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

Options

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

Explanation

To find the ratio of the energy of a photon with a wavelength of \( \lambda_1 = 2000 \, \text{Å} \) to that of a photon with a wavelength of \( \lambda_2 = 4000 \, \text{Å} \), we can follow these steps:

### Step 1: Understand the relationship between energy and wavelength
The energy \( E \) of a photon is given by the formula:
\[ E = \frac{hc}{\lambda} \]
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 2: Write the energy expressions for both wavelengths
For the first photon (with wavelength \( \lambda_1 = 2000 \, \text{Å} \)):
\[ E_1 = \frac{hc}{\lambda_1} \]
For the second photon (with wavelength \( \lambda_2 = 4000 \, \text{Å} \)):
\[ E_2 = \frac{hc}{\lambda_2} \]

### Step 3: Find the ratio of the energies
Now, we can find the ratio of the energies \( \frac{E_1}{E_2} \):
\[ \frac{E_1}{E_2} = \frac{\frac{hc}{\lambda_1}}{\frac{hc}{\lambda_2}} = \frac{\lambda_2}{\lambda_1} \]
Here, the constants \( h \) and \( c \) cancel out.

### Step 4: Substitute the values of the wavelengths
Substituting the values of \( \lambda_1 \) and \( \lambda_2 \):
\[ \frac{E_1}{E_2} = \frac{4000 \, \text{Å}}{2000 \, \text{Å}} = 2 \]

### Conclusion
Thus, the ratio of the energy of the photon with a wavelength of \( 2000 \, \text{Å} \) to that of the photon with a wavelength of \( 4000 \, \text{Å} \) is:
\[ \frac{E_1}{E_2} = 2:1 \]