Two coherent light sources A and B are at a d | vedclass

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Question

(B) Let the position of source $A$ be $(0, 0)$ and source $B$ be $(0, 3\lambda)$ on the $y$-axis.Let a point $P$ on the $x$-axis be at a distance $x$

Vedclass · Accepted Answer

$4\lambda$

Vedclass · Answer

(B) Let the position of source $A$ be $(0, 0)$ and source $B$ be $(0, 3\lambda)$ on the $y$-axis.Let a point $P$ on the $x$-axis be at a distance $x$ from $A$. The coordinates of $P$ are $(x, 0)$.The distance $AP = x$.The distance $BP = \sqrt{x^2 + (3\lambda)^2}$.For constructive interference, the path difference $\Delta p = |BP - AP| = n\lambda$, where $n = 0, 1, 2, \dots$.$\sqrt{x^2 + 9\lambda^2} - x = n\lambda$.$\sqrt{x^2 + 9\lambda^2} = x + n\lambda$.Squaring both sides: $x^2 + 9\lambda^2 = x^2 + 2nx\lambda + n^2\lambda^2$.$9\lambda^2 - n^2\lambda^2 = 2nx\lambda$.$x = \frac{(9 - n^2)\lambda}{2n}$.For $n = 1$: $x = \frac{(9 - 1)\lambda}{2} = 4\lambda$.For $n = 2$: $x = \frac{(9 - 4)\lambda}{4} = 1.25\lambda$.For $n = 3$: $x = 0$ (This is the point between the sources, but the question asks for points on the $x$-axis).Comparing with the given options, $4\lambda$ is a valid distance.

Two coherent light sources $A$ and $B$ are at a distance $3\lambda$ from each other (where $\lambda$ is the wavelength). Find the distances from $A$ on the $x$-axis at which constructive interference occurs.

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