Monochromatic light of frequency `5xx10^(14) Hz` travelling in vacuum enters a medium of refractive index 1.5. Its wavelength in the medium is

Options

A. 4000 Å
B. 5000 Å
C. 6000 Å
D. 5500 Å

Answer

To find the wavelength of monochromatic light in a medium with a given refractive index, we can follow these steps:

### Step 1: Identify the given values
- Frequency of light, \( f = 5 \times 10^{14} \, \text{Hz} \)
- Refractive index of the medium, \( n = 1.5 \)
- Speed of light in vacuum, \( c = 3 \times 10^8 \, \text{m/s} \)

### Step 2: Calculate the wavelength in vacuum
The wavelength of light in vacuum (\( \lambda_1 \)) can be calculated using the formula:
\[ \lambda_1 = \frac{c}{f} \]
Substituting the values:
\[ \lambda_1 = \frac{3 \times 10^8 \, \text{m/s}}{5 \times 10^{14} \, \text{Hz}} = \frac{3}{5} \times 10^{-6} \, \text{m} = 6 \times 10^{-7} \, \text{m} \]

### Step 3: Calculate the wavelength in the medium
The wavelength in the medium (\( \lambda_2 \)) can be calculated using the relationship between the wavelengths and the refractive index:
\[ \lambda_2 = \frac{\lambda_1}{n} \]
Substituting the values:
\[ \lambda_2 = \frac{6 \times 10^{-7} \, \text{m}}{1.5} = 4 \times 10^{-7} \, \text{m} \]

### Step 4: Convert the wavelength to nanometers
To express the wavelength in nanometers:
\[ \lambda_2 = 4 \times 10^{-7} \, \text{m} = 400 \, \text{nm} \]

### Final Answer
The wavelength of the monochromatic light in the medium is \( 400 \, \text{nm} \).

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