An intially parallel cyclindrical beam travels in a medium of refractive index `mu (I) = mu_(0) + mu_(2) I`, where `mu_(0)` and `mu_(2)` are positive constants and `I` is intensity of light beam. The intensity of the beam is decreasing with increasing radius.
Answer the following questions :
As the beam enters the medium, it will

Options

A. diverge
B. converge
C. diverge near the axis and converge near the periphery
D. travel as a cylindrical beam

Answer

To solve the problem, we need to analyze how the cylindrical beam behaves as it enters a medium with a varying refractive index. The refractive index is given by:

\[ \mu(I) = \mu_0 + \mu_2 I \]

where \(\mu_0\) and \(\mu_2\) are positive constants, and \(I\) is the intensity of the light beam. The intensity of the beam decreases with increasing radius.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
The beam is initially parallel and cylindrical. This means that all rays of light are traveling in the same direction and are equidistant from each other.

2. **Effect of Refractive Index on Light Propagation**:
As the beam enters the medium, the refractive index varies with intensity. Since the intensity decreases with increasing radius, we can infer that the refractive index will also vary across the beam. Specifically, the rays closer to the center (where intensity is higher) will experience a higher refractive index compared to those further away (where intensity is lower).

3. **Behavior of Light Rays**:
According to Snell's law, light rays will bend towards the normal when they enter a medium with a higher refractive index. In our case, since the center of the beam has a higher refractive index, the rays closer to the axis will bend more than those at the periphery.

4. **Convergence of the Beam**:
As a result of the varying refractive index, the rays that are closer to the axis will converge towards the axis while the rays at the periphery will diverge less. This means that as the beam enters the medium, it will start to converge towards the axis.

5. **Conclusion**:
Therefore, as the beam enters the medium, it will converge.

### Final Answer:
The beam will **converge** as it enters the medium.