A resistor is constructed as hollow cylinder with inner and outer radii `r_(a)=0.5 cm, r_(b)=1.0` cm respectively and resistivity `rho=3.5 xx10^(-5)Omega.` The resistance of the configuration for the length of 5 cm cylinder is _______`xx10^(-3)Omega`.

Options

A. 7.42
B. 10.56
C. 14.38
D. 16.48

Answer

To find the resistance of a hollow cylindrical resistor, we can use the formula for resistance:

\[ R = \frac{\rho L}{A} \]

Where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity,
- \( L \) is the length of the cylinder,
- \( A \) is the cross-sectional area.

### Step 1: Identify the parameters
Given:
- Inner radius \( r_a = 0.5 \, \text{cm} = 0.005 \, \text{m} \)
- Outer radius \( r_b = 1.0 \, \text{cm} = 0.01 \, \text{m} \)
- Length \( L = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Resistivity \( \rho = 3.5 \times 10^{-5} \, \Omega \cdot \text{m} \)

### Step 2: Calculate the cross-sectional area \( A \)
The cross-sectional area of the hollow cylinder is given by the difference between the area of the outer circle and the area of the inner circle:

\[ A = \pi r_b^2 - \pi r_a^2 = \pi (r_b^2 - r_a^2) \]

Calculating \( r_b^2 \) and \( r_a^2 \):

\[ r_b^2 = (0.01)^2 = 0.0001 \, \text{m}^2 \]
\[ r_a^2 = (0.005)^2 = 0.000025 \, \text{m}^2 \]

Now, substituting these values into the area formula:

\[ A = \pi (0.0001 - 0.000025) = \pi (0.000075) = 0.000075\pi \, \text{m}^2 \]

### Step 3: Substitute values into the resistance formula
Now, substituting \( \rho \), \( L \), and \( A \) into the resistance formula:

\[ R = \frac{3.5 \times 10^{-5} \times 0.05}{0.000075\pi} \]

### Step 4: Calculate the resistance
Calculating the numerator:

\[ 3.5 \times 10^{-5} \times 0.05 = 1.75 \times 10^{-6} \]

Now substituting into the resistance formula:

\[ R = \frac{1.75 \times 10^{-6}}{0.000075\pi} \]

Calculating \( 0.000075\pi \):

\[ 0.000075\pi \approx 0.000235619 \]

Now, substituting this value:

\[ R = \frac{1.75 \times 10^{-6}}{0.000235619} \approx 7.42 \times 10^{-3} \, \Omega \]

### Final Result
Thus, the resistance of the configuration is approximately:

\[ R \approx 7.42 \times 10^{-3} \, \Omega \]