A light of amplitude A and wavelength `lamda` is incident on a metallic surface, then saturation current flows is proportional to (assume cut off wave length = `lamda_0`) -

Options

A. `A^2, if lamdagtlamda_0`
B. `A^2, if lamdaltlamda_0`
C. `A, if lamdagtlamda_0`
D. `A, if lamdaltlamda_0`

Answer

To solve the problem, we need to analyze the relationship between the saturation current and the parameters given in the question, specifically the amplitude \( A \) and the wavelength \( \lambda \) of the incident light.

### Step-by-Step Solution:

1. **Understanding the Photoelectric Effect**:
The photoelectric effect states that when light of sufficient energy strikes a metallic surface, it can eject electrons from that surface. The saturation current is the maximum current that can flow when all photoelectrons are collected.

2. **Energy of Incident Light**:
The energy \( E \) of a photon is given by the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light. For the photoelectric effect to occur, this energy must be greater than the work function \( \phi \) of the metal.

3. **Condition for Photoelectric Effect**:
The condition for the photoelectric effect to take place can be expressed as:
\[ \frac{hc}{\lambda} > \phi \]
Rearranging this gives:
\[ \lambda < \frac{hc}{\phi} \]
We denote \( \lambda_0 = \frac{hc}{\phi} \) as the cutoff wavelength. Thus, the condition becomes:
\[ \lambda < \lambda_0 \]

4. **Saturation Current and Amplitude**:
The saturation current \( I \) is directly proportional to the number of photoelectrons emitted per unit time, which is proportional to the intensity of the incident light. The intensity \( I \) of light is related to the amplitude \( A \) of the light wave:
\[ I \propto A^2 \]
Therefore, the saturation current can be expressed as:
\[ I \propto A^2 \]

5. **Final Relationship**:
Since the saturation current is proportional to the amplitude squared and the condition \( \lambda < \lambda_0 \) must be satisfied for the photoelectric effect to occur, we can conclude:
\[ I \propto A^2 \quad \text{for} \quad \lambda < \lambda_0 \]

### Conclusion:
The saturation current flows is proportional to \( A^2 \) when the wavelength \( \lambda \) is less than the cutoff wavelength \( \lambda_0 \).