1. **Problem statement:** A monochromatic light beam of wavelength $\lambda = 594$ nm shines vertically on a wedge-shaped air film formed by two glass plates touching at the left end. Initially, a dark fringe is at the left end, a bright fringe at the right end, and 9 dark fringes in between. The plates are squeezed, changing the wedge angle and causing the fringe at the right end to alternate between bright and dark every 12.0 s. We need to find: (a) The rate of change of the spacing between the plates at the right end. (b) The change in spacing at the right end when both ends show dark fringes and there are 5 dark fringes in between. 2. **Key concepts and formulas:** - The condition for dark fringes (destructive interference) in a thin air film of thickness $d$ is: $$2d = m\lambda$$ where $m$ is an integer fringe order. - For bright fringes (constructive interference): $$2d = (m + \tfrac{1}{2})\lambda$$ - The number of dark fringes between two points corresponds to the difference in fringe orders. - The fringe at the right end alternates every 12.0 s, meaning the thickness changes by half a wavelength every 12.0 s. 3. **Part (a): Rate of change of spacing at the right end** - The fringe at the right end changes from bright to dark every 12.0 s, so the thickness changes by $\Delta d = \frac{\lambda}{4}$ in 12.0 s (because going from bright to dark corresponds to a change of $\frac{\lambda}{4}$ in thickness, half the path difference $\frac{\lambda}{2}$). - Rate of change of spacing: $$\frac{\Delta d}{\Delta t} = \frac{\lambda/4}{12.0} = \frac{594 \times 10^{-9} / 4}{12.0} = \frac{148.5 \times 10^{-9}}{12.0} = 1.2375 \times 10^{-8} \text{ m/s}$$ 4. **Part (b): Change in spacing when both ends show dark fringes and 5 dark fringes are in between** - Initially, left end is dark fringe $m=0$, right end bright fringe $m=9.5$ (since 9 dark fringes in between means 10 dark fringe positions, and bright fringe is halfway between dark fringes). - Now both ends show dark fringes and 5 dark fringes in between, so right end fringe order is $m=6$ (left end $m=0$, 5 dark fringes in between means 6 dark fringes total). - Change in fringe order at right end: $$\Delta m = 9.5 - 6 = 3.5$$ - Change in thickness: $$\Delta d = \frac{\lambda}{2} \Delta m = \frac{594 \times 10^{-9}}{2} \times 3.5 = 297 \times 10^{-9} \times 3.5 = 1.0395 \times 10^{-6} \text{ m}$$ **Final answers:** (a) Rate of change of spacing at right end is approximately $1.24 \times 10^{-8}$ m/s. (b) The spacing at the right end has changed by approximately $1.04 \times 10^{-6}$ m when both ends show dark fringes with 5 dark fringes in between.