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

Step 1: Given information
We are given the following values:
- \( R = 20 \, \Omega \) (resistance of the wire),
- \( l = 100 \, \text{m} \) (length of the wire),
- radius \( r = 5 \, \text{mm} = 0.005 \, \text{m} \) (radius of the wire).

Step 2: Calculate the area of the cross-section of the wire
The wire has a circular cross-section, so we use the formula for the area of a circle:
\[ A = \pi r^2 \] Substituting the value of the radius \( r = 0.005 \, \text{m} \):
\[ A = \pi (0.005)^2 = 2.5 \pi \times 10^{-5} \, \text{m}^2 \]

Step 3: Use the formula for resistivity
The formula for resistivity is:
\[ \rho = \frac{R A}{l} \] Substitute the given values for \( R = 20 \, \Omega \), \( A = 2.5 \pi \times 10^{-5} \, \text{m}^2 \), and \( l = 100 \, \text{m} \):
\[ \rho = \frac{20 \times 2.5 \pi \times 10^{-5}}{100} \]

Step 4: Simplify the expression
\[ \rho = 5 \pi \times 10^{-6} \, \Omega \cdot \text{m} \]

Step 5: Final answer
Therefore, the resistivity of the metal is approximately:
\[ \rho \approx 5 \pi \times 10^{-6} \, \Omega \cdot \text{m} \]