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 :
The speed of light in the medium is

Options

A. minimum on the axis of the beam
B. the same everywhere in the beam
C. directly proportional to the intensity l
D. maximum on the axis of the beam

Answer

To solve the problem, we need to find the speed of light in a medium with a refractive index that varies with the intensity of the light beam. 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.

### Step 1: Understand the relationship between speed of light and refractive index
The speed of light in a medium is related to the refractive index by the formula:

\[ v = \frac{c}{\mu} \]

where \( v \) is the speed of light in the medium, \( c \) is the speed of light in vacuum, and \( \mu \) is the refractive index of the medium.

### Step 2: Substitute the expression for refractive index
Substituting the expression for the refractive index into the speed of light formula gives:

\[ v = \frac{c}{\mu_0 + \mu_2 I} \]

### Step 3: Analyze the intensity variation with radius
The problem states that the intensity \( I \) decreases with increasing radius \( r \). This means that at the center of the beam (where \( r = 0 \)), the intensity \( I \) is at its maximum.

### Step 4: Determine the speed of light at maximum intensity
Since the intensity is maximum at the center (axis of the beam), we can say:

- At \( r = 0 \), \( I \) is maximum.
- As \( r \) increases, \( I \) decreases.

Thus, when \( r \) is minimum (at the axis), the refractive index \( \mu_i \) is minimum because it depends on \( I \).

### Step 5: Conclusion about the speed of light
Since the speed of light \( v \) is inversely proportional to the refractive index \( \mu_i \):

- When \( \mu_i \) is minimum (at the axis), \( v \) is maximum.
- When \( \mu_i \) is maximum (away from the axis), \( v \) is minimum.

Therefore, the speed of light in the medium is minimum on the axis of the beam.

### Final Answer
The speed of light in the medium is minimum on the axis of the beam.