Question

$\huge \sf{Question:}$
(a) (i) “Two independent monochromatic sources of light cannot produce a sustained interference pattern”. Give reason.

(ii) Light waves each of amplitude “a” and frequency “ω”, emanating from two coherent light sources superpose at a point. If the displacements due to these waves is given by y1 = a cos ωt and y2 = a cos(ωt + ϕ ) where ϕ is the phase difference between the two, obtain the expression for the resultant intensity at the point.


(b) In Young’s double slit experiment, using monochromatic light of wavelength λ, the intensity of light at a point on the screen where path difference is λ, is K units. Find out the intensity of light at a point where path difference is λ/3.

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Answer 1

$\ \huge \sf{solution}$

(a)(i) The condition for the sustained interference is that both the sources must be coherent (i.e. they must have the same wavelength and the same frequency, and they must have the same phase or constant phase difference). Two sources are monochromatic if they have the same frequency and wavelength. Since they are independent, i.e. they have different phases with irregular difference, they are not coherent sources.

Or

Two independent monochromatic sources of light cannot produce a sustained interference pattern. The phase difference between these two sources will continuously vary; and the positions of maxima and minima will change with time.

$\sf{ii)} \: \: \: \: \: \: \: \sf{y_1 = a \cos \: wt \: and \: y_2 = a \cos(wt + o) } \\ \\ \sf \: y = y_1 + y_2 = a( \cos \: wt + \cos(wt + o) \\ \\ \sf = 2a \cos( \frac{o}{2} ) \cos((wt + \frac{o}{2} )) \\ \\ \sf{ The \: resultant \: amplitude \: is \: a = 2a \cos( \frac{o}{2} ) } \\ \sf \: intensity(i) = 4 {a}^{2} \cos {}^{2} ( \frac{o}{2} ) \\ \sf = 4I_0 \cos {}^{2} ( \frac{o}{2} )$

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$\sf \:(iii) \: \: \: \: \: \: I=I_1+I_2 + 2 \sqrt{I_1I_2} \cos(o) \\ \sf \: let \: I_0 \: be \: the \: intensity \: of \: either \: source \: then \: \\ \sf \: I = I_2 = I_0 \: \: \: \: \: \: \: and \: \\ \sf \: I = 2I_0(1 + \cos(o) = 4I_0 \cos {}^{2} ( \frac{o}{2} ) \\ \\ \sf \: when \: p = lemda \: \: \: \: \: \: \: \: \: \: \: \: \pi =2 \pi \\ \\ \sf \: I = 4I_0 \cos {}^{2} ( \frac{o}{22} ) = 4I_0 \cos {}^{2} (\pi) = 4I_0 = k \\ \sf \: when \: p = \frac{lemda}{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: o = \frac{2\pi}{3} \\ \sf \: I = 4I_0 \cos {}^{2} ( \frac{\pi}{3} ) = 4I_0 \times \frac{1}{4} = I_0 = \frac{k}{4}$

$\: \: \: \: \: \: \: \: \: \: \huge \sf{@WildCat7083 }$