Answer the following questions:
$(a)$ In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band?
$(b)$ In what way is diffraction from each slit related to the interference pattern in a double-slit experiment?
$(c)$ When a tiny circular obstacle is placed in the path of light from a distant source, a bright spot is seen at the centre of the shadow of the obstacle. Explain why?
$(d)$ Two students are separated by a $7 \; m$ partition wall in a room $10 \; m$ high. If both light and sound waves can bend around obstacles, how is it that the students are unable to see each other even though they can converse easily?
$(e)$ Ray optics is based on the assumption that light travels in a straight line. Diffraction effects (observed when light propagates through small apertures/slits or around small obstacles) disprove this assumption. Yet the ray optics assumption is so commonly used in understanding location and several other properties of images in optical instruments. What is the justification?

(N/A) In a single slit diffraction experiment, if the width of the slit $(a)$ is doubled, the angular width of the central maximum $(2\lambda/a)$ reduces to half. The intensity of the central maximum is proportional to the square of the slit width $(I \propto a^2)$, so it increases by a factor of $4$.
$(b)$ The interference pattern in a double-slit experiment is the result of the superposition of two waves, but it is modulated by the diffraction pattern produced by each individual slit. The intensity distribution is the product of the interference pattern and the diffraction envelope.
$(c)$ This phenomenon is known as Poisson's spot. Light waves diffract around the edges of the circular obstacle and reach the center of the shadow in phase, resulting in constructive interference that creates a bright spot.
$(d)$ Diffraction is significant only when the wavelength $(\lambda)$ is comparable to the size of the obstacle $(d)$. For light, $\lambda$ is very small $(\sim 500 \; nm)$, so it does not diffract significantly around a wall. For sound, $\lambda$ is in the range of $0.1 \; m$ to $1 \; m$, which is comparable to the size of the wall, allowing it to bend around the obstacle.
$(e)$ The ray optics approximation is valid when the size of the apertures or obstacles is much larger than the wavelength of light. In most optical instruments, the apertures are large enough that diffraction effects are negligible, making the straight-line propagation assumption valid.