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

Answer

To find the speed of light in water, we can use the relationship between the speed of light in a medium, its refractive index, and the speed of light in vacuum. The formula we will use is:

\[ v = \frac{c}{n} \]

where:
- \( v \) is the speed of light in the medium,
- \( c \) is the speed of light in vacuum (approximately \( 3 \times 10^8 \) m/s),
- \( n \) is the refractive index of the medium.

### Step-by-Step Solution:

1. **Identify the given values**:
- Refractive index of glass, \( n_{glass} = \frac{3}{2} \)
- Refractive index of water, \( n_{water} = \frac{4}{3} \)
- Speed of light in glass, \( v_{glass} = 2.00 \times 10^8 \) m/s

2. **Calculate the speed of light in vacuum (c)**:
Using the formula for glass:
\[ v_{glass} = \frac{c}{n_{glass}} \]
Rearranging gives:
\[ c = v_{glass} \times n_{glass} \]
Substituting the known values:
\[ c = (2.00 \times 10^8 \, \text{m/s}) \times \left(\frac{3}{2}\right) \]
\[ c = 2.00 \times 10^8 \times 1.5 = 3.00 \times 10^8 \, \text{m/s} \]

3. **Calculate the speed of light in water**:
Now, using the speed of light in vacuum to find the speed of light in water:
\[ v_{water} = \frac{c}{n_{water}} \]
Substituting the values:
\[ v_{water} = \frac{3.00 \times 10^8 \, \text{m/s}}{\frac{4}{3}} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ v_{water} = 3.00 \times 10^8 \times \frac{3}{4} \]
\[ v_{water} = \frac{9.00 \times 10^8}{4} = 2.25 \times 10^8 \, \text{m/s} \]

4. **Final Answer**:
The speed of light in water is:
\[ v_{water} = 2.25 \times 10^8 \, \text{m/s} \]