91Ó°ÊÓ

${\mathrm{θ}}_1={70}^0$

1. Angle of incidence
2. Index of refraction for blue$n_{2b}=1.343$
3. Index of refraction for rad$n_{2r}=1.331$

We can use Snell’s law for each color to find the angle of refraction for red and blue colors at point A and at B. From this, the difference in angles of the red light and blue light that emerge at pointand B can be calculated.

Formula:

$n_1\sin {\mathrm{θ}}_1=n_2\sin {\mathrm{θ}}_2$

According to Snell’s law, for blue color,

$$\begin{gathered} n_{air}\sin {\mathrm{θ}}_1=n_{2b}\sin {\mathrm{θ}}_{2b} \\ {\mathrm{θ}}_{2b}={\sin }^{-1}\left(\frac{n_{air}\sin {\mathrm{θ}}_1}{n_{2b}}\right) \\ {\mathrm{θ}}_{2b}={\sin }^{-1}\left(\frac{\left(1.0\right)\sin {70}^0}{1.343}\right) \\ {\mathrm{θ}}_{2b}=44.{403}^0 \end{gathered}$$

Now, according to Snell’s law, for red color,

$$\begin{gathered} n_{air}\sin {\mathrm{θ}}_1=n_{2r}\sin {\mathrm{θ}}_{2r} \\ {\mathrm{θ}}_{2r}={\sin }^{-1}\left(\frac{n_{air}\sin {\mathrm{θ}}_1}{n_{2r}}\right) \\ {\mathrm{θ}}_{2r}={\sin }^{-1}\left(\frac{\left(1.0\right)\sin {70}^0}{1.331}\right) \\ {\mathrm{θ}}_{2r}=44.{911}^0 \end{gathered}$$

These are the refraction angles at the first surface.

Now, for the second refraction,

According to Snell’s law, for blue color,

$$\begin{gathered} n_{2b}\sin ({90}^0-{\mathrm{θ}}_{2b})=n_{3b}\sin {\mathrm{θ}}_{3b} \\ {\mathrm{θ}}_{3b}={\sin }^{-1}\left(\frac{1.343\sin ({90}^0-{44.403}^0)}{1.0}\right) \\ {\mathrm{θ}}_{3b}=73.{636}^0 \end{gathered}$$

Now, according to Snell’s law, for red color,

$$\begin{gathered} {\mathrm{θ}}_{3r}={\sin }^{-1}\left(\frac{\left(1.331\right)\sin ({90}^0-{44.911}^0)}{1.0}\right) \\ {\mathrm{θ}}_{3r}=70.{497}^0 \end{gathered}$$

Therefore,

$$\begin{gathered} ∆\mathrm{θ}=73.{636}^0-70.{497}^0 \\ ∆\mathrm{θ}=3.1^0 \end{gathered}$$

Therefore, the difference in angles of the red light and blue light that emerge at point A is $3.1^0$.

The angle of incidence and angle of emergence is the same. Therefore, the difference in angles of the red light and blue light that emerge at point B is 0⁰.