Practicing Success
In a double slit experiment, Io is the intensity of the central bright fringe obtained with monochromatic light of $λ = 6000Å$. Determine the intensity at a distance of $4.8×10^{−5}m$ from the central maximum if the separation between the slits is 0.25cm and the distance between the screen and the double slit is 1.20m. |
$0.25I_o$ $0.50I_o$ $0.75I_o$ $I_o$ |
$0.75I_o$ |
From the principle of superposition the resultant intensity $I = I_1 + I_2 + 2\sqrt{I_1I_2} \cos δ$, where $δ$ = phase difference = $\frac{2π}{λ}$ × path difference. At the central fringe, $δ = 0$, thus $I_0 = I_1+I_2+2I_1 = 4I_1$ At a distance x from the central bright fringe, the path difference = $\frac{xd}{D}$ and the corresponding phase difference $δ =\frac{2π}{λ}\frac{xd}{D}$ Hence the resultant intensity at the assigned position will be $I = I_1 + I_2 + 2\sqrt{I_1I_2} \cos δ =\frac{I_o}{4}+\frac{I_o}{4}+\frac{2I_o}{4}\cos δ$ $\frac{I_o}{4}[1+1+2\cos δ]=\frac{I_o}{2}[1+\cos δ]=I_o \cos^2(\frac{δ}{2})$ Here, $\frac{δ}{2}=\frac{1}{2}\frac{2π}{λ}\frac{xd}{D}=\frac{π}{6}$ Thus $I=\frac{I_o}{2}×(\frac{\sqrt{3}}{2})^2=\frac{3I_o}{4}=0.75I_o$ |