Analytical expression of intensity for constructive and destructive interference due to Young's double slit

Analytical expression of intensity for interference due to Young's double slit:

Let us consider two waves from slit $S_{1}$ and $S_{2}$ having amplitude $a_{1}$ and $a_{2}$ respectively superimpose on each other at point $P$ . If the displacement of waves is $y_{1}$ and $y_{2}$ and the phase difference is $\phi$ then

$y_{1}=a_{1} \: sin \omega t \qquad(1)$

$y_{2}=a_{2} \: sin \left( \omega t + \phi \right) \qquad(2)$

According to the principle of superposition:

$y=y_{1}+y_{2} \qquad(3)$

Now substitute the value of $y_{1}$ and $y_{2}$ in the above equation $(3)$

$y=a_{1} \: sin \omega t + a_{2} \: sin \left( \omega t + \phi \right)$

$y=a_{1} \: sin \omega t + a_{2} \left( sin \omega t \: cos \phi + cos \omega t \: sin \phi \right) $

$y=a_{1} \: sin \omega t + a_{2} \: sin \omega t \: cos \phi + a_{2}\: cos \omega t \: sin \phi $

$y= \left( a_{1} + a_{2} \: cos \phi \right) \: sin \omega t + a_{2} \: sin \phi \: cos \omega t \qquad(4)$

Let

$a_{1} + a_{2} \: cos \phi = A\: cos\theta \qquad(5)$

$a_{2} \: sin \phi = A\: sin\theta \qquad(6)$

Now the above equation $(4)$ can be written as

$y= A\: cos\theta \: sin \omega t + A\: sin\theta \: cos \omega t $

$y= A \left( cos\theta \: sin \omega t + sin\theta \: cos \omega t \right)$

$y= A \: sin \left( \omega t + \theta \right) \qquad(7)$

This is the equation of the resultant displacement of the waves from slit $S_{1}$ and $S_{2}$ at point $P$ on the screen.

Now square the equation $(5)$ and equation $(6)$ and then add to them so

$ A^{2} \: sin^{2}\theta + A^{2}\: cos^{2}\theta = \left( a_{1} + a_{2} \: cos \phi \right)^{2} + a^{2}_{2} \: sin^{2} \phi $

$ A^{2} \left( sin^{2}\theta + cos^{2}\theta \right) = a^{2}_{1} + a^{2}_{2} \: cos^{2} \phi + 2a_{1} \: a_{2} \: cos\phi + a^{2}_{2} \: sin^{2} \phi $

$ A^{2} = a^{2}_{1} + a^{2}_{2} \left( cos^{2}\phi + sin^{2}\phi \right) + 2a_{1} \: a_{2} \: cos\phi $

$ A^{2} = a^{2}_{1} + a^{2}_{2} + 2a_{1} \: a_{2} \: cos\phi \qquad(8)$

$ A = \sqrt{ a^{2}_{1} + a^{2}_{2} + 2a_{1} \: a_{2} \: cos\phi }\qquad(9)$

Resultant Intensity:

The resultant intensity at point $P$ due waves from slit $S_{1}$ and $S_{2}$ can be find by flollowing formula:

$I=A^{2}$

Now subtitute the va;ue of $A^{2}$ from equation $(8)$ in the above equation

$I= a^{2}_{1} + a^{2}_{2} + 2a_{1} \: a_{2} \: cos\phi \qquad(10)$

$I= I_{1} + I_{2} + 2\sqrt{I_{1} \: I_{2}} \: cos\phi \qquad(11)$

Constructive Interference:

For constructive interference the $cos\phi$ should be equal to +1 i.e. $cos\phi =1$

Phase Difference in Constructive Interference:

We know that for constructive interference $cos\phi =1$

$cos\phi = cos( 2n\pi)$

$\phi = 2n\pi \qquad(12)$

Path difference in Constructive Interference:

We know that the path difference

$\Delta x = \frac{\lambda}{2 \pi} \phi$

Now subtitute the value of $\phi $ from equation $(12)$ in the above equation

$\Delta x = \frac{\lambda}{2 \pi} 2n\pi$

$\Delta x = n \lambda \qquad(13)$

Resultant Amplitude due to Constructive Interference:

For constructive interference, the resultant amplitude is maximum at point $P$. So from equation $(9)$

$ A_{max} = \sqrt{ a^{2}_{1} + a^{2}_{2} + 2a_{1} \: a_{2} } \qquad \left( \because cos\phi =+1 \right)$

$ A_{max} = \sqrt{ \left (a_{1} + a_{2} \right)^{2}} $

$ A_{max} = \left( a_{1} + a_{2} \right) \qquad(14)$

For constructive interference, the resultant amplitude of the waves is the sum of the amplitude of individual waves.

Resultant Intensity due to Constructive Interference:

For constructive interference, the resultant intensity is maximum at point $P$. So from equation $(10)$

$I_{max}= a^{2}_{1} + a^{2}_{2} + 2a_{1} \: a_{2} \qquad \left( \because cos\phi =+1 \right)$

$I_{max}=\left( a_{1} + a_{2} \right)^{2} \qquad(15)$

From equation $(11)$

$I_{max}= I_{1} + I_{2} + 2\sqrt{I_{1} \: I_{2}} \qquad(16)$

Destructive Interference:

For destructive interference the $cos\phi$ should be equal to -1 i.e. $cos\phi = -1$

Phase Difference in Destructive Interference:

We know that for destructive interference $cos\phi = -1$

$cos\phi = cos \left( \left( 2n \pm 1 \right)\pi \right)$

$\phi = \left( 2n \pm 1 \right)\pi \qquad(17)$

Path difference in Destructive Interference:

We know that the path difference

$\Delta x = \frac{\lambda}{2 \pi} \phi$

Now subtitute the value of $\phi $ from equation $(17)$ in the above equation

$\Delta x = \frac{\lambda}{2 \pi} \left( 2n \pm 1 \right)\pi$

$\Delta x = \left( 2n \pm 1 \right) \frac{\lambda}{2} \qquad(18)$

Resultant Amplitude due to Destructive Interference:

For the destructive interference, the resultant amplitude is minimum at point $P$. So from equation $(9)$

$ A_{min} = \sqrt{ a^{2}_{1} + a^{2}_{2} - 2a_{1} \: a_{2} } \qquad \left( \because cos\phi =-1 \right)$

$ A_{min} = \sqrt{ \left (a_{1} - a_{2} \right)^{2}} $

$ A_{min} = \left( a_{1} - a_{2} \right)$

For constructive interference, the resultant amplitude of the waves is the difference in amplitude of individual waves.

Resultant Intensity due to Destructive Interference:

For the destructive interference, the resultant intensity is minimum at point $P$. So from equation $(10)$

$I_{min}= a^{2}_{1} + a^{2}_{2} - 2a_{1} \: a_{2} \qquad \left( \because cos\phi =-1 \right)$

$I_{min}=\left( a_{1} - a_{2} \right)^{2}$

From equation $(11)$

$I_{min}= I_{1} + I_{2} - 2\sqrt{I_{1} \: I_{2}} $