Question

If $\hat{i}$ denotes a unit vector along an incident ray, $\hat{r}$ a unit vector along the refracted ray into a medium of refractive index '$\mu$' and $\hat{n}$ a unit vector normal to the boundary of the media directed towards the incident medium, then
$[$consider the first medium to be air$]$

• A. $\hat{i}.\hat{n}=\mu (\hat{r}.\hat{n})$
• B. $\hat{i}\times \hat{n}=\mu (\hat{n}\times \hat{r})$
• C. $\hat{i}\times \hat{n}=\mu (\hat{r}\times \hat{n})$
• D. $\mu (\hat{i}\times \hat{n})=\hat{r}\times \hat{n}$

Answer 1

The correct option is C $\hat{i}\times \hat{n}=\mu (\hat{r}\times \hat{n})$

As per given conditions,

$\hat{i}$ be the unit vector along the incident ray.
$\hat{r}$ be the unit vector along the refracted ray.
$\hat{n}$ is the unit vector along normal to the interface of two medium.
Applying Snell's law,
${\mu }_1\sin i={\mu }_2\sin r$
$\Rightarrow 1\times \sin ({180}^{\circ }-i)=\mu \sin ({180}^{\circ }-r)$
$\Rightarrow |\hat{i}||\hat{n}|\sin ({180}^{\circ }-i)=\mu \left(|\hat{r}||\hat{n}|\sin \right({180}^{\circ }-r\left)\right)$
$\Rightarrow \hat{i}\times \hat{n}=\mu (\hat{r}\times \hat{n})$

Why this question? Caution: Magnitude wise $(\hat{r}\times \hat{n})$ and $(\hat{n}\times \hat{r})$ will yield the same. However, the direction of $(\hat{i}\times \hat{n})$ will be opposite to the direction of $(\hat{n}\times \hat{r})$.