The ratio of energy of photon of `lambda = 2000 Å` to that of `lambda = 4000 Å` is

Options

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

Answer

To find the ratio of the energy of photons with wavelengths \( \lambda_1 = 2000 \, \text{Å} \) and \( \lambda_2 = 4000 \, \text{Å} \), we can use the formula for the energy of a photon, which is given by:

\[ E = \frac{hc}{\lambda} \]

where:
- \( E \) is the energy of the photon,
- \( h \) 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 of the photon.

### Step 1: Define the energies
Let:
- \( E_1 \) be the energy of the photon with wavelength \( \lambda_1 = 2000 \, \text{Å} \),
- \( E_2 \) be the energy of the photon with wavelength \( \lambda_2 = 4000 \, \text{Å} \).

### Step 2: Write the energy equations
Using the formula for energy:
\[ E_1 = \frac{hc}{\lambda_1} = \frac{hc}{2000 \, \text{Å}} \]
\[ E_2 = \frac{hc}{\lambda_2} = \frac{hc}{4000 \, \text{Å}} \]

### Step 3: Find the ratio of energies
To find the ratio \( \frac{E_1}{E_2} \):
\[ \frac{E_1}{E_2} = \frac{\frac{hc}{2000 \, \text{Å}}}{\frac{hc}{4000 \, \text{Å}}} \]

### Step 4: Simplify the ratio
The \( hc \) terms cancel out:
\[ \frac{E_1}{E_2} = \frac{4000 \, \text{Å}}{2000 \, \text{Å}} = \frac{4000}{2000} = 2 \]

### Conclusion
The ratio of the energy of the photon with wavelength \( 2000 \, \text{Å} \) to that of \( 4000 \, \text{Å} \) is:
\[ \frac{E_1}{E_2} = 2 \]

Thus, the final answer is \( 2 \).

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