Two stream of photons, possessing energies equal to twice and ten times the work function of metal are incident on the metal surface successively. The value of ratio of maximum velocities of the photoelectrons emitted in the two respective cases is x : y. The value of x is ...............

Answer

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

### Step-by-Step Solution:

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

2. **Photon Energies**:
- For the first stream of photons:
\[ E_1 = 2\phi \]
- For the second stream of photons:
\[ E_2 = 10\phi \]

3. **Energy Conservation in Photoelectric Effect**:
The kinetic energy (KE) of the emitted photoelectrons can be expressed as:
\[ KE = E - \phi \]
where \( E \) is the energy of the incident photon.

4. **Calculate Kinetic Energy for Each Case**:
- For the first case (energy \( E_1 = 2\phi \)):
\[ KE_1 = E_1 - \phi = 2\phi - \phi = \phi \]
- For the second case (energy \( E_2 = 10\phi \)):
\[ KE_2 = E_2 - \phi = 10\phi - \phi = 9\phi \]

5. **Relate Kinetic Energy to Velocity**:
The kinetic energy of a particle is also given by:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is its velocity. Thus, we can write:
- For the first case:
\[ \frac{1}{2} mv_1^2 = \phi \quad \text{(1)} \]
- For the second case:
\[ \frac{1}{2} mv_2^2 = 9\phi \quad \text{(2)} \]

6. **Express Velocities in Terms of Kinetic Energy**:
From equation (1):
\[ v_1^2 = \frac{2\phi}{m} \]
From equation (2):
\[ v_2^2 = \frac{18\phi}{m} \]

7. **Calculate the Ratio of Velocities**:
To find the ratio of the maximum velocities \( v_1 \) and \( v_2 \):
\[ \frac{v_1^2}{v_2^2} = \frac{\frac{2\phi}{m}}{\frac{18\phi}{m}} = \frac{2}{18} = \frac{1}{9} \]

8. **Taking the Square Root for Velocity Ratio**:
Taking the square root gives us:
\[ \frac{v_1}{v_2} = \sqrt{\frac{1}{9}} = \frac{1}{3} \]

9. **Expressing the Ratio**:
Thus, the ratio of the maximum velocities \( v_1 : v_2 = 1 : 3 \). Here, \( x = 1 \) and \( y = 3 \).

### Final Answer:
The value of \( x \) is \( 1 \).