The resistance of 100m of copper wire of diameter `0.056 xx 10^(-2)` m is 6.62`Omega`. Calculate (i) resistivity, (ii) Electrical conductivity of copper.

Answer

To solve the problem, we need to calculate the resistivity and the electrical conductivity of the copper wire given its resistance, length, and diameter.

### Given:
- Length of the wire (L) = 100 m
- Diameter of the wire (d) = \(0.056 \times 10^{-2}\) m
- Resistance (R) = 6.62 Ω

### Step 1: Calculate the radius of the wire
The radius (R) can be calculated from the diameter using the formula:
\[ R = \frac{d}{2} \]
Substituting the value of the diameter:
\[ R = \frac{0.056 \times 10^{-2}}{2} = 0.028 \times 10^{-2} \text{ m} \]

### Step 2: Calculate the cross-sectional area (A) of the wire
The cross-sectional area (A) of the wire can be calculated using the formula for the area of a circle:
\[ A = \pi R^2 \]
Substituting the radius:
\[ A = \pi \left(0.028 \times 10^{-2}\right)^2 \]
Calculating this gives:
\[ A \approx \pi \times (0.000000784) \approx 2.465 \times 10^{-6} \text{ m}^2 \]

### Step 3: Calculate the resistivity (ρ) of the copper wire
Using the formula for resistance:
\[ R = \frac{\rho L}{A} \]
We can rearrange this to find resistivity (ρ):
\[ \rho = \frac{R \cdot A}{L} \]
Substituting the known values:
\[ \rho = \frac{6.62 \cdot (2.465 \times 10^{-6})}{100} \]
Calculating this gives:
\[ \rho \approx \frac{1.63 \times 10^{-7}}{1} = 1.63 \times 10^{-8} \text{ Ω m} \]

### Step 4: Calculate the electrical conductivity (σ) of copper
The electrical conductivity (σ) is the reciprocal of resistivity:
\[ \sigma = \frac{1}{\rho} \]
Substituting the value of resistivity:
\[ \sigma = \frac{1}{1.63 \times 10^{-8}} \approx 6.13 \times 10^{7} \text{ S/m} \]

### Final Answers:
(i) Resistivity (ρ) = \(1.63 \times 10^{-8} \text{ Ω m}\)

(ii) Electrical conductivity (σ) = \(6.13 \times 10^{7} \text{ S/m}\)