Q.Describe Young's double slit interference experiment and derive conditions for occurrence of dark and bright fringes on the screen. Define fringe width and derive a formula for it.

[!TLDR] Two coherent slits $S_1,S_2$ (separation d) illuminated by a plane wavefront give path difference $\approx yd/D$ at screen height y; bright fringes at $y_n=n\lambda D/d$, dark fringes at $y_n=(n-\tfrac12)\lambda D/d$, both equally spaced with fringe width $\beta=\lambda D/d$. [!ANSWER] Bright: $\Delta l=n\lambda \R …

In Young's double slit experiment, a plane wavefront is made to fall on an opaque screen AB pierced by two narrow, identical, closely-spaced (a few mm apart) parallel slits $S_1$ and $S_2$, with their lengths perpendicular to the plane of the diagram; a linear source S (or a source at the focus of a converging lens) supplies the plane wavefront, and $S_1,S_2$ are taken equidistant from S so both receive the light in phase. An observing screen PQ is placed behind AB, at distance D. Reaching $S_1$ and $S_2$ in phase, both slits act as coherent secondary sources emitting cylindrical wavelets that overlap and interfere in the region beyond AB, producing alternating bright and dark FRINGES on PQ.

To locate the fringes: set x along the original propagation direction, with the screen along the y-z plane; O is the midpoint of $S_1S_2$ (separation d) and O' the point on the screen directly opposite O, at distance D (with $D\gg d$). For a general screen point P at height y above O', $(S_2P)^2-(S_1P)^2=\left[D^2+(y+d/2)^2\right]-\left[D^2+(y-d/2)^2\right]=2yd$. Factoring the left side as $(S_2P-S_1P)(S_2P+S_1P)$, and using $S_2P+S_1P\approx2D$ (valid since $d\ll D$), gives the path difference $\Delta l=S_2P-S_1P\approx\dfrac{yd}{D}$. …

Set up the double-slit geometry with axes at O (midpoint of the slits) and O' (screen centre), derive the small-angle path-difference formula $\Delta l\approx yd/D$ from the triangle geometry, then apply the bright ($n\lambda$) and dark ($(n-\tfrac12)\lambda$) pa …

Mixing up the bright- and dark-fringe conditions (writing $n\lambda$ for dark and $(n-\tfrac12)\lambda$ for bright), or forgetting the $D\gg d$ approximation $S_2P+S_1P\approx2D$ that is what actually lets …