Two radiations of photons energies 1 eV and 2.5 eV, successively illuminate a photosensitive metallic surface of work function 0.5 eV. The ratio of the maximum speeds of the emitted electrons is

Options

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

Explanation

To solve the problem, we need to determine the ratio of the maximum speeds of the emitted electrons when two different photon energies illuminate a photosensitive metallic surface. The work function of the surface is given, and we will use the photoelectric effect principles to find the solution.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Photon energy 1 (E1) = 1 eV
- Photon energy 2 (E2) = 2.5 eV
- Work function (Φ) = 0.5 eV

2. **Calculate the Kinetic Energy of Electrons:**
The kinetic energy (KE) of the emitted electrons can be calculated using the formula:
\[ KE = E - \Phi \]
where \(E\) is the energy of the incident photon.

- For the first photon (E1 = 1 eV):
\[ KE_1 = E1 - \Phi = 1 \, \text{eV} - 0.5 \, \text{eV} = 0.5 \, \text{eV} \]

- For the second photon (E2 = 2.5 eV):
\[ KE_2 = E2 - \Phi = 2.5 \, \text{eV} - 0.5 \, \text{eV} = 2.0 \, \text{eV} \]

3. **Relate Kinetic Energy to Speed:**
The kinetic energy of an electron is also given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \(m\) is the mass of the electron and \(v\) is its speed.

4. **Set Up the Ratio of Kinetic Energies:**
We can set up the ratio of the kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{0.5 \, \text{eV}}{2.0 \, \text{eV}} = \frac{1}{4} \]

5. **Relate the Speeds:**
Using the relationship between kinetic energy and speed, we can write:
\[ \frac{KE_1}{KE_2} = \frac{\frac{1}{2} m v_1^2}{\frac{1}{2} m v_2^2} = \frac{v_1^2}{v_2^2} \]
Since the mass \(m\) and the factor \(\frac{1}{2}\) cancel out.

6. **Calculate the Ratio of Speeds:**
From the ratio of kinetic energies:
\[ \frac{v_1^2}{v_2^2} = \frac{1}{4} \]
Taking the square root of both sides gives:
\[ \frac{v_1}{v_2} = \frac{1}{2} \]

### Final Result:
The ratio of the maximum speeds of the emitted electrons is:
\[ \frac{v_1}{v_2} = \frac{1}{2} \]