A planar structure of length L and width W is made of two different optical media of refractive indices n₁ = 1.5 and n₂ = 1.44 as shown in figure. If L >> W, a ray entering from end AB will emerge from end CD. CD only if the total internal reflection condition is met inside the structure. For L = 9.6 m, if the incident angle $$\theta $$ is varied, the maximum time taken by a ray to exit the plane CD is t $$ \times $$ 10^-9 s, where, t is ................

[Speed of light, c = 3 $$ \times $$ 10⁸ m/s]

Sunlight of intensity $$1.3$$ $$kW\,{m^{ - 2}}$$ is incident normally on a thin convex lens of focal length $$20$$ cm. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in $$kW\,{m^{ - 2}},$$ at a distance $$22$$ cm from the lens on the other side is ___________.

A monochromatic light is travelling in a medium of refractive index $$n=1.6.$$ It enters a stack of glass layers from the bottom side at an angle $$\theta = {30^ \circ }.$$ The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $${n_m} = n - m\Delta n,$$ where $${n_m}$$ is the refractive index of the $${m^{th}}$$ slab and $$\Delta n = 0.1$$ (see the figure). The ray is refracted out parallel to the interface between the $${\left( {m - 1} \right)^{th}}$$ and $${m^{th}}$$ slabs from the right side of the stack. What is the value of $$m$$?

A monochromatic beam of light is incident at 60$$^\circ$$ on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle $$\theta$$(n) with the normal (see figure). For n = $$\sqrt 3 $$ the value of $$\theta$$ is 60$$^\circ$$ and $${{d\theta } \over {dn}} = m$$. The value of m is