Question

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.

Answer 1

Proof:

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

\[Now  R =  \infty \] 

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

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

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

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

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

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