$A$ paraxial beam is incident on a glass $(n = 1.5)$ hemisphere of radius $R = 6\, cm$ in air as shown. The distance of the point of convergence $F$ from the plane surface of the hemisphere is......$cm$.

$A$ vessel of depth $d$ is half filled with oil of refractive index $n_1$ and the other half is filled with water of refractive index $n_2$. The apparent depth of this vessel when viewed from above will be

An ice cube has a bubble inside. When viewed from one side,the apparent distance of the bubble is $12 \ cm$. When viewed from the opposite side,the apparent distance of the bubble is observed as $4 \ cm$. If the side of the ice cube is $24 \ cm$,the refractive index of the ice cube is $.....$.

The back surface of a glass slab (refractive index $n$ and thickness $t$) is polished to act as a mirror,as shown in the figure. $A$ laser beam is incident on it,which is partially reflected and partially refracted at the air-glass interface. The refracted part is fully reflected at the mirror surface. $A$ pattern of discrete spots of light is observed on a screen placed perpendicular to the glass surface. The spacing between the spots on the screen is:

$A$ fish rising vertically upward with a uniform velocity of $8\,ms^{-1}$ observes that a bird is diving vertically downward towards the fish with a velocity of $12\,ms^{-1}$. If the refractive index of water is $\frac{4}{3}$,then the actual velocity of the diving bird to pick the fish will be $.......\,ms^{-1}$.

$A$ ray of light is incident at an angle of incidence $60^{\circ}$ on a glass slab of refractive index $\sqrt{3}$. After refraction,the light ray emerges out from the other parallel face,and the lateral shift between the incident ray and the emergent ray is $4 \sqrt{3} \, cm$. The thickness of the glass slab is . . . $cm$.