To solve the problem of finding the relationship between the energy \( E \) of radiation with wavelengths \( 8000 \, \text{Å} \) and \( 16000 \, \text{Å} \), we can follow these steps:
### Step 1: Understand the relationship between energy and wavelength
The energy \( E \) of a photon is given by the equation:
\[
E = \frac{hc}{\lambda}
\]
where:
- \( E \) is the energy,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength.
### Step 2: Write the energy equations for both wavelengths
Let \( E_1 \) be the energy corresponding to the wavelength \( \lambda_1 = 8000 \, \text{Å} \) and \( E_2 \) be the energy corresponding to the wavelength \( \lambda_2 = 16000 \, \text{Å} \).
Using the formula:
\[
E_1 = \frac{hc}{\lambda_1} \quad \text{and} \quad E_2 = \frac{hc}{\lambda_2}
\]
### Step 3: Substitute the values of the wavelengths
Substituting the values:
- \( \lambda_1 = 8000 \, \text{Å} = 8000 \times 10^{-10} \, \text{m} \)
- \( \lambda_2 = 16000 \, \text{Å} = 16000 \times 10^{-10} \, \text{m} \)
Thus, we have:
\[
E_1 = \frac{hc}{8000 \times 10^{-10}} \quad \text{and} \quad E_2 = \frac{hc}{16000 \times 10^{-10}}
\]
### Step 4: Find the ratio of the energies
To find the relationship between \( E_1 \) and \( E_2 \), we can take the ratio:
\[
\frac{E_1}{E_2} = \frac{\frac{hc}{8000 \times 10^{-10}}}{\frac{hc}{16000 \times 10^{-10}}}
\]
The \( hc \) terms cancel out:
\[
\frac{E_1}{E_2} = \frac{16000 \times 10^{-10}}{8000 \times 10^{-10}} = \frac{16000}{8000} = 2
\]
### Step 5: Conclude the relationship
Thus, we find that:
\[
E_1 = 2E_2
\]
This means the energy of the radiation with a wavelength of \( 8000 \, \text{Å} \) is twice that of the radiation with a wavelength of \( 16000 \, \text{Å} \).
### Final Answer
The relationship between the energy of the radiation with a wavelength of \( 8000 \, \text{Å} \) and \( 16000 \, \text{Å} \) is:
\[
E_1 = 2E_2
\]
