Path Difference ($\Delta x$): For a point P on the screen at a distance $y$ from the central axis: $\Delta x = S_2P - S_1P$ Using geometry and binomial approximation $(1+x)^n \approx 1+nx$ for small $x$: $S_1P = \sqrt{D^2 + (y - d/2)^2} \approx D + \frac{(y - d/2)^2}{2D}$ $S_2P = \sqrt{D^2 + (y + d/2)^2} \approx D + \frac{(y + d/2)^2}{2D}$ $\Delta x = \frac{(y + d/2)^2 - (y - d/2)^2}{2D} = \frac{4y(d/2)}{2D} = \frac{yd}{D}$ This approximation is valid when $D \gg d$ and $D \gg y$. Alternatively, using angles: $\Delta x = d \sin\theta$. For small $\theta$, $\sin\theta \approx \tan\theta \approx \frac{y}{D}$, so $\Delta x = \frac{yd}{D}$.

Condition for Bright Fringes (Constructive Interference): Occurs when $\Delta x = n\lambda$ (where $n = 0, \pm 1, \pm 2, ...$) $\frac{yd}{D} = n\lambda \implies y_n = \frac{n\lambda D}{d}$

For $n=0$, $y_0 = 0$ (Central Bright Fringe)
For $n=\pm 1$, $y_{\pm 1} = \pm \frac{\lambda D}{d}$ (First Bright Fringes)

Condition for Dark Fringes (Destructive Interference): Occurs when $\Delta x = (2n+1)\frac{\lambda}{2}$ (where $n = 0, \pm 1, \pm 2, ...$) $\frac{yd}{D} = (2n+1)\frac{\lambda}{2} \implies y_n = (2n+1)\frac{\lambda D}{2d}$

For $n=0$, $y_0 = \frac{\lambda D}{2d}$ (First Dark Fringe)
For $n=1$, $y_1 = \frac{3\lambda D}{2d}$ (Second Dark Fringe)

Fringe Width ($\beta$): The distance between the centers of two consecutive bright fringes or two consecutive dark fringes. $\beta = y_{n+1} - y_n = \frac{(n+1)\lambda D}{d} - \frac{n\lambda D}{d} = \frac{\lambda D}{d}$ All bright and dark fringes have the same width in YDSE.

Angular Fringe Width ($\theta$): The angular separation between consecutive fringes. $\theta = \frac{\beta}{D} = \frac{\lambda}{d}$

Intensity Distribution: The intensity at a point $P$ is $I = I_0 \cos^2(\frac{\phi}{2})$, where $\phi = \frac{2\pi}{\lambda}\Delta x = \frac{2\pi}{\lambda}\frac{yd}{D}$. $I = I_0 \cos^2(\frac{\pi yd}{\lambda D})$ Maximum Intensity ($I_{max} = I_0$) at bright fringes, Minimum Intensity ($I_{min} = 0$) at dark fringes.

Effect of Thin Transparent Sheet: If a thin sheet of thickness $t$ and refractive index $\mu$ is placed in the path of one of the beams (say, from $S_1$):

The optical path length for $S_1$ changes from $S_1P$ to $S_1P - t + \mu t = S_1P + (\mu-1)t$.
The new path difference: $\Delta x' = S_2P - (S_1P + (\mu-1)t) = \Delta x - (\mu-1)t = \frac{yd}{D} - (\mu-1)t$.
For the central bright fringe ($n=0$), $\Delta x' = 0 \implies \frac{y_{shift}d}{D} - (\mu-1)t = 0$. So, the shift of the central bright fringe is $\Delta y = y_{shift} = \frac{D}{d}(\mu-1)t$.
The entire fringe pattern shifts towards the slit in front of which the sheet is placed.

4. Diffraction of Light

Definition: Bending of light waves around obstacles or through small apertures.
Fraunhofer Diffraction (Single Slit):
- Condition for minima: $a \sin\theta = n\lambda$ ($n = \pm 1, \pm 2, ...$) where $a$ is slit width, $\theta$ is angle from center.
- Condition for maxima: $a \sin\theta = (2n+1)\frac{\lambda}{2}$ ($n = \pm 1, \pm 2, ...$) (approximate for secondary maxima