An electron in a hydrogen atom undergoes a transition from an orbit with quantum number $${n_i}$$ to another with quantum number $${n_f}$$. $${V_i}$$ and $${V_f}$$ are respectively the initial and final potential energies of the electron. If $${{{V_i}} \over {{V_f}}} = 6.25$$, then the smallest possible $${n_f}$$ is

A drop of liquid of radius $$R = {10^{ - 2}}\,m$$ having surface tension $$S = {{0.1} \over {4\pi }}N{m^{ - 1}}$$ divides itself into $$K$$ identical drops. In this process the total change in the surface energy $$\Delta U = {10^{ - 3}}\,J.$$ If $$K = {10^\alpha }$$ then the value of $$\alpha $$ 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 stationary source emits sound of frequency $${f_0} = 492\,Hz.$$ The sound is reflected by a large car approaching the source with a speed of $$2\,m{s^{ - 1.}}$$ The reflected signal is received by the source and superposed with the original.

What will be the beat frequency of the resulting signal in $$Hz$$? (Given that the speed of sound in air is $$330\,m{s^{ - 1}}$$ and the car reflects the sound at the frequency it has received).