The hollow cylinder of length l and inner and outer radius `R_1` and `R_2` respectively. Find resistance of cylinder if current flows radially outward in the cylinder. Resistivity of material of cylinder id `rho` .

Options

A. `((rho/(piel)ln(R_1/R_2))`
B. `((rho/(4piel)ln(R_2/R_1))`
C. `((rho/(3piel)ln(R_2/R_1))`
D. `((rho/(2piel)ln(R_2/R_1))`

Answer

To find the resistance of a hollow cylinder with inner radius \( R_1 \), outer radius \( R_2 \), length \( L \), and resistivity \( \rho \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a hollow cylinder with inner radius \( R_1 \) and outer radius \( R_2 \). The current flows radially outward from the inner surface to the outer surface.

2. **Define the Resistance Formula**:
- The resistance \( R \) of a conductor is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity, \( L \) is the length of the conductor, and \( A \) is the cross-sectional area through which the current flows.

3. **Identify the Length and Area**:
- For a thin cylindrical shell at a radius \( r \) with thickness \( dr \):
- The length \( L \) of the cylinder remains constant.
- The cross-sectional area \( A \) for the current flowing through this shell is the circumference of the shell times its length:
\[ A = 2 \pi r L \]

4. **Calculate the Resistance of the Thin Shell**:
- The resistance \( dR \) of a thin shell of thickness \( dr \) at radius \( r \) is given by:
\[ dR = \frac{\rho \, dr}{A} = \frac{\rho \, dr}{2 \pi r L} \]

5. **Integrate to Find Total Resistance**:
- To find the total resistance \( R \) of the hollow cylinder, integrate \( dR \) from \( R_1 \) to \( R_2 \):
\[ R = \int_{R_1}^{R_2} dR = \int_{R_1}^{R_2} \frac{\rho \, dr}{2 \pi r L} \]
- This simplifies to:
\[ R = \frac{\rho}{2 \pi L} \int_{R_1}^{R_2} \frac{dr}{r} \]
- The integral \( \int \frac{dr}{r} \) evaluates to \( \ln(r) \):
\[ R = \frac{\rho}{2 \pi L} \left[ \ln(r) \right]_{R_1}^{R_2} = \frac{\rho}{2 \pi L} \left( \ln(R_2) - \ln(R_1) \right) \]

6. **Final Expression**:
- Using the properties of logarithms, this can be expressed as:
\[ R = \frac{\rho}{2 \pi L} \ln\left(\frac{R_2}{R_1}\right) \]

### Final Result:
\[ R = \frac{\rho}{2 \pi L} \ln\left(\frac{R_2}{R_1}\right) \]