Question 69: Waves Optics

Question

In the Young's double slit experiment the intensity produced by each one of the individual slits is $I_{\mathrm{o}}$. The distance between two slits is 2 mm . The distance of screen from slits is 10 m . The wavelength of light is $6000 \mathrm{~A}^{\circ}$. The intensity of light on the screen in front of one of the slits is $\_\_\_\_$

Accepted Answer

The intensity from each slit is equal, $\mathrm{I}_1=\mathrm{I}_2=\mathrm{I}_0$
The slit width is equal to distance between slits $d=2 \mathrm{~mm}=2 	imes 10^{-3} \mathrm{~m}$
Distance of screen from slits $\mathrm{D}=10 \mathrm{~m}$
Wavelength of light $\lambda=6000 \mathop A\limits^o =6 	imes 10^{-7} \mathrm{~m}$
The distance from the central maximum to a point directly in front of one slit is :
$$ \mathrm{y}=\frac{\mathrm{d}}{2} $$
The path difference for a point at height $y$ on the screen is given by the formula :
$$ \Delta \mathrm{x}=\frac{\mathrm{dy}}{\mathrm{D}} $$
$$\Rightarrow $$ $\Delta \mathrm{x}=\frac{\mathrm{d}\left(\frac{\mathrm{d}}{2}ight)}{\mathrm{D}}=\frac{\mathrm{d}^2}{2 \mathrm{D}}$
$$ \Delta \mathrm{x}=\frac{\left(2 	imes 10^{-3}ight)^2}{2 	imes 10}=\frac{4 	imes 10^{-6}}{20}=2 	imes 10^{-7} \mathrm{~m} $$
The relation between path difference and phase difference is :
$\phi=\frac{2 \pi}{\lambda} 	imes \Delta x$
$$\Rightarrow $$ $$ \phi=\frac{2 \pi}{6 	imes 10^{-7}} 	imes\left(2 	imes 10^{-7}ight)=\frac{4 \pi}{6}=\frac{2 \pi}{3} $$
The formula for resultant intensity when two waves of equal intensity $\mathrm{I}_0$ interfere is :
$$ \mathrm{I}=4 \mathrm{I}_0 \cos ^2\left(\frac{\phi}{2}ight) $$
$$\Rightarrow $$ $I=4 I_0 \cos ^2\left(\frac{\frac{2 \pi}{3}}{2}ight)=4 I_0 \cos ^2\left(\frac{\pi}{3}ight)$
$$\Rightarrow $$ $\mathrm{I}=4 \mathrm{I}_0\left(\frac{1}{2}ight)^2=4 \mathrm{I}_0 	imes \frac{1}{4}$
$$\Rightarrow $$ $$ \mathrm{I}=\mathrm{I}_0 $$
The intensity of light on the screen directly in front of one of the slits is equal to the intensity of a single slit, i.e., $\mathrm{I}_0$.
Hence, the correct option is (B).

Answer

$\frac{I_o}{2}$

Answer

$I_{\mathrm{o}}$

Answer

$2 I_{\mathrm{o}}$

Answer

$4 I_{\mathrm{o}}$

Waves Optics

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