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 \( R_2 \), and length \( l \), where the current flows radially outward, we can follow these steps:

### Step 1: Understand the Resistance Formula
The resistance \( R \) of a conductor is given by the formula:
\[ R = \frac{\rho \cdot L}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the conductor,
- \( A \) is the cross-sectional area.

### Step 2: Consider an Elemental Thickness
For a hollow cylinder, we consider an elemental thickness \( dr \) at a distance \( r \) from the center. The length \( L \) of this elemental section is \( l \), and the area \( A \) at this distance is the lateral surface area of the cylindrical shell:
\[ A = 2 \pi r \cdot l \]

### Step 3: Expression for Resistance of the Element
The resistance \( dR \) of this elemental thickness \( dr \) can be expressed as:
\[ dR = \frac{\rho \cdot dr}{A} = \frac{\rho \cdot dr}{2 \pi r \cdot l} \]

### Step 4: Integrate to Find Total Resistance
To find the total resistance \( R \) of the hollow cylinder from radius \( R_1 \) to \( R_2 \), we integrate \( dR \):
\[ R = \int_{R_1}^{R_2} dR = \int_{R_1}^{R_2} \frac{\rho \cdot dr}{2 \pi r \cdot l} \]
This simplifies to:
\[ R = \frac{\rho}{2 \pi l} \int_{R_1}^{R_2} \frac{dr}{r} \]

### Step 5: Solve the Integral
The integral \( \int \frac{dr}{r} \) evaluates to \( \ln r \):
\[ R = \frac{\rho}{2 \pi l} \left[ \ln r \right]_{R_1}^{R_2} = \frac{\rho}{2 \pi l} \left( \ln R_2 - \ln R_1 \right) \]
Using properties of logarithms, this can be rewritten as:
\[ R = \frac{\rho}{2 \pi l} \ln \left( \frac{R_2}{R_1} \right) \]

### Final Result
Thus, the total resistance \( R \) of the hollow cylinder is:
\[ R = \frac{\rho}{2 \pi l} \ln \left( \frac{R_2}{R_1} \right) \]