  Question

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

A
20  B
22  C
24  D
26  Solution

The correct option is A $$20$$Path difference at $$P$$ is$$\Delta x=2\left( \dfrac { x }{ 2 } \cos { \theta } \right) =x\cos { \theta }$$For intensity to be maximum$$\Delta x=n\lambda$$             $$\left( n=0,1,\dots \right)$$$$\Rightarrow x\cos { \theta } =n\lambda$$$$\cos { \theta } =\dfrac { n\lambda }{ x }$$ or $$\cos { \theta } \ngtr 1$$$$\therefore \dfrac { n\lambda }{ x } \ngtr 1$$$$\therefore n\ngtr \dfrac { x }{ \lambda }$$Substituting $$x=5\lambda$$$$n\ngtr 5$$ or $$n=1,2,3,4,5,\dots$$Therefore, in all four quadrants, there can be $$2$$ maximas. There are more maximas at $$\theta ={ 0 }^{ o }$$ and $$\theta ={ 180 }^{ o }$$.But $$n=5$$ corresponds to $$\theta ={ 90 }^{ o }$$ and $$\theta ={ 270 }^{ o }$$ which are coming only twice. While we have multiplied it four times. Therefore, total number of maximas still $$20$$. Physics

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