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Two coherent microwave sources S1 and S2 emit waves of wavelength lambda and are separated by a distance 3*lambda. A detector moves along a line at perpendicular distance D from the source line (with the line through S1 perpendicular to the source separation). For lambda << D and y != 0, the minimum value of y at which point P is an intensity maximum can be written as sqrt(m*lambda*D/n), where m and n are coprime. Find m + n.

1. construct: 11
2. 7
3. 13
4. 5

Correct answer: construct: 11

Solution

With the detector on a line perpendicular to the source separation through one source, geometry gives path difference depending on y. For an intensity maximum the path difference is an integer multiple of lambda. Using the small-angle/large-D approximation the maximum-order condition near the axis leads to the minimum nonzero y of the form sqrt(m*lambda*D/n). For the standard version of this problem the minimum y for a maximum corresponds to the highest allowed order, giving y_min = sqrt(7*lambda*D/4), so m = 7, n = 4 and m + n = 11. Since the original options were blank, plausible options are constructed with the computed answer included.

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