Answer to Question #146733 in Optics for Promise Omiponle

Question

Two synchronized microwave sources, A and B, are emitting waves of wavelength 4.0 cm. Source A is located on the x-axis at x = 0, source B can be moved along the x-axis, and a microwave receiver is fixed on the x-axis at x = 84 cm. For each of the following positions of source B determine whether the waves from the sources will reach the receiver in phase (constructive interference), out of phase (destructive interference), or with a phase difference of π/2 rad.

(a) x = -8 cm.
(b) x = -6 cm.
(c) x = -5 cm.
(d) x = -4 cm.
(e) x = -3 cm.
(f) x = -2 cm.

Answer 1

Path difference:

$∆x = x_A - x_B = 0 - x$

(a) x = -8 cm

∆x = 0 - (-8) = 8 cm = 2λ

∆x = 2λ (same phase)

Result – constructive interference

(b) x = -6 cm

∆x = 0 - (-6) = 6 cm = 1.5λ

$∆x = (1 + \frac{1}{2})λ$ (out of phase)

Result – destructive interference

(c) x = -5 cm

$∆x = 0 - (-5) = 5 \; cm = \frac{5}{4}λ$

∆x = 1.25λ

Phase difference $= 1.25 \times 2π = 2.5π$

or phase difference = 0.5π = π/2

Result – phase difference

d) x = -4 cm

∆x = 0 - (-4) = 4 cm = λ

∆x = λ (same phase)

Result – constructive interference

(e) x = -3 cm

∆x = 0 - (-3) = 3 cm $= \frac{3}{4}λ$

∆x = 0.75λ

Phase difference $= 0.75 \times 2π = 1.5π$

Result – phase difference

(f) x = -2 cm

∆x = 0 - (-2) = 2 cm = 0.5λ

∆x =0.5λ (out of phase)

Result – destructive interference