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.

Explanation

Let us consider a portionh of a ray betweehn x and x+dx inside the liquid . Let the angle of incidence at x be `theta` and let it enter it thin column at height y. Because of the bending it shall emerge at x+dx with and angle `theta+dtheta` and at a height y+ dy. From Snell's law

`" " mu(y)sin theta=y (y+dy) sin (theta +d theta)`
or `" " mu(y)sintheta,(mu(y)+(dmu)/(dy)dy)(sinthetacosd theta+ cos theta sind theta)`
or `" " mu(y)sin theta mu(y) cos theta d theta+(dmu)/(dy)dysintheta`
or `" "mu(y)cos thetad theta,(dmu)/(dy)dysintheta`
`" " d theta,(-dmu)/(mudy)dytantheta`
But `" " tan theta =(dx)/(dy)" "` (from the figure)
On solving , we have
`:. " " d theta=(-1dmu)/(mudy)dx`
Solving this variable separable form of differential equation.
`:. " " theta=(-1dmu)/(mudy)int _(o)^(d)dx=(-1dmu)/(mudy)d`