The equation of refraction at a spherical surface is \[\frac{\mu_2}{\nu} - \frac{\mu_1}{\mu} = \frac{\mu_2 - \mu_1}{R}\]

Taking \[R = \infty\] show that this equation leads to the equation

\[\frac{\text{ Real depth }}{\text{ Apparent depth }} = \frac{\mu_2}{\mu_1}\]

for refraction at a plane surface.

#### Solution

Proof:

\[\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}\]

\[Now R = \infty \]

\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{\infty}\]

\[\frac{\mu_2}{v} - \frac{\mu_1}{u} = 0\]

\[ \frac{\mu_2}{v} = \frac{\mu_1}{u}\]

\[\frac{\mu_1}{\mu_2} = \frac{u}{v}\]

\[But \frac{u}{v} = \frac{\text{ Real depth/height}}{\text{ Apparent depth/height}} \]

\[ \therefore \frac{\text{ Real depth }/height}{\text{ Apparent depth/height }} = \frac{\mu_1}{\mu_2}\]