The maximum intensity in Young's double slit experiment is \${I}_{0}\$. Distance between the slits is \$d=5\lambda\$, where \$\lambda\$ is the wavelength of monochromatic light used in the experiment. What will be the intensity of light in front of one of the slits on a screen at a distance \$D=10d\$?

Path difference \$=\Delta x=\dfrac{xd}{D}\$-------(i)In front of one of the slits,\$x=\dfrac{d}{2}\$ but \$d=5\lambda\$\$x=\dfrac{5\lambda}{2}\$ and \$D=10d=50\lambda\$Now, from equation (i), we have:\$\Delta x=\dfrac{5\lambda}{2}\times\dfrac{5\lambda}{50\lambda}=\dfrac{\lambda}{4}\$So, corresponding phas defference:\$\phi=\dfrac{2\pi}{\lambda}.(\Delta x)=\dfrac{2\pi}{\lambda}\times\dfrac{\lambda}{4}=\dfrac{\pi}{2}\$As, \$I=I_0\cos^2(\dfrac{\phi}{2})\$So, \$I=I_0\cos^2(\dfrac{\pi}{4})=\dfrac{I_0}{2}\$

Class 12 Wave Optics - Young's Double Slit Experiment

Q: In a double slit experiment, the two slits are 1 mm apart and the screen is place 1 m away. A monochromatic light of wavelength 500 nm is used. What will be the width of each slit for obtaining ten maxima of double slit within the central maxima of single slit pattern?

\$d = 10^{-3}\:m\$\$D=1\:m\$\$ \lambda = 500 \times 10^{-9}\:m\$The width of central maxima in a single slit diffraction: \$\dfrac{2 \lambda D}{a}\$Fringe width in double slit pattern: \$\beta = \dfrac{\lambda D}{d}\$Given,\$10 \beta = \dfrac{2 \lambda D}{a}\$\$ \Rightarrow 10 \dfrac{\lambda D}{d} = \dfrac{2 \lambda D}{a}\$\$ \Rightarrow a =\dfrac{d}{5}\:mm= 0.2\:mm\$

Q: A beam of light consisting of two wavelengths, \$650 nm\$ and \$520 nm\$, is used to obtain interference fringes in a Young's double-slit experiment.(a) Find the distance of the third bright fringe on the screen from the central maximum for wavelength \$650 nm\$.(b) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide? The distance between the two slits is \$0.28mm\$ and the screen is at a distance of \$1.4m\$ from the slits.

GivenThe wavelength of the light is \$ \lambda_1 = 650 nm\$The wavelength of seond light,\$ \lambda_2 = 520 nm\$Distance between the slit and the screen is \$1.4m\$.Distance between the slits is \$0.28mm\$.\$(a)\$The relation between the \$n^{th}\$ bright fringe and the width of fringe is:\$x = n\lambda_1\dfrac Dd\$For third bright fringe, \$n = 3\$\$ x = 3 \times 650\dfrac {1.4}{0.28\times10^{-3}}= 1950 \times5\times10^3 nm\$\$x=9.75\times10^{-3} m\$\$=9.75mm\$\$(b)\$We can consider that \$n^{th}\$ bright frind of \$\lambda_2\$ and the \$(n-1)^{th}\$ bright fringe of wavelength \$\lambda_1\$ coincide with each other.\$n\lambda_2 = (n - 1) \lambda_1\$\$520n = 650n - 650\$\$650 = 130n\$\$ n = 5\$therefore, the least distance from the central maximum can be obtained as:\$x' = n\lambda_2 \dfrac Dd \$\$x'= 5 \times 520 \dfrac Dd = 2600 \dfrac {1.4}{0.28\times10^{-3}} nm\$\$x'=1.30\times10^{-2}m=1.3cm\$

Q: Two coherent monochromatic light beams of intensities \$I\$ and \$4I\$ are superposed. The maximum and minimum possible resulting intensities are :

Maximum intensity, \$I_{max} = (\sqrt {I_{1}} + \sqrt {I_{2}})^{2}\$\$= (\sqrt {I} + \sqrt {4I})^{2}\$\$= 9I\$Minimum intensity \$I_{min} = (\sqrt {I_{1}} - \sqrt {I_{2}})^{2}\$\$(\sqrt {I} - \sqrt {4I})^{2}\$\$= I\$

Q: In a Youngs double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used in the experiment.

