Derive an expression used to obtain the refractive index of a liquid using air cell.

Derive an expression used to obtain the refractive index of a liquid using air cell.

Ray SO is refracted along OB in glass. However, at B total internal reflection just begins.

Suppose i1 is the angle of incidence in the liquid, i2 is the angle of incidence in the glass  while n and ng are the corresponding refractive indices, Then applying the relation n sin i = a constant gives

n sin i1 = ng sin i2 = nasin 90°

=>n sin i1 =na sin 90°

=> n sin i1 = 1

but i1 = θ

thus n = 1/sinθ

(where θ is the averages angle through which the air cell is turned for light to disappear)