A metal surface is illuminated by the protons of energy 5 eV and 2.5 eV respectively . The ratio of their wavelength is

Options

A. ` 1 : 3 `
B. ` 1 : 4 `
C. ` 2 : 5 `
D. ` 1 : 2 `

Answer

To solve the problem of finding the ratio of the wavelengths of two photons with energies of 5 eV and 2.5 eV, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between energy and wavelength**:
The energy (E) of a photon is related to its wavelength (λ) by the equation:
\[ E = \frac{hc}{\lambda} \]
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 (\(3 \times 10^8 \, \text{m/s}\)),
- \(\lambda\) is the wavelength of the photon.

2. **Express wavelength in terms of energy**:
Rearranging the equation gives:
\[ \lambda = \frac{hc}{E} \]
This shows that the wavelength is inversely proportional to the energy of the photon.

3. **Set up the ratio of wavelengths**:
For two photons with energies \(E_1 = 5 \, \text{eV}\) and \(E_2 = 2.5 \, \text{eV}\), we can write:
\[ \frac{\lambda_1}{\lambda_2} = \frac{E_2}{E_1} \]

4. **Substitute the values**:
Substituting the given energies into the ratio:
\[ \frac{\lambda_1}{\lambda_2} = \frac{2.5 \, \text{eV}}{5 \, \text{eV}} = \frac{1}{2} \]

5. **Express the ratio of wavelengths**:
Thus, we can express the ratio of the wavelengths as:
\[ \lambda_1 : \lambda_2 = 1 : 2 \]

### Final Answer:
The ratio of the wavelengths of the two photons is \(1 : 2\).

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