[2013EM/HMW-13]

PHY 102: Introduction to Physics-2
Tutorial-13
(Wave optics, interference)

1. Intensity distribution in interference: A point source $S$ located at the origin of a coordinate system emits a spherical sinusoidal wave in which the electric field $E_1$ is given by $$ E_1=A\left({D\over r}\right)\cos\left(\omega t-{2\pi r\over\lambda}\right) $$ where $r$ is the distance from $S$ (see figure below). In addition, there is a plane wave propagating along the $x$-axis. This wave is given by $$ E_2=A\cos\left(\omega t-{2\pi x\over\lambda}\right) $$ (Note: we treat both $E_1$ and $E_2$ as scalar waves in this problem). Both waves are incident on a flat screen as shown. Compute the resultant intensity $I$ at the screen as a function of the distance $y$ from the $x$-axis for values of $y$ small compared with $D$. Express $I$ in terms of $y$, $D$, $\lambda$ and the intensity $I_0$ at $y=0$. 
2. Double slit interference: (a)~White light falls at an angle of 30$^\circ$ to the normal of a plane containing a pair of slits separated by 2.50 $\mu$m. What visible wavelengths give a bright interference maximum in the transmitted light in the direction normal to the plane? (b)~Using a conventional two-slit apparatus and light that has a 589-nm wavelength, 28 bright fringes per centimeter are observed near the center of a screen 3.00 m away. What is the slit separation?
3. Newton's ring: A Newton's ring apparatus consists of a plano-convex glass lens with radius of curvature $R$ that rests on a flat glass plate. The think film is air of variable thickness. The apparatus is illuminated from above by light from a sodium lamp that has a wavelength of 590 nm. (a)~Show that for $t\ll R$, the radius $r$ of a fringe is related to film thickness $t$ by $r=(2tR)^{1/2}$. (b)~For a radius of curvature of 10.0 m and a lens diameter of 4.00 cm. How many bright fringes would you see in the reflected light? (c)~What would be the diameter of the sixth bright fringe? (d)~ If the glass used in the apparatus has n index of refraction $n=1.5$ and water replaces the air between the two pieces of glass, explain qualitatively the changes that will take place in the bright-fringe pattern.