Find the ratio of frequencies of light waves of wavelengths 4000 A and 8000 A.

Answer

To find the ratio of frequencies of light waves with wavelengths of 4000 Å (angstroms) and 8000 Å, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between wavelength and frequency**:
The speed of light (c) is related to the wavelength (λ) and frequency (ν) by the equation:
\[ c = \lambda \cdot \nu \]
where:
- \( c \) = speed of light = \( 3 \times 10^8 \) m/s
- \( \lambda \) = wavelength
- \( \nu \) = frequency

2. **Convert wavelengths from angstroms to meters**:
Since 1 Å = \( 1 \times 10^{-10} \) m, we convert the given wavelengths:
- \( \lambda_1 = 4000 \, \text{Å} = 4000 \times 10^{-10} \, \text{m} = 4 \times 10^{-7} \, \text{m} \)
- \( \lambda_2 = 8000 \, \text{Å} = 8000 \times 10^{-10} \, \text{m} = 8 \times 10^{-7} \, \text{m} \)

3. **Calculate frequencies using the speed of light**:
Using the formula \( \nu = \frac{c}{\lambda} \):
- For \( \lambda_1 \):
\[ \nu_1 = \frac{c}{\lambda_1} = \frac{3 \times 10^8 \, \text{m/s}}{4 \times 10^{-7} \, \text{m}} = 7.5 \times 10^{14} \, \text{Hz} \]
- For \( \lambda_2 \):
\[ \nu_2 = \frac{c}{\lambda_2} = \frac{3 \times 10^8 \, \text{m/s}}{8 \times 10^{-7} \, \text{m}} = 3.75 \times 10^{14} \, \text{Hz} \]

4. **Find the ratio of the frequencies**:
The ratio of the frequencies \( \nu_1 \) and \( \nu_2 \) is given by:
\[ \text{Ratio} = \frac{\nu_1}{\nu_2} = \frac{7.5 \times 10^{14}}{3.75 \times 10^{14}} = 2 \]

5. **Express the ratio**:
Therefore, the ratio of the frequencies of light waves of wavelengths 4000 Å and 8000 Å is:
\[ \text{Ratio} = 2:1 \]

### Final Answer:
The ratio of frequencies of light waves of wavelengths 4000 Å and 8000 Å is \( 2:1 \).