The distance travelled by a ray of light in two media, in the same time are in the ratio 2 : 3. The ratio of refractive index of the first medium to second medium is

Options

A. `3 : 2`
B. `4 : 9`
C. `2 : 3`
D. `9 : 8`

Answer

To solve the problem, we need to find the ratio of the refractive indices of two media based on the distances traveled by light in those media in the same time.

### Step-by-Step Solution:

1. **Understanding the Given Ratio**:
We are given that the distances traveled by a ray of light in two media are in the ratio of 2:3. Let's denote:
- Distance in medium 1 (d1) = 2x
- Distance in medium 2 (d2) = 3x
where x is a common factor.

2. **Using the Relation of Speed**:
The speed of light in a medium is given by the formula:
\[ v = \frac{d}{t} \]
Since the time (t) taken in both media is the same, we can write:
\[ \frac{v_1}{v_2} = \frac{d_1}{d_2} \]
Substituting the distances:
\[ \frac{v_1}{v_2} = \frac{2x}{3x} = \frac{2}{3} \]

3. **Relating Refractive Index to Speed**:
The refractive index (μ) of a medium is defined as:
\[ \mu = \frac{c}{v} \]
where c is the speed of light in vacuum (or air) and v is the speed of light in the medium. For two media, we can write:
\[ \frac{\mu_1}{\mu_2} = \frac{v_2}{v_1} \]

4. **Substituting the Speed Ratio**:
From our earlier step, we found that:
\[ \frac{v_1}{v_2} = \frac{2}{3} \]
Therefore, the inverse gives:
\[ \frac{v_2}{v_1} = \frac{3}{2} \]

5. **Finding the Ratio of Refractive Indices**:
Now substituting this into the refractive index ratio:
\[ \frac{\mu_1}{\mu_2} = \frac{v_2}{v_1} = \frac{3}{2} \]

### Final Answer:
The ratio of the refractive index of the first medium to the second medium is:
\[ \frac{\mu_1}{\mu_2} = \frac{3}{2} \]