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Two identical coherent sources placed on a diameter of a circle of radius \mathrm{R} at separation \mathrm{\mathrm{x}(<<\mathrm{R})} symmetrically about the centre of the circle. The sources emit identical wavelength \mathrm{\lambda} each. The number of points on the circle with maximum intensity is \mathrm{(\mathrm{x}=5 \lambda)} :

Option: 1

24


Option: 2

20


Option: 3

22


Option: 4

26


Answers (1)

best_answer

As is clear from figure, path difference between the light waves reaching P from \mathrm{S_1} and \mathrm{S_2} is

\mathrm{ \Delta x=2\left(\frac{x}{2} \cos \theta\right)=x \cos \theta }
For intensity to be maximum at P,

Path difference \mathrm{\Delta \mathrm{x}=\mathrm{n} \lambda}, where \mathrm{ \mathrm{n}=0,1,2, \ldots.}
\mathrm{ \mathrm{x} \cos \theta=\mathrm{n} \lambda \quad \text { or } \quad \cos \theta=\frac{\mathrm{n} \lambda}{\mathrm{x}} }
As \mathrm{ \cos \theta}cannot be greater than one, therefore,

\mathrm{ \frac{\mathrm{n} \lambda}{\mathrm{x}} \ngtr 1 \quad \text { or } \quad \mathrm{n} \ngtr \frac{\mathrm{x}}{\lambda} }
As \mathrm{ \mathrm{x}=5 \lambda}, therefore, \mathrm{\mathrm{n} \ngtr \frac{5 \lambda}{\lambda}}

\mathrm{i.e., \mathrm{n} \ngtr 5 \, \, or \, \, \mathrm{n}=1,2,3,4,5.}
Therefore, in all the four quadrants, there can be 20 maxima.

 

Posted by

Rishi

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