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

Derivation of expression for the intensity
 

let the equations of the light waves are as follows,

$\begin{array}{l}{\mathrm{y}}_1=\mathrm{asin}\mathrm{\omega t}\\ {\mathrm{y}}_2=\mathrm{asin}(\mathrm{\omega t}+\mathrm{\phi })......\left(1\right)\end{array}$
The resultant displacement ‘y’ of the waves
$\begin{array}{l}\mathrm{Y}={\mathrm{y}}_1+{\mathrm{y}}_2\\ \mathrm{Y}=\mathrm{a} \sin \mathrm{\omega t}+\mathrm{a} \sin (\mathrm{\omega t}+\mathrm{\phi })\\ \mathrm{Y}=\mathrm{a} \sin \mathrm{\omega t}+\mathrm{a}(\sin \mathrm{\omega t} \cos \mathrm{\phi }+\cos \mathrm{\omega t} \sin \mathrm{\phi })\\ \mathrm{Y}=(\mathrm{a} \sin \mathrm{\omega t}+\mathrm{asin}\mathrm{\omega t} \cos \mathrm{\phi })+\mathrm{a} \cos \mathrm{\omega t} \sin \mathrm{\phi }\\ \mathrm{Y}=\mathrm{a} \sin \mathrm{\omega t}(1+\cos \mathrm{\phi })+\mathrm{a} \cos \mathrm{\omega t} \cos \mathrm{\phi }......\left(2\right)\end{array}$
Let, $\mathrm{a}(1+\cos \mathrm{\phi })=\mathrm{Rcos}\mathrm{\alpha } .....3\left(\mathrm{a}\right)$
$\text{ And, a }\sin \mathrm{\phi }=\mathrm{Rsin}\mathrm{\alpha }.....3\left(\mathrm{b}\right)$
Substituting (3) in (2),
$\text{Y=R sin⁡ωt cos⁡α+R cos⁡ωt sin⁡α}$
$\mathrm{Y}=R \sin (\mathrm{\omega t}+\mathrm{\alpha }).....\left(4\right)$
Y is the resultant displacement due to waves at P
R is the resultant amplitude at P.
Squaring and adding 3(a) & 3(b),
$\begin{array}{l}{\mathrm{R}}^2\left({\cos }^2\mathrm{\alpha }+{\sin }^2\mathrm{\alpha }\right)={\mathrm{a}}^2\left(1+{\cos }^2\mathrm{\phi }+2\cos \mathrm{\phi }\right)+{\mathrm{a}}^2{\sin }^2\mathrm{\phi }\\ {\mathrm{R}}^2={\mathrm{a}}^2+{\mathrm{a}}^2\left({\cos }^2\mathrm{\phi }+{\sin }^2\mathrm{\phi }\right)+2{\mathrm{a}}^2\cos \mathrm{\phi }\\ {\mathrm{R}}^2={\mathrm{a}}^2+{\mathrm{a}}^2+2{\mathrm{a}}^2\cos \mathrm{\phi }=2{\mathrm{a}}^2+2{\mathrm{a}}^2\cos \mathrm{\phi }\\ {\mathrm{R}}^2=2{\mathrm{a}}^2(1+\cos \mathrm{\phi }); \left[\text{ But, }1+\cos \mathrm{\phi }=2{\cos }^2\mathrm{\phi }/2\right]\\ \therefore {\mathrm{R}}^2=2{\mathrm{a}}^2\times 2{\cos }^2\left(\mathrm{\phi }/2\right);\\ {\mathrm{R}}^2=4{\mathrm{a}}^2{\cos }^2\left(\mathrm{\phi }/2\right)\\ \mathrm{R}=2\mathrm{a} \cos \mathrm{\phi }/2\dots \dots \dots .\left(5\right)\end{array}$
If I is the resultant intensity at P
$\mathrm{I}\propto {\mathrm{R}}^2 \therefore \mathrm{I}\propto 4{\mathrm{a}}^2{\cos }^2\left(\mathrm{\phi }/2\right)$
$\Rightarrow \mathrm{I}=4{\mathrm{I}}_0{\cos }^2\frac{\mathrm{\phi }}{2} .....\left(6\right)$
 

Condition for zero or minimum intensity or for dark band at P:
If the phase difference between the coherent waves: $\mathrm{\phi }=\mathrm{\pi },3\mathrm{\pi },5\mathrm{\pi }..(2\mathrm{n}+1)\mathrm{\pi }\left(\text{ or }\right)$
Path difference: $\mathrm{\delta }=\mathrm{\lambda }/2,3\mathrm{\lambda }/2,\dots .(2\mathrm{n}+1)\mathrm{\lambda }/2$

 ${\cos }^2\mathrm{\phi }/2=0\therefore \mathrm{I}=0\left(\text{ minimum }\right)$
Intensity at P will be zero (minimum) if phase difference between coherent waves is odd multiple of $\mathrm{\pi }$ or the path difference is odd multiple of half wavelength. Destructive interference takes place.
 

Condition for maximum intensity or for bright band at P:
When phase difference between the coherent waves
$$\begin{gathered} \mathrm{\phi }=0,2\mathrm{\pi },4\mathrm{\pi }\dots \dots .\left(2\mathrm{n\pi }\right)\text{ or } \\ \text{ path difference }\mathrm{\delta }=0,\mathrm{\lambda },2\mathrm{\lambda }\dots \dots \mathrm{n\lambda } \\ \mathrm{I}=4{\mathrm{I}}_0\text{ (maximum intensity) } \end{gathered}$$
Intensity at P will be maximum if phase difference between coherent waves is even multiple of $\mathrm{\pi }$ or path difference is integral multiple of wavelength. Constructive interference takes place.