Blue light of wavelength 470 nm emerges through a liquid column of refractive index 1.42. Calculate the wavelength and frequency of blue light in the liquid.

Answer

To solve the problem of finding the wavelength and frequency of blue light in a liquid with a refractive index of 1.42, we can follow these steps:

### Step 1: Understand the relationship between wavelength, frequency, and speed of light
The speed of light in a medium is given by the formula:
\[ v = f \cdot \lambda \]
where:
- \( v \) is the speed of light in the medium,
- \( f \) is the frequency of the light,
- \( \lambda \) is the wavelength of the light in the medium.

### Step 2: Calculate the wavelength of blue light in the liquid
The relationship between the wavelength in air (\( \lambda_{air} \)) and the wavelength in the medium (\( \lambda_{liquid} \)) is given by:
\[ \lambda_{liquid} = \frac{\lambda_{air}}{n} \]
where \( n \) is the refractive index of the medium.

Given:
- \( \lambda_{air} = 470 \, \text{nm} \)
- \( n = 1.42 \)

Substituting the values:
\[ \lambda_{liquid} = \frac{470 \, \text{nm}}{1.42} \]
\[ \lambda_{liquid} \approx 330.98 \, \text{nm} \]

### Step 3: Calculate the frequency of blue light in the liquid
The speed of light in a vacuum (or air) is approximately \( c = 3 \times 10^8 \, \text{m/s} \).

Using the relationship:
\[ v = f \cdot \lambda \]
In the liquid:
\[ v = \frac{c}{n} \]
Substituting for \( v \):
\[ f = \frac{v}{\lambda_{liquid}} \]
\[ f = \frac{c/n}{\lambda_{liquid}} \]

Now substituting the values:
\[ f = \frac{(3 \times 10^8 \, \text{m/s}) / 1.42}{330.98 \times 10^{-9} \, \text{m}} \]

Calculating \( v \):
\[ v = \frac{3 \times 10^8}{1.42} \approx 2.11 \times 10^8 \, \text{m/s} \]

Now substituting \( v \) into the frequency formula:
\[ f = \frac{2.11 \times 10^8}{330.98 \times 10^{-9}} \]
\[ f \approx 6.38 \times 10^{14} \, \text{Hz} \]

### Final Answers:
- Wavelength of blue light in the liquid: \( \lambda_{liquid} \approx 330.98 \, \text{nm} \)
- Frequency of blue light in the liquid: \( f \approx 6.38 \times 10^{14} \, \text{Hz} \)