Distance between the slits, \$d=0.28\times10^{-3}\ m\$Distance between the slits and the screen, \$D = 1.4 m\$Distance between the central fringe and the fourth \$(n = 4)\$ fringe, \$u=1.2\times10^{-2}\ m\$In case of a constructive interference, we have the relation for the distance between the two fringes as:\$u=n\lambda D/d\$\$\Rightarrow \lambda=ud/nD=6\times10^{-7}m=600nm\$

Q: The interference pattern is obtained with two coherent light source of intensity ratio n. In the interference pattern, the ratio \$\dfrac{I_{max}-I_{min}}{I_{max}+I_{min}}\$ will be

Given : \$I_2 = n I_1\$Maximum intensity of interference \$I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2\$\$\therefore\$ \$I_{max} = (\sqrt{I_1} + \sqrt{nI_1})^2 = (1+\sqrt{n})^2 I_1 = (1+ n +2\sqrt{n } ) I_1\$Minimum intensity of interference \$I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2\$\$\therefore\$ \$I_{min} = (\sqrt{I_1} - \sqrt{nI_1})^2 = (1-\sqrt{n})^2 I_1 = (1+ n - 2\sqrt{n})I_1\$\$\therefore\$ \$\dfrac{I_{max} -I_{min}}{I_{max} + I_{min}} = \dfrac{2\sqrt{n} - (-2\sqrt{n})}{2(1+n) } = \dfrac{2\sqrt{n}}{1+n}\$

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

Let \$y_1\$ and \$y_2\$ be displacement of two waves having same amplitude a and phase difference \$\phi\$ between them. \$y_1=a\sin\omega t\$\$y_2=a\sin(\omega t+\phi)\$Resultant displacement is: \$y=y_1+y_2\$\$y=a\sin\omega t+a\sin(\omega t+\phi)=a\sin\omega t(1+\cos\phi)+\cos\omega t(a\sin\phi) \$\$R\cos \theta=a(1+\cos\phi)\$\$R\sin \theta=a\sin\phi\$\$y=R\sin(\omega t +\theta)\$Where, \$R\$ is resultant amplitude at \$P\$, \$I\$ is intensity, squaring the equations we get,\$I=R^2=a^2(1+\cos\phi)^2+a^2(\sin\phi)^2=2a^2(1+\cos\phi)=4a^2\cos^2\dfrac{\phi}{2}\$Maximum intensity:\$\cos^2\dfrac{\phi}{2}=1\$\$\phi=2n\pi\$ where \$n=0,1,2,3,\$....Therefore, \$I_{max}=4a^2\$Minmum intensity:\$\cos^2\dfrac{\phi}{2}=0\$\$\phi=(2n+1)\pi\$ where \$n=0,1,2,3,\$....Therefore, \$I_{max}=0\$

Q: A parallel beam of light of wavelength 600 nm is incident normally on a slit of width d. If the distance between the slits and the screen is 0.8 m and the distance of 2nd order maximum from the center of the screen is 15 mm. The width of the slit is

Distance of \$2^{nd}\$ order maximum form the centre of the screen \$x \, = \, \dfrac{5}{2} \, \dfrac{D\lambda}{d}\$Here, \$D \, = \, 0.8 \, m, \, x \, = \, 15 \, mm \, = \, 15 \, \times \, 10^{-3} \, m\$\$\lambda \, = \, 600 \, nm \, = \, 600 \, nm \, = \, 600 \, \times \, 10^{-9} \, m\$\$\therefore \, d \, = \, \dfrac{5}{2} \cdot \, \dfrac{D \lambda}{x} \, = \, \dfrac{5}{2} \, \times \, \dfrac{0.8 \, \times \, 600 \, \times \, 10^{-9}}{15 \, \times \, 10^{-3}} \, = \, 80 \, \mu m\$