A photo cell is illuminated by a small bright source placed 1m away When the same source of light is placed 2m away, the electrons emitted by photo cathode

Options

A. carry one quarter of their previous energy
B. carry one quarter of their previous momenta
C. are half as numerous
D. are one quarter as numerous

Explanation

To solve the problem, we need to understand the relationship between the intensity of light, the distance from the light source, and the number of electrons emitted by the photocathode in a photo cell.

### Step-by-Step Solution:

1. **Understanding Intensity and Distance**:
The intensity \( I \) of light from a point source is inversely proportional to the square of the distance \( d \) from the source. Mathematically, this is expressed as:
\[ I \propto \frac{1}{d^2} \]

2. **Calculating Initial Intensity**:
When the light source is at a distance of 1 meter, let’s denote the initial intensity as \( I_0 \):
\[ I_0 = \frac{P}{4\pi(1)^2} = \frac{P}{4\pi} \]
where \( P \) is the power of the light source.

3. **Calculating New Intensity**:
When the light source is moved to a distance of 2 meters, the new intensity \( I \) becomes:
\[ I = \frac{P}{4\pi(2)^2} = \frac{P}{16\pi} \]

4. **Finding the Ratio of Intensities**:
To find the ratio of the new intensity to the initial intensity:
\[ \frac{I}{I_0} = \frac{\frac{P}{16\pi}}{\frac{P}{4\pi}} = \frac{1/16}{1/4} = \frac{1}{4} \]
This means that the new intensity \( I \) is one-fourth of the initial intensity \( I_0 \).

5. **Relating Intensity to Number of Electrons**:
The intensity of light is directly proportional to the number of electrons emitted by the photocathode. Therefore, if the intensity decreases to one-fourth, the number of emitted electrons will also decrease to one-fourth:
\[ n' = \frac{n_0}{4} \]
where \( n_0 \) is the initial number of emitted electrons and \( n' \) is the new number of emitted electrons.

6. **Conclusion**:
Hence, when the light source is moved from 1 meter to 2 meters away, the number of electrons emitted by the photocathode becomes one-fourth of the original number.

### Final Answer:
The electrons emitted by the photocathode are one quarter as numerous.

---