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Three coherent point sources S1, S2, and S3 are placed along the same straight line and each produces intensity I0 at point P on this line. When only S1 and S2 are turned on together, the resultant intensity at P is 2*I0. When only S2 and S3 are turned on together, the resultant intensity at P is also 2*I0. Which of the following statements is/are correct?

1. When S1 and S3 are switched on together, the intensity at P can be 2*I0
2. When S1 and S3 are switched on together, the intensity at P can be 0
3. When all three sources are switched on together, the intensity at P can be 2*I0
4. When all three sources are switched on together, the intensity at P can be 3*I0

Correct answer: When S1 and S3 are switched on together, the intensity at P can be 0

Solution

Since each source gives I0 (amplitude A), the intensity from two sources is I = 2*I0 + 2*I0*cos(phi). For S1+S2 = 2*I0: cos(phi12) = 0 so phi12 = +/-90 deg. Similarly phi23 = +/-90 deg. The phase difference phi13 = phi12 + phi23. Depending on the sign combination, phi13 = 0 or 180 deg. When phi13 = 180 deg, S1 and S3 cancel completely, giving intensity 0. When phi13 = 0 they add constructively giving 4*I0. 2*I0 is not possible for S1+S3. For all three: if phi12=90 and phi23=90, phasors are at 0, 90, 180 deg; the resultant amplitude = A (only S2 contributes net), giving intensity I0, not 2*I0 or 3*I0.

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