Question

It is found for any type wave; say an earthquake wave that if it reaches a boundary beyond which its speed is increased, there is a maximum incident angle. If there is to be a transmitted refracted wave. The maximum incident angle ${\theta _M}$ corresponds to an angle of refraction equal to $90^\circ$. If ${\theta _1} > {\theta _M}$, all the wave reflected at the boundary and non is refracted ( because this would correspond to $\sin {\theta _r} > 1$, where ${\theta _r}$ is the angle of refraction, which is impossible). This phenomenon is referred to as total internal reflection. Find a formula for ${\theta _M}$.

Answer 1

Hint The above problem is based on the concept of the total internal reflection. It is the phenomenon in which the incident angle always remains more than the critical angle of the medium. The medium of the incident ray must be denser than the other medium for this phenomenon.

Complete step by step answer
Given: The angle of refraction is ${\theta _r} = 90^\circ$.
The incident angle is, ${\theta _i}$.
Let us assume that the refractive index of the incident medium is ${n_i}$ and the refractive index of the refracted medium is ${n_r}$.
Apply the Snell’s Law to find the formula for maximum incident angle.
${n_i}\sin {\theta _M} = {n_r}\sin r$
Substitute $90^\circ$for ${\theta _r}$ and ${\theta _M}$ for ${\theta _i}$ in the above expression to find the maximum incident angle.
${n_i}\sin {\theta _M} = {n_r}\sin 90^\circ$
$\sin {\theta _M} = \dfrac{{{n_r}}}{{{n_i}}}$
${\theta _M} = {\sin ^{ - 1}}\left( {\dfrac{{{n_r}}}{{{n_i}}}} \right)$
Thus, the formula for maximum incident angle is ${\sin ^{ - 1}}\left( {\dfrac{{{n_r}}}{{{n_i}}}} \right)$.

Additional Information The angle of incidence of the ray at which the refraction angle becomes $90^\circ$is called the critical angle. It depends on the refractive index of the medium. The denser the incident medium lesser becomes the critical angle. If the critical angle for some medium is very less then it shows that the incident ray refracted totally in the incident medium easily.

Note Refractive index is the ratio of the speed of wave in rarer medium to denser medium. It describes the decrease in the speed of the wave. The phenomenon of the total internal reflection is used in the optical fibers and reflecting prisms. The Snell’ law can also be expressed in terms of speed of ray in medium as ${v_r}\sin i = {v_i}\sin r$ . Here, ${v_i}$ is the speed of the ray in the incident medium and ${v_r}$ is the speed of the light in the refractive medium.