Question

The resistance of a metallic wire of 100 m is 20 \(\Omega\). If the radius of the wire is 5 mm, find the resistivity of the metal of the wire.

Answer 1

Given:
Resistance \( R = 20 \, \Omega \), Length \( l = 100 \, \text{m} \), Radius \( r = 5 \, \text{mm} = 0.005 \, \text{m} \).

Calculate cross-sectional area:
Area \( A = \pi r^2 \).
\( A = \pi (0.005 \, \text{m})^2 = 2.5 \pi \times 10^{-5} \, \text{m}^2 \).

Calculate resistivity:
Resistivity \( \rho = \frac{R A}{l} \).
\( \rho = \frac{(20 \, \Omega) (2.5 \pi \times 10^{-5} \, \text{m}^2)}{100 \, \text{m}} \).

Simplify:
\( \rho = 5 \pi \times 10^{-6} \, \Omega \cdot \text{m} \).

Result:
The resistivity is \( \rho \approx 5 \pi \times 10^{-6} \, \Omega \cdot \text{m} \).