question_answer

Two identical coherent sources are placed on a diameter of a circle of radius R at separation x (<< R) symmetrical about the center of the circle. The sources emit identical wavelength\[\lambda \] each. The number of points on the circle of maximum intersity is\[(x=5\lambda )\]    

A) 20

B) 22   

C) 24

D) 26

Correct Answer: A

Solution :

[a] Path difference at P is \[\Delta x=2\left( \frac{x}{2}\cos \theta  \right)=x\cos \theta \] For intensity to be maximum, \[\Delta x=n\lambda \]    (n=0, 1, 2, 3, ....)    or\[x\cos \theta =n\lambda \]   or \[x\,\cos \,\theta =\frac{n\lambda }{x}\ge 1\] \[\therefore \,\,\,n\ge \frac{x}{\lambda }\] Subsituting x= 51, we get \[n\ge 5\]or n= 1, 2, 3, 4, 5....... Therefore in all four quadrants there can be 20 maxima. There are more maxima at \[\theta =0{}^\circ \]and\[\theta =180{}^\circ \]. But n = 5 corresponds to \[\theta =90{}^\circ \] and \[\theta =270{}^\circ \] which are coming only twice while we have multuplies it four times. Therefore, total number of maxima are still 20, i.e., n = 1 to 4 in four quadrants (total 16) plus more at \[\theta =0{}^\circ \], \[90{}^\circ \], \[180{}^\circ \]and\[270{}^\circ \].