A hollow cylinder `(rho=2.2xx10^(-8)Omega-m)` of length 3 m has inner and outer diameters are 2 mm and 4 mm respectively. The resistance of the cylinder is

Options

A. `0.35xx10^(-3)Omega`
B. `3xx10^(-3)Omega`
C. `7xx10^(-3)Omega`
D. `3.1xx10^(-3)Omega`

Answer

To find the resistance of a hollow cylinder, we can use the formula for resistance given by:

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

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

### Step 1: Identify the given values
- Resistivity, \( \rho = 2.2 \times 10^{-8} \, \Omega \cdot m \)
- Length, \( L = 3 \, m \)
- Inner diameter, \( d_2 = 2 \, mm = 2 \times 10^{-3} \, m \)
- Outer diameter, \( d_1 = 4 \, mm = 4 \times 10^{-3} \, m \)

### Step 2: Calculate the inner and outer radii
- Inner radius, \( r_2 = \frac{d_2}{2} = \frac{2 \times 10^{-3}}{2} = 1 \times 10^{-3} \, m \)
- Outer radius, \( r_1 = \frac{d_1}{2} = \frac{4 \times 10^{-3}}{2} = 2 \times 10^{-3} \, m \)

### Step 3: Calculate the cross-sectional area \( A \)
The cross-sectional area of a hollow cylinder is given by:

\[ A = \pi (r_1^2 - r_2^2) \]

Substituting the values:

\[ A = \pi \left((2 \times 10^{-3})^2 - (1 \times 10^{-3})^2\right) \]

Calculating the squares:

\[ A = \pi \left(4 \times 10^{-6} - 1 \times 10^{-6}\right) \]
\[ A = \pi \left(3 \times 10^{-6}\right) \]

Now, substituting the value of \( \pi \approx 3.14 \):

\[ A \approx 3.14 \times 3 \times 10^{-6} \]
\[ A \approx 9.42 \times 10^{-6} \, m^2 \]

### Step 4: Substitute values into the resistance formula
Now we can substitute \( \rho \), \( L \), and \( A \) into the resistance formula:

\[ R = \frac{2.2 \times 10^{-8} \cdot 3}{9.42 \times 10^{-6}} \]

### Step 5: Calculate the resistance
Calculating the numerator:

\[ 2.2 \times 10^{-8} \cdot 3 = 6.6 \times 10^{-8} \]

Now, divide by the area:

\[ R = \frac{6.6 \times 10^{-8}}{9.42 \times 10^{-6}} \]

Calculating this gives:

\[ R \approx 7 \times 10^{-3} \, \Omega \]

### Final Result
Thus, the resistance of the hollow cylinder is:

\[ R \approx 7 \times 10^{-3} \, \Omega \]