The ratio of energy of a photon of `2000Å` wavelength radiation to that of `4000Å` radiation is

Options

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

Answer

To find the ratio of the energy of a photon with a wavelength of 2000 Å to that of a photon with a wavelength of 4000 Å, we can follow these steps:

### Step 1: Understand the formula for energy of a photon
The energy (E) of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength of the radiation.

### Step 2: Calculate the energy for the photon with wavelength 4000 Å
Convert the wavelength from angstroms to meters:
\[ 4000 \, \text{Å} = 4000 \times 10^{-10} \, \text{m} = 4 \times 10^{-7} \, \text{m} \]

Now, substitute the values into the energy formula:
\[ E_1 = \frac{hc}{\lambda_1} = \frac{hc}{4 \times 10^{-7}} \]

### Step 3: Calculate the energy for the photon with wavelength 2000 Å
Convert the wavelength from angstroms to meters:
\[ 2000 \, \text{Å} = 2000 \times 10^{-10} \, \text{m} = 2 \times 10^{-7} \, \text{m} \]

Now, substitute the values into the energy formula:
\[ E_2 = \frac{hc}{\lambda_2} = \frac{hc}{2 \times 10^{-7}} \]

### Step 4: Find the ratio of the energies
Now we can find the ratio of the energies \( E_1 \) to \( E_2 \):
\[ \frac{E_1}{E_2} = \frac{\frac{hc}{4 \times 10^{-7}}}{\frac{hc}{2 \times 10^{-7}}} \]

### Step 5: Simplify the ratio
The \( hc \) terms cancel out:
\[ \frac{E_1}{E_2} = \frac{1}{4 \times 10^{-7}} \times \frac{2 \times 10^{-7}}{1} = \frac{2}{4} = \frac{1}{2} \]

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