The ratio of energies of photons with wavelengths `2000A^(0)` and `4000A^(0)` is

Options

A. `1:4`
B. `4:1`
C. `1:2`
D. `2:1`

Answer

To find the ratio of energies of photons with wavelengths \(2000 \, \text{Å}\) and \(4000 \, \text{Å}\), we can use the formula that relates energy to wavelength:

1. **Understanding the Energy-Wavelength Relationship**:
The energy \(E\) of a photon is given by the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \(E\) = energy of the photon,
- \(h\) = Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \(c\) = speed of light (\(3.00 \times 10^8 \, \text{m/s}\)),
- \(\lambda\) = wavelength of the photon.

2. **Calculating Energies for Both Wavelengths**:
Let’s denote:
- \(E_1\) as the energy of the photon with wavelength \(\lambda_1 = 2000 \, \text{Å}\),
- \(E_2\) as the energy of the photon with wavelength \(\lambda_2 = 4000 \, \text{Å}\).

Using the energy formula:
\[ E_1 = \frac{hc}{\lambda_1} \quad \text{and} \quad E_2 = \frac{hc}{\lambda_2} \]

3. **Finding the Ratio of Energies**:
The ratio of the energies \(E_1\) and \(E_2\) can be expressed as:
\[ \frac{E_1}{E_2} = \frac{\frac{hc}{\lambda_1}}{\frac{hc}{\lambda_2}} = \frac{\lambda_2}{\lambda_1} \]
Here, the \(hc\) terms cancel out.

4. **Substituting the Wavelengths**:
Now substituting the values of \(\lambda_1\) and \(\lambda_2\):
\[ \frac{E_1}{E_2} = \frac{4000 \, \text{Å}}{2000 \, \text{Å}} = 2 \]

5. **Final Ratio**:
Thus, we find that:
\[ E_1 = 2E_2 \]
Therefore, the ratio of the energies of the photons is:
\[ \frac{E_1}{E_2} = 2:1 \]

### Final Answer:
The ratio of energies of photons with wavelengths \(2000 \, \text{Å}\) and \(4000 \, \text{Å}\) is \(2:1\).