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 solve the problem of finding the resistance of a hollow cylindrical resistor, we can follow these steps:

### Step 1: Identify the given parameters
- 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 of the cylinder \( l = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Resistivity \( \rho = 3.5 \times 10^{-5} \, \Omega \cdot \text{m} \)

### Step 2: Calculate the cross-sectional area of the hollow cylinder
The cross-sectional area \( A \) of the hollow cylinder can be calculated using the formula:
\[ A = \pi (r_b^2 - r_a^2) \]
Substituting the values:
\[ A = \pi \left( (0.01)^2 - (0.005)^2 \right) \]
Calculating the squares:
\[ A = \pi \left( 0.0001 - 0.000025 \right) = \pi \left( 0.000075 \right) = \frac{3\pi}{4} \times 10^{-4} \, \text{m}^2 \]

### Step 3: Use the formula for resistance
The resistance \( R \) of the hollow cylinder can be calculated using the formula:
\[ R = \frac{\rho l}{A} \]
Substituting the known values:
\[ R = \frac{3.5 \times 10^{-5} \times 0.05}{\frac{3\pi}{4} \times 10^{-4}} \]

### Step 4: Simplify the expression
First, calculate the numerator:
\[ 3.5 \times 10^{-5} \times 0.05 = 1.75 \times 10^{-6} \]
Now, substitute this back into the resistance equation:
\[ R = \frac{1.75 \times 10^{-6}}{\frac{3\pi}{4} \times 10^{-4}} = \frac{1.75 \times 10^{-6} \times 4}{3\pi \times 10^{-4}} \]
This simplifies to:
\[ R = \frac{7 \times 10^{-6}}{3\pi \times 10^{-4}} = \frac{7}{3\pi} \times 10^{-2} \, \Omega \]

### Step 5: Calculate the numerical value
Using \( \pi \approx 3.14 \):
\[ R \approx \frac{7}{3 \times 3.14} \times 10^{-2} \approx \frac{7}{9.42} \times 10^{-2} \approx 0.0742 \, \Omega \]
Thus, the resistance \( R \) is approximately:
\[ R \approx 7.42 \times 10^{-3} \, \Omega \]

### Final Answer
The resistance of the configuration is \( 7.42 \times 10^{-3} \, \Omega \).

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