Question

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 \(2.4 \times 10^8\) m/s and \(2.7 \times 10^8\) m/s, respectively, then the value of critical angle is :

• A. sin⁻¹(9/8)
• B. cos⁻¹(8/9)
• C. tan⁻¹(8/√17)
• D. cot⁻¹(3/√15)

Hint:

If you know $\sin \theta = a/b$, you can always find other trigonometric ratios by drawing a right-angled triangle. This is a common trick in JEE options.

Answer 1

The Correct Option is C

To find the critical angle for light travelling from medium A to medium B, where light undergoes total internal reflection, we can use Snell's Law, which in terms of critical angle can be expressed as:

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

Here, \(n_A\) and \(n_B\) are the refractive indices of mediums A and B, respectively, and \(\theta_c\) is the critical angle. Given that sin(90°) = 1, the equation simplifies to:

\(\sin(\theta_c) = \frac{n_B}{n_A}\)

The refractive index of a medium is the ratio of the speed of light in vacuum to the speed of light in the medium. We are given that the speed of light in medium A, \(v_A\), is \(2.4 \times 10^8 \text{ m/s}\), and in medium B, \(v_B\), is \(2.7 \times 10^8 \text{ m/s}\). The speed of light in vacuum, \(c\), is \(3 \times 10^8 \text{ m/s}\).

Therefore, the refractive indices are:

• \(n_A = \frac{c}{v_A} = \frac{3 \times 10^8}{2.4 \times 10^8} = \frac{5}{4}\)
• \(n_B = \frac{c}{v_B} = \frac{3 \times 10^8}{2.7 \times 10^8} = \frac{10}{9}\)

Now, substituting these values into the equation for \(\sin(\theta_c)\):

\(\sin(\theta_c) = \frac{\frac{10}{9}}{\frac{5}{4}} = \frac{10}{9} \times \frac{4}{5} = \frac{8}{9}\)

We want the critical angle \(\theta_c\) such that:

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

Given the options, \(\tan^{-1}\left(\frac{8}{\sqrt{17}}\right)\) is an equivalent expression when simplifying trigonometric identities. Therefore, the correct answer is:

tan⁻¹(8/√17)