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Question

Two coherent waves, coming from sources at different locations, move along the x axis. Their wave functions are $$E_1 = 860 \sin \left [ \frac{2 \pi x_1}{650} - 924 \pi t + \frac{\pi}{6} \right ]$$
and $$E_1 = 860 \sin \left [ \frac{2 \pi x_1}{650} - 924 \pi t + \frac{\pi}{6} \right ]$$ where $$E_1$$ and $$E_2$$ are in volts per meter, $$x_1$$ and $$x_2$$ are in nanometers, and t is in picoseconds. When the two waves are superposed, determine the relationship between $$x_1$$ and $$x_2$$ that produces constructive interference.


Solution

Constructive interference occurs where the phases of the waves differ by integral multiples m of $$2 \pi$$:
$$\left ( \frac{2 \pi x_1}{650} - 924 \pi t + \frac{\pi}{6} \right ) - \left ( \frac{2 \pi x_2}{650} - 924 \pi t + \frac{\pi}{8} \right ) = 2 \pi m$$
which becomes
$$\frac{2 \pi (x_1 - x_2)}{650} + \left ( \frac{\pi}{6} - \frac{\pi}{8} \right ) = 2 \pi m$$
$$\frac{(x_1 - x_2)}{650} + \frac{1}{12} - \frac{1}{16} = m$$
$$x_1 - x_2 = (m - \frac{1}{48}) 650$$, where $$x_1$$ and $$x_2$$ are in nanometers and $$m = 0, 1, -1, 2, -2, 3, -3, ....$$

Physics
NCERT
Standard XII

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