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An interference is observed due to two c...

An interference is observed due to two coherent sources `S_1` placed at origin and `S_2` placed at `(0, 3lambda, 0)`. Here, lambda is the wavelength of the sources. A detector D is moved along the positive x-axis. Find x-coordinates on the x-axis (excluding `x = 0 and x= oo`) where maximum intensity is observed.

Text Solution

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At x=0, path difference is `3lambda`. Hence, third order maxima will be obtained. At `x=oo`, path difference is zero. Hence, zero order maxima is obtained. In between first and second order maximas will be obtained.
First order maxima : `S_(2)P-S_(1)P= lambda` (or)
`sqrt(x^(2)+9lambda^(2))-x= lambda`
or `sqrt(x^(2)+9lambda^(2))= x+ lambda` Squaring both sides, we get `x^(2)+9lambda^(2)= x^(2)+lambda^(2)+2x lambda` Solving this, we get `x= 4 lambda`
Second order maxima : `S_(2)P-S_(1)P= 2 lambda`
(or) `sqrt(x^(2)+9lambda^(2))- 2lambda" (or) "sqrt(x^(2)+9lambda^(2))= (x+2lambda)`
Squaring both sides, we get `x^(2)+9lambda^(2)= x^(2)+4lambda^(2)+ 4x lambda`
Solving, we get `x= (5)/(4)lambda= 1.25 lambda`
Hence, the desired x coordinates are,
`x= 1.25 lambda" and "x= 4lambda`.
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