Refractive index of glass is `(3)/(2)` and refractive index of water is `(4)/(3)`. If the speed of light in glass is `2.00 xx 10^(8)` m/s, the speed in water will be

Options

A. `2.67 xx 10^(8)` m/s
B. `2.25xx 10^(8)` m/s
C. `1.78 xx 10^(8)` m/s
D. `1.50 xx 10^(8)` m/s

Answer

To find the speed of light in water given the refractive indices of glass and water and the speed of light in glass, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between refractive index and speed of light**:
The refractive index (μ) of a medium is defined as the ratio of the speed of light in vacuum (c) to the speed of light in that medium (v):
\[ \mu = \frac{c}{v} \]
Therefore, we can also express the speed of light in a medium as:
\[ v = \frac{c}{\mu} \]

2. **Write down the given values**:
- Refractive index of glass (μ_glass) = \( \frac{3}{2} \)
- Refractive index of water (μ_water) = \( \frac{4}{3} \)
- Speed of light in glass (v_glass) = \( 2.00 \times 10^8 \) m/s

3. **Set up the relationship between the refractive indices and speeds**:
Using the relationship of refractive indices, we can write:
\[ \frac{\mu_{\text{glass}}}{\mu_{\text{water}}} = \frac{v_{\text{water}}}{v_{\text{glass}}} \]

4. **Substitute the known values**:
Substitute the values of the refractive indices and the speed of light in glass into the equation:
\[ \frac{\frac{3}{2}}{\frac{4}{3}} = \frac{v_{\text{water}}}{2.00 \times 10^8} \]

5. **Simplify the left side**:
To simplify the left side:
\[ \frac{3}{2} \div \frac{4}{3} = \frac{3}{2} \times \frac{3}{4} = \frac{9}{8} \]
So, we have:
\[ \frac{9}{8} = \frac{v_{\text{water}}}{2.00 \times 10^8} \]

6. **Cross-multiply to solve for \( v_{\text{water}} \)**:
Cross-multiplying gives us:
\[ v_{\text{water}} = \frac{9}{8} \times 2.00 \times 10^8 \]

7. **Calculate \( v_{\text{water}} \)**:
\[ v_{\text{water}} = \frac{9 \times 2.00 \times 10^8}{8} = \frac{18.00 \times 10^8}{8} = 2.25 \times 10^8 \text{ m/s} \]

### Final Answer:
The speed of light in water is \( 2.25 \times 10^8 \) m/s.

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