In the double-slit experiment of Fig. 35-10, the viewing screen is at distance $\mathit{D}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{\text{m}}$, point P lies at distance role="math" localid="1663143982922" $\mathit{y}\mathbf{=}\mathbf{20}\mathbf{.}\mathbf{5}\mathbf{\text{cm}}$from the center of the pattern, the slit separation d is $\mathbf{4}\mathbf{.}\mathbf{50}\mathbf{\text{mm}}$, and the wavelength $\mathit{\lambda }$is $\mathbf{\text{580 nm}}$. (a) Determine where point P is in the interference pattern by giving the maximum or minimum on which it lies, or the maximum and minimum between which it lies. (b) What is the ratio of the intensity${\mathit{l}}_{\mathbf{P}}$at point P to the intensity${\mathit{l}}_{\mathbf{c}\mathbf{e}\mathbf{n}}$ at the centerof the pattern?

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 $d=4.50\mathrm{\mu m}$

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

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:

$\tan \theta =\frac{y}{D}$

Substitute all the values in the equation.

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

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

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:

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

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

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

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 $0.101$.