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

In Fig., we have shown a ray of light entering the column at (x,y) at `90^@` to the vertical. A portion of ray between `x and (x + dx)` deviates through an angle `d theta`, emerging at `(x + dx, y + dy)` at angle `(theta + d theta)`, while the angle at the entry is `theta`.
According to Snell's law,
`mu(y) sin theta = mu (y + dy) sin (theta + d theta)`
=`[mu(y) + (d mu)/(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)sin theta ~~ mu (y) sin theta + mu (y) cos theta d theta + (d mu)/(d y) dy sin theta` (fourth term is negligibly small)
`mu (y) cos theta d theta = - (d mu)/(dy) dy sin theta`
`d theta = -(1)/(mu) (d mu)/(dy) dy tan theta`
From Fig., `tan theta = (dx)/(dy)` or `dy tan theta = dx`
`d theta = -(1)/(mu) (d mu)/(dy) dx`
`theta = -(1)/(mu) (d mu)/(dy) int_0^d dx = - (1)/(mu) (d mu)/(dy) (d)`
This is the required deviation in travelling a horizontal distance `d`.
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