# The mixture a pure liquid and a solution in a long vertical column (i.e, horizontal dimensions << vertical dimensions)

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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 is deviated from its original path. Find the deviation in travelling a horizontal distance d << h, the height of the column.

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Consider a portion of a ray between x and x + dx inside the liquid. Let the angle of incidence at x be θ and let it enter the thin column at height y. Because of the bending, it shall emerge at x + dx with an angle θ + dθ and at a height y + dy. From Snell’s law