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Question

Two points sources separated by 2.0 m are radiating in phase with λ=0.50 m. A detector moves in a circular path around the two sources in a plane containing them. How many maxima are detected ?


A
16
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B
20
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C
24
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D
32
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Solution

The correct option is A 16

Condition of path difference for maxima is given by

Δx=nλ

According to the figure shown, path difference at any point is given by,

Δx=S1P=dcosθ

dcosθ=nλ .......(1)

cosθ=nλd=n(0.502.0)=0.25n

As cosθ lies between 1 and 1, so we wish to find all values of n for which 1(0.25n)1.

So, corresponding n values are 4,3,2,1,0,+1,+2,+3,+4.

For each of these, there are two different values of θ except for 4 and +4.

A single value of θ, 900 and +900, is associated with n=4 and n=+4, respectively. Thus, there are 16 different angles in all which satisfies equation (1), and therefore 16 maxima will be detected by the detector.

Hence, option (A) is correct.
Alternate solution:
At the end of vertical diameter, zeroth maxima (Δx=0) is obtained, and at the end of horizontal diameter 4th maxima (Δx=4λ) is obtained. Hence, there are three maxima in each quadrant and another four maxima at the ends of horizontal and vertical diameters.
Therefore, total maxima =(3×4)+4=16.

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