Question

Two transparent media A and B are separated by a plane boundary. The speed of light in medium A is $2.0\times {10}^8{\mathrm{ms}}^{-1}$ and in medium B is $2.5\times {10}^8{\mathrm{ms}}^{-1}$. The critical angle for which ray of light going from A to B suffers total internal reflection is

• A. ${\sin }^{-1}\left(\frac{1}{2}\right)$
• B. ${\sin }^{-1}\left(\frac{2}{5}\right)$
• C. ${\sin }^{-1}\left(\frac{4}{5}\right)$
• D. ${\sin }^{-1}\left(\frac{3}{5}\right)$

Answer 1

The correct option is C

${\sin }^{-1}\left(\frac{4}{5}\right)$

Step 1: The critical angle for total internal reflection

1. The critical angle is the angle of incidence above which the total internal reflection occurs.
2. Total internal reflection can be seen in various types of waves, such as light waves, electromagnetic waves, sound waves etcetera.
3. The diagram for the critical angle can be shown as follows:

Step 2: Calculation of ratio of the velocities of light in media A.

$n_A=\frac{c}{v_A}$

Here, $n_A$ is the ratio of the velocities of light in media A, $c$ is the velocity of light in air, and $v_A$ is the velocity of light in media A.

$$\begin{gathered} n_A=\frac{3\times {10}^8{\mathrm{ms}}^{-1}}{2\times {10}^8{\mathrm{ms}}^{-1}} \\ {n}_A=1.5 \end{gathered}$$

Step 3: Calculation of ratio of the velocities of light in media B.

$n_B=\frac{c}{v_B}$

Here, $n_B$ is the ratio of the velocities of light in media B, and $v_B$ is the velocity of light in media B.

$$\begin{gathered} n_B=\frac{3\times {10}^8{\mathrm{ms}}^{-1}}{2.5\times {10}^8{\mathrm{ms}}^{-1}} \\ {n}_B=1.2 \end{gathered}$$

Step 4: Calculation of critical angle

The critical angle can be calculated as follows:

$\sin \left(i_c\right)=\frac{n_B}{n_A}$

Here, $i_c$ is the critical angle.

$$\begin{gathered} \sin \left(i_c\right)=\frac{1.2}{1.5} \\ {i}_c={\sin }^{-1}\left(\frac{4}{5}\right) \end{gathered}$$

Hence, the critical angle for which the ray suffers total internal reflection is ${\sin }^{-1}\left(\frac{4}{5}\right)$.

Therefore, option (C) is correct.