91Ó°ÊÓ

The distance of point P from the center pattern is $y=20.5\text{cm}$

The distance of the screen from the double slit is $D=4\text{m}$

The slit separation for double slit is $d=4.50\mathrm{μ³¾}$

The wavelength of light is$\mathrm{λ}=580\text{nm}$

The phase difference of the fringe pattern varies with the path difference. For minimum phase difference path difference should be minimum and vice versa.

The angular position of point P is given as:

$\tan \mathrm{θ}=\frac{y}{D}$

Substitute all the values in the equation.

$\tan \mathrm{θ}=\frac{\left(20.5\text{cm}\right)\left(\frac{1\text{m}}{100\text{cm}}\right)}{4\text{m}}$

role="math" localid="1663145625550" $\begin{array}{rcl}\mathrm{θ}& =& {\text{tan}}^{-1}\left(\text{0.05125}\right)\\ & =& 2.934°\end{array}$

The phase difference for point P is given as

The value of phase difference for point P is lying between 0 and 0.5 which means pointP is lying between center of pattern and first minimum.

Therefore, the point P is lying between central maximum and first minimum.

The phase difference of point p in degree is converted as:

$\begin{array}{rcl}\mathrm{ϕ}& =& 2\mathrm{π}\left(0.397°\right)\left(\frac{180°}{\mathrm{π}}\right)\\ & =& 142.92°\end{array}$

The ratio of intensity at point P to intensity at center of interference pattern is given as:

$\frac{I_P}{I_{cen}}={\cos }^2\left(\frac{\mathrm{Ï•}}{2}\right)$

Substitute all the values in equation.

$\begin{array}{rcl}\frac{I_P}{I_{cen}}& =& {\cos }^2\left(\frac{142.92°}{2}\right)\\ & =& {\cos }^2\left(71.46°\right)\\ & =& 0.101\end{array}$

Therefore, theratio of intensity at point P to intensity at center of interference pattern is $0.101$.