A photon of energy `h upsilon` is absorbed by a free electron of a metal having work function `phi lt h upsilon`

Options

A. The electron is sure to come out
B. The electron is sure to come out with a kinetic energy `(hv-w)`
C. Either the electron does not come out or it comes with a kinetic energy `(hv-w)`
D. It may come out with a kinetic energy less than `(hv-w)`

Explanation

To solve the given problem, we will analyze the interaction between a photon and a free electron in a metal. The key concepts involved are the energy of the photon, the work function of the metal, and the kinetic energy of the emitted electron.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- A photon with energy \( E = h\nu \) is absorbed by a free electron.
- The work function of the metal is denoted as \( \phi \).
- It is given that \( \phi < h\nu \).

2. **Photon Energy and Work Function**:
- The work function \( \phi \) is the minimum energy required to remove an electron from the surface of the metal.
- Since \( h\nu > \phi \), the photon has enough energy to overcome the work function and can potentially release the electron from the metal.

3. **Calculate the Kinetic Energy of the Electron**:
- When the photon is absorbed, the energy of the photon is used to do two things:
- Overcome the work function \( \phi \).
- Provide kinetic energy to the emitted electron.
- The relationship can be expressed as:
\[ \text{Kinetic Energy (KE)} = h\nu - \phi \]

4. **Determine the Maximum Kinetic Energy**:
- The maximum kinetic energy of the emitted electron occurs when all the excess energy (the energy of the photon minus the work function) is converted into kinetic energy.
- Therefore, the maximum kinetic energy is given by:
\[ KE_{\text{max}} = h\nu - \phi \]

5. **Consider the Options**:
- The question provides four options regarding the behavior of the electron after absorbing the photon.
- Since the electron can come out with kinetic energy less than the maximum kinetic energy \( KE_{\text{max}} \) (for example, if it loses some energy to other processes), the correct interpretation is:
- The electron is sure to come out, but it may come out with kinetic energy less than \( h\nu - \phi \).

6. **Conclusion**:
- Based on the analysis, the correct answer is that the electron may come out with a kinetic energy less than \( h\nu - \phi \).

### Final Answer:
The answer is **option b**: The electron may come out with a kinetic energy less than \( h\nu - \phi \).