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: Name the absolute indices.
Let air, water and glass have absolute refractive indices $\mu_a$, $\mu_w$ and $\mu_g$. A relative index is the ratio of the new medium's index to the old.
Step 2: Write $X$.
Water with respect to air: $X = \dfrac{\mu_w}{\mu_a}$.
Step 3: Write $Y$.
Glass with respect to water: $Y = \dfrac{\mu_g}{\mu_w}$.
Step 4: Write $Z$.
Air with respect to glass: $Z = \dfrac{\mu_a}{\mu_g}$.
Step 5: Multiply the three.
$XYZ = \dfrac{\mu_w}{\mu_a}\cdot\dfrac{\mu_g}{\mu_w}\cdot\dfrac{\mu_a}{\mu_g}$.
Step 6: Cancel everything.
Each index appears once on top and once on the bottom, so $XYZ = 1$, which is option (2).
\[ \boxed{XYZ = 1} \]