Two identical sources S1 and S2, separated by distance d, coherently emit light of wavelength λu

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Two identical sources S1 and S2, separated by distance d,...

Two identical sources $\mathrm{S}_{1}$ and $\mathrm{S}_{2},$ separated by distance $d,$ coherently emit light of wavelength $\lambda$ uniformly in all direc- tions. Defining the $x$ axis with its origin at $S_{1}$ as shown in Fig. $32,$ find the locations (expressed as multiples of $\lambda )$ where the signals from the two sources are out of phase along this axis for $x>0$ , if $d=3 \lambda.$

Two identical sources $\mathrm{S}_{1}$ and $\mathrm{S}_{2},$ separated by distance $d,$
coherently emit light of wavelength $\lambda$ uniformly in all direc-
tions. Defining the $x$ axis with its origin at $S_{1}$ as shown in
Fig. $32,$ find the locations (expressed as multiples of $\lambda )$
where the signals from the two sources are out of phase
along this axis for $x>0$ , if $d=3 \lambda.$

Key Concepts

Interference Patterns

Interference patterns form when two or more waves superimpose. Constructive interference occurs when the phase difference is a multiple of 2? (or integer multiples of the wavelength), leading to amplitude reinforcement, while destructive interference occurs when the waves are out of phase, typically by ? radians, leading to cancellation of the wave amplitudes.

Destructive Interference

Destructive interference, or the condition in which waves are out of phase, occurs when the path difference between the waves is such that it equals an odd multiple of half the wavelength. This results in a phase difference of ?, causing the wave peaks of one source to coincide with the troughs of another, thus canceling each other out.

Coherent Sources

Coherent sources are those that produce waves with a constant phase relation, identical frequency, and wavelength. This property allows their emitted waves to interfere predictably because any phase differences at a given point are solely due to differences in the path lengths traveled by the waves.

Phase Difference and Path Difference

The phase difference between two interfering waves at a point depends directly on the difference in the distances from each source to that point; this is known as the path difference. A path difference corresponding to a half wavelength, or an odd multiple of a half wavelength, results in a phase difference of ? radian (180°).

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