Red light of wavelength 6400 Å in air has a wavelength of 4000 Å in glass. If the wavelength of violet light in air is 4400 Å, then the wavelength in glass is

Options

A. `2570Å`
B. `2750Å`
C. `1600Å`
D. `2510Å`

Answer

To solve the problem, we need to find the wavelength of violet light in glass, given the relationship between the wavelengths of light in air and glass.

### Step-by-Step Solution:

1. **Understanding the Relationship**: The refractive index \( n \) of a medium is inversely proportional to the wavelength \( \lambda \) of light in that medium. This can be expressed as:
\[ n_a / n_g = \lambda_g / \lambda_a \]
where:
- \( n_a \) = refractive index of air
- \( n_g \) = refractive index of glass
- \( \lambda_a \) = wavelength in air
- \( \lambda_g \) = wavelength in glass

2. **Using Given Data for Red Light**: We know the wavelength of red light in air (\( \lambda_{a, red} = 6400 \, \text{Å} \)) and in glass (\( \lambda_{g, red} = 4000 \, \text{Å} \)). We can set up the equation:
\[ \frac{n_a}{n_g} = \frac{\lambda_{g, red}}{\lambda_{a, red}} = \frac{4000}{6400} \]
Simplifying this gives:
\[ \frac{n_a}{n_g} = \frac{4000}{6400} = \frac{40}{64} = \frac{5}{8} \]

3. **Finding Wavelength of Violet Light in Glass**: Now we need to find the wavelength of violet light in glass (\( \lambda_{g, violet} \)). The wavelength of violet light in air is given as \( \lambda_{a, violet} = 4400 \, \text{Å} \). Using the same relationship:
\[ \frac{n_a}{n_g} = \frac{\lambda_{g, violet}}{\lambda_{a, violet}} \]
Substituting the known values:
\[ \frac{5}{8} = \frac{\lambda_{g, violet}}{4400} \]

4. **Solving for \( \lambda_{g, violet} \)**: Rearranging the equation to solve for \( \lambda_{g, violet} \):
\[ \lambda_{g, violet} = 4400 \times \frac{5}{8} \]
Calculating this gives:
\[ \lambda_{g, violet} = 4400 \times 0.625 = 2750 \, \text{Å} \]

5. **Final Answer**: Therefore, the wavelength of violet light in glass is:
\[ \lambda_{g, violet} = 2750 \, \text{Å} \]