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(D) The distance between the two coherent sources is $d = \frac{5a}{2} - (-\frac{5a}{2}) = 5a$.For constructive interference (maxima), the path differ

Vedclass · Accepted Answer

$14$

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(D) The distance between the two coherent sources is $d = \frac{5a}{2} - (-\frac{5a}{2}) = 5a$.For constructive interference (maxima), the path difference $\Delta x$ must be an integer multiple of the wavelength $\lambda$, i.e., $\Delta x = n\lambda$, where $n$ is an integer.The path difference at any point on the circle is given by $\Delta x = d \cos 	heta$, where $	heta$ is the angle with the $x$-axis.Since the circle is of large radius, $	heta$ varies from $0$ to $2\pi$.The condition for maxima is $d \cos 	heta = n\lambda$.Substituting the values, $5a \cos 	heta = n(\frac{4a}{3}) \implies \cos 	heta = \frac{4n}{15}$.Since $-1 \le \cos 	heta \le 1$, we have $-1 \le \frac{4n}{15} \le 1$, which gives $-3.75 \le n \le 3.75$.The possible integer values for $n$ are $n = 0, \pm 1, \pm 2, \pm 3$.For $n = 0$, $\cos 	heta = 0$, which gives $2$ points $(	heta = \frac{\pi}{2}, \frac{3\pi}{2})$.For $n = \pm 1, \pm 2, \pm 3$, each value of $n$ gives $2$ points on the circle (one in the upper half and one in the lower half).Total number of maxima = $2 + 2 	imes (3 + 3) = 2 + 12 = 14$.

Two coherent sources of light are placed at points $(-\frac{5a}{2}, 0)$ and $(+\frac{5a}{2}, 0)$. The wavelength of the light is $\lambda = \frac{4a}{3}$. How many maxima will be obtained on a planar circle of large radius with its center at the origin?

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