JEE Physics 2025 Question with Solution

Given: Two interfaces involving refractive indices $n_1, n_2, n_3$ with $n_1 > n_2 > n_3$, $\frac{n_2}{n_3} = \frac{2}{5}$, and $\sin\theta_{c2} - \sin\theta_{c1} = \frac{1}{2}$.

Find: $\theta_{c1}$.

For critical angle from a denser medium to a rarer medium,

$$\sin\theta_c = \frac{n_{\text{rarer}}}{n_{\text{denser}}}$$

Hence,

$$\sin\theta_{c1} = \frac{n_2}{n_1}, \qquad \sin\theta_{c2} = \frac{n_3}{n_1}$$

Using the given relation,

$$\sin\theta_{c2} - \sin\theta_{c1} = \frac{n_3}{n_1} - \frac{n_2}{n_1} = \frac{1}{2}$$

So,

$$\frac{n_3 - n_2}{n_1} = \frac{1}{2}$$

Given

$$\frac{n_2}{n_3} = \frac{2}{5} \Rightarrow n_2 = \frac{2}{5}n_3$$

Substituting,

$$\frac{n_3 - \frac{2}{5}n_3}{n_1} = \frac{1}{2}$$ $$\frac{\frac{3}{5}n_3}{n_1} = \frac{1}{2}$$ $$\frac{n_3}{n_1} = \frac{5}{6}$$

Then,

$$\sin\theta_{c1} = \frac{n_2}{n_1} = \frac{2}{5} \cdot \frac{n_3}{n_1} = \frac{2}{5} \cdot \frac{5}{6} = \frac{1}{3}$$

Thus,

$$\theta_{c1} = \sin^{-1}\left(\frac{1}{3}\right)$$

the solution working gives this value, but it does not match the listed options exactly. The solution still marks Option A as correct. Therefore, the extracted answer is kept as A, while noting the discrepancy between the working and the options.

Given: the solution's explicitly states The Correct Option is A.

Find: Whether that agrees with the algebra shown in the solution.

The first approach in the solution uses

$$\sin\theta_{c1} = \frac{n_2}{n_1}, \qquad \sin\theta_{c2} = \frac{n_3}{n_1}$$

and from

$$\sin\theta_{c2} - \sin\theta_{c1} = \frac{1}{2}$$

it arrives at

$$\frac{n_3 - n_2}{n_1} = \frac{1}{2}$$

Using

$$\frac{n_2}{n_3} = \frac{2}{5}$$

we get

$$n_2 = \frac{2}{5}n_3$$

So,

$$\frac{3n_3}{5n_1} = \frac{1}{2} \Rightarrow \frac{n_3}{n_1} = \frac{5}{6}$$

and therefore

$$\sin\theta_{c1} = \frac{n_2}{n_1} = \frac{2}{5}\cdot\frac{5}{6} = \frac{1}{3}$$

Hence the working implies

$$\theta_{c1} = \sin^{-1}\left(\frac{1}{3}\right)$$

which is not present among the options. The second approach in the source HTML is internally inconsistent because it writes the critical-angle relations incorrectly. Since the page explicitly labels A as correct, the extracted final answer is A, but the algebra shown on the page suggests an options mismatch.