Question

The critical angle for light going from medium '$x$' to medium '$Y$' is $\theta$. The speed of light in medium '$x$' is $v$. The speed of light in medium '$Y$' is

• A. $v \sin\theta$
• B. $\frac{v}{\sin\theta}$
• C. $v \cos\theta$
• D. $\frac{v}{\cos\theta}$

Hint:

Total internal reflection only occurs when light moves into a faster medium, so the velocity in the rarer medium $Y$ must be strictly greater than $v$. Since $\sin\theta \le 1$, dividing $v$ by $\sin\theta$ scales it up ($\frac{v}{\sin\theta} \gt v$), while multiplying scales it down. This instantly rules out option (A)!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A light ray travels from an optically denser medium $x$ to an optically rarer medium $Y$, with a critical angle $\theta$. The velocity of light in medium $x$ is given as $v$. We need to express the velocity of light in medium $Y$, denoted as $v_Y$, in terms of $v$ and $\theta$.

Step 2: Key Formula or Approach:
The critical angle $\theta$ for total internal reflection between two media is defined by: $$\sin\theta = \frac{n_{\text{rarer}}}{n_{\text{denser}}} = \frac{n_Y}{n_x}$$ Since the refractive index $n$ is inversely proportional to the speed of light in that medium ($n = \frac{c}{v}$), we can rewrite the sine relation directly in terms of velocities: $$\sin\theta = \frac{v_x}{v_Y}$$

Step 3: Detailed Explanation:
From our velocity relation, substitute the given speed of light in medium $x$ ($v_x = v$): $$\sin\theta = \frac{v}{v_Y}$$ Isolate the target variable $v_Y$ by cross-multiplying: $$v_Y = \frac{v}{\sin\theta}$$

Step 4: Final Answer:
The speed of light in medium $Y$ is $\frac{v}{\sin\theta}$, which corresponds to option (B).