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Chapter 16: Problem 2262

A ray of light from denser medium strikes a rarer medium at angle of incidence i. The reflected and refracted rays make an angle of \(90^{\circ}\) with each other. The angle of reflection and refraction are r and \(\mathrm{r}^{\prime}\) respectively. The critical angle is (A) \(\sin ^{-1}(\tan i)\) (B) \(\tan ^{-1}(\tan \mathrm{r})\) (C) \(\tan ^{-1}(\sin i)\) (D) \(\sin ^{-1}(\tan \mathrm{r})\)

Short Answer

Expert verified
The critical angle is given by option (A) \(\sin^{-1}(\tan i)\).

Step by step solution

01

Write down the Snell's law equation

Snell's law states that: \[\frac{\sin{i}}{\sin{r'}}=\frac{n_2}{n_1}\] Where i is the angle of incidence, r' is the angle of refraction, and n1 and n2 are refractive indices of the denser and rarer media, respectively.
02

Calculate the angle of reflection (r)

Since the angle of reflection is equal to the angle of incidence, we have: \[r = i\]
03

Use the given angle between reflected and refracted rays

We are given that the angle between the reflected and refracted rays is 90°. Therefore, we can write: \[r + r' = 90°\] Since the angle of reflection is equal to the angle of incidence (r = i), we have: \[i + r' = 90°\] \[r' = 90° - i\]
04

Substitute for r' in Snell's law equation

Now, substitute the value of r' from Step 3 into the Snell's law equation from Step 1: \[\frac{\sin{i}}{\sin{90°-i}}=\frac{n_2}{n_1}\] Recall that \(\sin{90°-i} = \cos{i}\), so we have: \[\frac{\sin{i}}{\cos{i}}=\frac{n_2}{n_1}\]
05

Write the expression for the critical angle

The critical angle (C) is the angle of incidence for which the angle of refraction is 90°. Given that, we can deduce the equation, \[\frac{\sin{C}}{\cos{C}}=\frac{n_1}{n_2}\]
06

Compare the given options

Now, we have to find which of the given options matches the expression we derived for the critical angle (C). Comparing the options, the correct one is: (A) \(\sin^{-1}(\tan{i})\) Since, \(C = \sin^{-1}\left(\frac{n_1}{n_2}\right) = \sin^{-1}(\tan{i})\) Hence, the critical angle is given by option (A) \(\sin^{-1}(\tan i)\).

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