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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
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B
22
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C
24
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D
26
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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 $$20$$.
728043_669997_ans_a5b2b51115d644e89ae0242c7904c84e.jpg

Physics

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