Radiation two photons having energies twice and five times the work function of a metal are incident successively on the metal surface. The ratio of the maximum velocity of photoelectrons emitted in the two cases will be

Options

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

Answer

To solve the problem, we need to determine the ratio of the maximum velocities of photoelectrons emitted when two different photon energies are incident on a metal surface. The energies of the photons are given as twice and five times the work function of the metal.

### Step-by-Step Solution:

1. **Define the Work Function**:
Let the work function of the metal be represented as \( W \).

2. **Photon Energies**:
- For the first photon, the energy is \( E_1 = 2W \).
- For the second photon, the energy is \( E_2 = 5W \).

3. **Apply the Photoelectric Effect Equation**:
The kinetic energy of the emitted photoelectrons can be expressed using the equation:
\[ KE = E - W \]
where \( KE \) is the kinetic energy of the emitted photoelectrons, \( E \) is the energy of the incident photon, and \( W \) is the work function.

4. **Calculate Kinetic Energy for Each Case**:
- For the first case (with energy \( E_1 \)):
\[ KE_1 = E_1 - W = 2W - W = W \]
- For the second case (with energy \( E_2 \)):
\[ KE_2 = E_2 - W = 5W - W = 4W \]

5. **Relate Kinetic Energy to Velocity**:
The kinetic energy can also be expressed in terms of the velocity of the emitted electrons:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

6. **Set Up Equations for Velocities**:
- For the first case:
\[ \frac{1}{2} mv_1^2 = W \quad \Rightarrow \quad v_1^2 = \frac{2W}{m} \]
- For the second case:
\[ \frac{1}{2} mv_2^2 = 4W \quad \Rightarrow \quad v_2^2 = \frac{8W}{m} \]

7. **Find the Ratio of Velocities**:
To find the ratio \( \frac{v_1}{v_2} \):
\[ \frac{v_1^2}{v_2^2} = \frac{\frac{2W}{m}}{\frac{8W}{m}} = \frac{2}{8} = \frac{1}{4} \]
Taking the square root gives:
\[ \frac{v_1}{v_2} = \frac{1}{2} \]

8. **Express the Ratio**:
Therefore, the ratio of the maximum velocities of the photoelectrons emitted in the two cases is:
\[ v_1 : v_2 = 1 : 2 \]

### Final Answer:
The ratio of the maximum velocity of photoelectrons emitted in the two cases is \( 1 : 2 \).