A hollow cyindrical conductor has length of 3.14 m ,while its inner and outer diametes are 4 mm and 8 mm respectively. The resistance of the conductor is `n xx 10^(-3) Omega`. If the resistivity of the material is `2.4 times 10^(-8)Omegam`.The valueof n is_____.

Answer

To solve the problem, we need to find the resistance of the hollow cylindrical conductor using the given parameters. The resistance \( R \) of a conductor can be calculated using 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, and
- \( A \) is the cross-sectional area.

### Step 1: Identify the given values
- Length \( L = 3.14 \, \text{m} \)
- Inner diameter \( d_i = 4 \, \text{mm} = 0.004 \, \text{m} \) (convert mm to m)
- Outer diameter \( d_o = 8 \, \text{mm} = 0.008 \, \text{m} \)
- Resistivity \( \rho = 2.4 \times 10^{-8} \, \Omega \cdot \text{m} \)

### Step 2: Calculate the inner and outer radii
- Inner radius \( r_i = \frac{d_i}{2} = \frac{0.004}{2} = 0.002 \, \text{m} \)
- Outer radius \( r_o = \frac{d_o}{2} = \frac{0.008}{2} = 0.004 \, \text{m} \)

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

\[ A = \pi (r_o^2 - r_i^2) \]

Substituting the values:

\[ A = \pi \left((0.004)^2 - (0.002)^2\right) \]

Calculating the squares:

\[ A = \pi \left(0.000016 - 0.000004\right) = \pi \left(0.000012\right) \]

Now calculating \( A \):

\[ A = \pi \times 0.000012 \approx 3.7699 \times 10^{-5} \, \text{m}^2 \]

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

\[ R = \frac{(2.4 \times 10^{-8}) \cdot (3.14)}{3.7699 \times 10^{-5}} \]

Calculating the numerator:

\[ (2.4 \times 10^{-8}) \cdot (3.14) \approx 7.5396 \times 10^{-8} \]

Now calculate \( R \):

\[ R = \frac{7.5396 \times 10^{-8}}{3.7699 \times 10^{-5}} \approx 2.0005 \times 10^{-3} \, \Omega \]

### Step 5: Express \( R \) in the form \( n \times 10^{-3} \)
From the calculation, we find:

\[ R \approx 2.0005 \times 10^{-3} \, \Omega \]

Thus, \( n \approx 2.0005 \).

### Final Answer
The value of \( n \) is approximately \( 2 \).

---