The wave length of two photons A and B are 100 Å and 100 nm. The ratio of their energies will be

Options

A. `1:1`
B. `2:1`
C. `10:1`
D. `1:10`

Answer

To find the ratio of the energies of two photons A and B with wavelengths of 100 Å and 100 nm, we can follow these steps:

### Step 1: Understand the relationship between energy and wavelength
The energy (E) of a photon is related to its wavelength (λ) by the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength of the photon.

### Step 2: Convert wavelengths to meters
We need to convert the given wavelengths from Ångströms and nanometers to meters:
- Wavelength of photon A (λ₁) = 100 Å = \( 100 \times 10^{-10} \, \text{m} = 1 \times 10^{-8} \, \text{m} \)
- Wavelength of photon B (λ₂) = 100 nm = \( 100 \times 10^{-9} \, \text{m} = 1 \times 10^{-7} \, \text{m} \)

### Step 3: Write the energy formulas for both photons
Using the energy formula, we can express the energies of photons A and B:
- Energy of photon A (E₁):
\[ E_1 = \frac{hc}{\lambda_1} = \frac{hc}{1 \times 10^{-8}} \]
- Energy of photon B (E₂):
\[ E_2 = \frac{hc}{\lambda_2} = \frac{hc}{1 \times 10^{-7}} \]

### Step 4: Find the ratio of their energies
Now, we can find the ratio of the energies:
\[ \frac{E_1}{E_2} = \frac{\frac{hc}{\lambda_1}}{\frac{hc}{\lambda_2}} = \frac{\lambda_2}{\lambda_1} \]
Substituting the values of λ₁ and λ₂:
\[ \frac{E_1}{E_2} = \frac{1 \times 10^{-7}}{1 \times 10^{-8}} = 10 \]

### Step 5: Conclusion
The ratio of the energies of the two photons A and B is:
\[ \frac{E_1}{E_2} = 10 \]

### Final Answer
The ratio of their energies \( E_1 : E_2 = 10 : 1 \).

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