The mixture a pure liquid and a solution in a long vertical column

Question:
The mixture a pure liquid and a solution in a long vertical column (i.e, horizontal dimensions << 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 deviates from its original path.

Find the deviation in travelling a horizontal distance d << h, the height of the column.

Solution:

Let the height of the long vertical column with transparent liquid be h and dx be the thickness

The angle at which the ray AB enters is θ

Let y be the new height of the liquid

(θ + d θ) is the emerging angle while (y + dy) is the height.

From Snell’s law,

μ(y) sin θ = μ (y + dy) sin (θ + d θ)

After integration we get,

θ = -1/ μ d μ/dy d

The mixture a pure liquid and a solution in a long vertical column

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