The mixture of a pure liquid and a solution in a along vertical column (i.e., horizontal dimensions `lt lt` vertical dimensions) produces diffusion of solute particles and hence a refractive index gradient along the vertical dimension. A ray of light entering the column at right angles to the vertical is deviated from its original path. Find the deviation in travelling a horizontal distance `d lt lt h`, the height of the column.

Answer

D

Explanation

Let us consider a light ray entering the column at `(x,y)` at `90^(@)` to the vertical . `theta` be the angle of incidence at x and it enetr the thin coloum at height `y. d theta` is the deviation of ray betweem `x` and `(x+dx)` emerging at `(x+dx,y+dy)` at angle `(theta+d theta)`.

By snell's law,
`mu(y) sin theta = mu(y+dy) sin (theta+d theta)`
`[mu(y)+(dmu)/(dy)dy] (sin theta cos d theta+ cos theta sin d theta)`
As `d theta` is small, `cos d theta ~~ 1` and `sin d theta ~~ d theta`.
`:. mu(y) cos theta = (dmu)/(dy)dysin theta, d theta = (-1)/(mu) (d mu dy tantheta)/(dy)`
`[tan theta= (dx)/(dy)` and replacing `mu(y)` by `mu`]
`theta = - 1/mu (d mu)/(dy)overset(d)underset(o)(int)dx = - (1)/(mu)((dmu)/(dy)) d`.