A monochromatic source of light is placed at a distance d from metal plate. The photoelectrons are ejected at a rate of n per second, and with maximum kinetic energy E . If the source is carried away from a distance d to a distance 3 d , then the rate of emission and the maximum kinetic energy of the photoelectron emitted will become nearly

Options

A. 3n and 3E, respectively
B. `n/9 and E/3` respectively
C. `n/9 and E` , respectively
D. n and E respectively

Explanation

To solve the problem, we need to analyze how the rate of emission of photoelectrons and their maximum kinetic energy change when the distance from the light source to the metal plate is increased from \( d \) to \( 3d \).

### Step-by-Step Solution:

1. **Understanding the Relationship Between Intensity and Photoelectron Emission**:
- The intensity \( I \) of light is directly proportional to the number of photoelectrons emitted per second \( n \).
- The intensity is also inversely proportional to the square of the distance \( d \) from the light source to the metal plate. Therefore, we can express this relationship as:
\[ n \propto \frac{1}{d^2} \]
- This means that if we increase the distance, the number of emitted photoelectrons will decrease.

2. **Calculating the New Rate of Emission at Distance \( 3d \)**:
- Initially, at distance \( d \), we have:
\[ n = k \cdot \frac{1}{d^2} \]
- When the distance is changed to \( 3d \), the new rate of emission \( n' \) can be expressed as:
\[ n' = k \cdot \frac{1}{(3d)^2} = k \cdot \frac{1}{9d^2} = \frac{n}{9} \]
- Thus, the new rate of emission of photoelectrons is \( n' = \frac{n}{9} \).

3. **Analyzing the Maximum Kinetic Energy of Photoelectrons**:
- The maximum kinetic energy \( E \) of the emitted photoelectrons is given by the photoelectric effect equation:
\[ E = hf - \phi \]
- Here, \( hf \) is the energy of the incident photons and \( \phi \) is the work function of the metal. The maximum kinetic energy depends on the frequency \( f \) of the light, not on the intensity or distance.
- Since the frequency of the light source remains unchanged when moving the source, the maximum kinetic energy of the photoelectrons remains the same:
\[ E' = E \]

4. **Final Results**:
- After moving the source from distance \( d \) to \( 3d \):
- The rate of emission of photoelectrons becomes \( n' = \frac{n}{9} \).
- The maximum kinetic energy remains \( E' = E \).

### Conclusion:
The rate of emission of photoelectrons will be \( \frac{n}{9} \) and the maximum kinetic energy will remain \( E \).