Question

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?

Short answer

(a) The point P is lying between the central maximum and first minimum.

(b) The ratio of intensity at point P to the intensity at the center of the interference pattern is $0.101$.

Step-by-step solution

Identification of given data

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$\lambda =580\text{nm}$

Understanding the concept

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

(a) Determination of the position of point P between maximum and minimum

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.

(b) Determination of ratio of intensity at point P to intensity at center of interference pattern

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 $0.101$.