Consider light travelling from a medium $A$ to medium $B$ separated by a plane interface. If the light undergoes total internal reflection during its travel from medium $A$ to $B$ and the speed of light in media $A$ and $B$ are $2.4 \times 10^8\text{ m/s}$ and $2.7 \times 10^8\text{ m/s}$, respectively, then the value of critical angle is :

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Consider light travelling from a medium $A$ to medium $B$ separated by a plane interface. If the light undergoes total internal reflection during its travel from medium $A$ to $B$ and the speed of light in media $A$ and $B$ are $2.4 \times 10^8\text{ m/s}$ and $2.7 \times 10^8\text{ m/s}$, respectively, then the value of critical angle is :

$\sin^{-1}\left(\frac{9}{8}\right)$

$\cos^{-1}\left(\frac{8}{9}\right)$

$\tan^{-1}\left(\frac{8}{\sqrt{17}}\right)$

$\cot^{-1}\left(\frac{3}{\sqrt{13}}\right)$

The correct answer is

$\cos^{-1}\left(\frac{8}{9}\right)$

Understanding Total Internal Reflection

Total Internal Reflection (TIR) occurs when light travels from an optically denser medium (medium A) to an optically rarer medium (medium B) at an angle of incidence greater than the critical angle ($\theta_c$).

For TIR to occur from A to B, the refractive index of A must be greater than B ($n_A > n_B$), which implies the speed of light in A is less than in B ($v_A < v_B$). The given speeds ($v_A = 2.4 \times 10^8\text{ m/s}$ and $v_B = 2.7 \times 10^8\text{ m/s}$) satisfy this condition.

Critical Angle Formula

The critical angle ($\theta_c$) is defined by Snell's Law when the angle of refraction is $90^\circ$:

$ n_A \sin(\theta_c) = n_B \sin(90^\circ) = n_B $

Using the relationship between refractive index and speed of light ($n = c/v$), we can write:

$ \frac{c}{v_A} \sin(\theta_c) = \frac{c}{v_B} $

Simplifying this gives the formula for the sine of the critical angle:

$ \sin(\theta_c) = \frac{v_A}{v_B} $

Calculating the Critical Angle

Substitute the given speeds of light:

$ \sin(\theta_c) = \frac{2.4 \times 10^8\text{ m/s}}{2.7 \times 10^8\text{ m/s}} $

$ \sin(\theta_c) = \frac{2.4}{2.7} = \frac{24}{27} = \frac{8}{9} $

Thus, the sine of the critical angle is $\sin(\theta_c) = 8/9$.

Selecting the Correct Option

The question asks for the value of the critical angle. Based on the calculation, $\sin(\theta_c) = 8/9$. We need to choose the option that represents this critical angle.

The correct option is B: $\cos^{-1}\left(\frac{8}{9}\right)$.

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