Question

A ray of light travels from air to water to glass and again from glass to air. Refractive index of water w.r.t. air is 'X', glass w.r.t. water is 'Y' and air w.r.t. glass is 'Z'. Which one of the following is correct?

• A. $YZ = X$
• B. $XYZ = 1$
• C. $XY = Z$
• D. $XZ = Y$

Hint:

For any closed, continuous optical loop that starts and finishes in the exact same medium, the product of all relative refractive indices across the successive boundaries will always equal 1 ($_{1}\mu_{2} \cdot _{2}\mu_{3} \dots _{n}\mu_{1} = 1$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A light ray cycles through three distinct media (air $\rightarrow$ water $\rightarrow$ glass $\rightarrow$ air). We are given the relative refractive indices between consecutive interfaces labeled as variables $X$, $Y$, and $Z$, and we must find the correct mathematical equation connecting them.

Step 2: Key Formula or Approach:
The relative refractive index of a medium 2 with respect to medium 1 is defined as the ratio of their absolute refractive indices:
$$_{1}\mu_{2} = \frac{\mu_2}{\mu_1}$$ According to the principle of reversibility and cyclic chain rules for multiple interfaces, the product of sequential relative refractive indices wrapping back to the original medium equals unity:
$$_{a}\mu_{w} \times _{w}\mu_{g} \times _{g}\mu_{a} = 1$$

Step 3: Detailed Explanation:
Let's express the absolute refractive indices for air, water, and glass as $\mu_a$, $\mu_w$, and $\mu_g$ respectively. Now, rewrite each given relative parameter using these absolute components:
1. Refractive index of water w.r.t. air:
$$X = _{a}\mu_{w} = \frac{\mu_w}{\mu_a}$$ 2. Refractive index of glass w.r.t. water:
$$Y = _{w}\mu_{g} = \frac{\mu_g}{\mu_w}$$ 3. Refractive index of air w.r.t. glass:
$$Z = _{g}\mu_{a} = \frac{\mu_a}{\mu_g}$$ Multiply all three expressions together:
$$XYZ = \left(\frac{\mu_w}{\mu_a}\right) \times \left(\frac{\mu_g}{\mu_w}\right) \times \left(\frac{\mu_a}{\mu_g}\right)$$ Rearranging the terms shows complete cancellation:
$$XYZ = \frac{\mu_w \cdot \mu_g \cdot \mu_a}{\mu_a \cdot \mu_w \cdot \mu_g} = 1$$ This cyclical multiplication product simplifies cleanly to 1.

Step 4: Final Answer:
The correct relationship is $XYZ = 1$, which corresponds to option (B).