Question

A light ray is moving from denser(refractive index=u) to air. if angle of incidence is half angle of refraction, find out angle of refraction

Answer 1

Please find solutions

Answer 2

Answer:

The angle of refraction will be $r=2 cos^{-1}\dfrac{\mu}{2}$.

Explanation:

A light ray is moving from denser to air.

If angle of incidence is half angle of refraction.

$i= \dfrac{r}{2}$

The refractive index is $\mu$ for denser medium and 1 for air .

We know that,

The refractive index is

$\dfrac{sin\ i}{sin\ r}=\dfrac{\mu_{2}}{\mu_{1}}$

$\dfrac{sin\ i}{sin\ r}=\dfrac{1}{\mu}$

$\dfrac{sin\dfrac{r}{2}}{sin\ r}=\dfrac{1}{\mu}$

Using formula of $sin\ 2A=2sinA\ cosA$

$\dfrac{sin\dfrac{r}{2}}{2sin\dfrac{r}{2}cos\dfrac{r}{2}}=\dfrac{1}{\mu}$

$\dfrac{1}{2cos\dfrac{r}{2}}=\dfrac{1}{\mu}$

$\mu=2cos\dfrac{r}{2}$

$r=2 cos^{-1}\dfrac{\mu}{2}$

Hence, The angle of refraction will be $r=2 cos^{-1}\dfrac{\mu}{2}$.