Question

A light ray is moving from denser (refractive index$\mu$) to air. If the angle of incidence is half the angle of refraction, find out the angle of refraction.

Answer 1

As a first step, one could understand the condition given in the question. You could then recall Snell's law and then substitute the given values into the expression for Snell’s law.
Formula used:
Snell’s law,
${{\mu }_{1}}\sin i={{\mu }_{2}}\sin r$

Complete step by step answer:
In the question, we are discussing a light ray that is travelling from denser medium to rarer medium that is air. The refractive index of the medium is given as$\mu$. We are also being said that the angle of incidence is one half the angle of refraction. We are supposed to find the angle of refraction with the given information.
So,
$i=\dfrac{r}{2}$
We know that according to Snell’s law,
${{\mu }_{1}}\sin i={{\mu }_{2}}\sin r$
But here one medium is air $\left( {{\mu }_{2}}=1 \right)$ and the other medium has a refractive index$\mu$.
$\Rightarrow \dfrac{\sin i}{\sin r}=\dfrac{1}{\mu }$
$\Rightarrow \dfrac{\sin \dfrac{r}{2}}{\sin r}=\dfrac{1}{\mu }$
Now we have a standard trigonometric formula that is given by,
$\sin 2A=2\sin A\cos A$
$\Rightarrow \dfrac{\sin \dfrac{r}{2}}{2\sin \dfrac{r}{2}\cos \dfrac{r}{2}}=\dfrac{1}{\mu }$
$\Rightarrow \mu =2\cos \dfrac{r}{2}$
$\therefore r=2{{\cos }^{-1}}\left( \dfrac{\mu }{2} \right)$
Therefore, we found the angle of refraction to be given by,
$r=2{{\cos }^{-1}}\left( \dfrac{\mu }{2} \right)$

Note: There are certain standard values that have to be known by any science students. One among them is the refractive index of air that is 1. The rest of the conditions used to solve this question is directly given in the question and all we have simply substituted accordingly and hence solved the question.