A solid cylinder of length l and cross-sectional area A is made of a material whose resistivity depends on the distance r from the axis of the cylinder as `rho = k//r^(2)` where k is constant . The resistance of the cylinder is -

Options

A. `(pi kl)/(A^(2))`
B. `(2pi kl)/(A^(2))`
C. `(pi kl)/(A)`
D. None

Explanation

To find the resistance of a solid cylinder with a resistivity that depends on the distance from the axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given information**:
- The resistivity of the material is given by \( \rho = \frac{k}{r^2} \), where \( k \) is a constant and \( r \) is the distance from the axis of the cylinder.
- The cylinder has a length \( l \) and a cross-sectional area \( A \).

2. **Consider a differential element**:
- We will consider a thin cylindrical shell of thickness \( dr \) at a distance \( r \) from the axis. The area of this shell is \( dA = 2\pi r \, dr \).

3. **Calculate the resistance of the differential element**:
- The resistance \( dR \) of this thin shell can be expressed as:
\[ dR = \frac{\rho \, dl}{dA} \]
- Substituting the values, we have:
\[ dR = \frac{\frac{k}{r^2} \cdot l}{2\pi r \, dr} \]
- Simplifying this, we get:
\[ dR = \frac{k \cdot l}{2\pi r^3} \, dr \]

4. **Integrate to find total resistance**:
- To find the total resistance \( R \), we need to integrate \( dR \) from \( r = R_1 \) (inner radius) to \( r = R_2 \) (outer radius):
\[ R = \int_{R_1}^{R_2} dR = \int_{R_1}^{R_2} \frac{k \cdot l}{2\pi r^3} \, dr \]
- The integral of \( \frac{1}{r^3} \) is:
\[ \int \frac{1}{r^3} \, dr = -\frac{1}{2r^2} \]
- Thus, we have:
\[ R = \frac{k \cdot l}{2\pi} \left[-\frac{1}{2r^2}\right]_{R_1}^{R_2} \]
- Evaluating the limits:
\[ R = \frac{k \cdot l}{2\pi} \left(-\frac{1}{2R_2^2} + \frac{1}{2R_1^2}\right) \]
- This simplifies to:
\[ R = \frac{k \cdot l}{4\pi} \left(\frac{1}{R_1^2} - \frac{1}{R_2^2}\right) \]

5. **Final expression for resistance**:
- If we assume the cylinder is solid (i.e., \( R_1 = 0 \) and \( R_2 = R \)), we can express the total resistance as:
\[ R = \frac{2k l}{A} \]
- Where \( A \) is the cross-sectional area of the cylinder.

### Final Answer:
The resistance of the cylinder is given by:
\[ R = \frac{2k l}{A} \]