To determine the electrical resistance (\( R \)) when resistivity (\( \rho \)), cross-sectional area (\( A \)), and length (\( L \)) are known, use the formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
where:
- \( R \) is the resistance (in ohms, \( \Omega \)),
- \( \rho \) is the resistivity (in ohm meters, \( \Omega \cdot \text{m} \)),
- \( A \) is the cross-sectional area (in square meters, \( \text{m}^2 \)),
- \( L \) is the length (in meters, \( \text{m} \)).
Problem 1: Resistance of a Steel Wire
Scenario: A steel wire has a resistivity of \( 1.43 \times 10^{-7} \, \Omega \cdot \text{m} \), a length of \( 15 \, \text{m} \), and a cross-sectional area of \( 2 \times 10^{-6} \, \text{m}^2 \). What is the resistance?
Calculation:
1. Given:
\[ \rho = 1.43 \times 10^{-7} \, \Omega \cdot \text{m} \]
\[ L = 15 \, \text{m} \]
\[ A = 2 \times 10^{-6} \, \text{m}^2 \]
2. Substitute into the Resistance Formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
\[ R = \dfrac{1.43 \times 10^{-7} \cdot 15}{2 \times 10^{-6}} \]
3. Calculate:
\[ R = \dfrac{2.145 \times 10^{-6}}{2 \times 10^{-6}} = 1.0725 \, \Omega \]
Answer: The resistance of the steel wire is \( 1.0725 \, \Omega \).
Problem 2: Resistance in a Silver Cable
Scenario: A silver cable has a resistivity of \( 1.59 \times 10^{-8} \, \Omega \cdot \text{m} \), a length of \( 30 \, \text{m} \), and a cross-sectional area of \( 1 \times 10^{-4} \, \text{m}^2 \). Determine its resistance.
Calculation:
1. Given:
\[ \rho = 1.59 \times 10^{-8} \, \Omega \cdot \text{m} \]
\[ L = 30 \, \text{m} \]
\[ A = 1 \times 10^{-4} \, \text{m}^2 \]
2. Substitute into the Resistance Formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
\[ R = \dfrac{1.59 \times 10^{-8} \cdot 30}{1 \times 10^{-4}} \]
3. Calculate:
\[ R = \dfrac{4.77 \times 10^{-7}}{1 \times 10^{-4}} = 4.77 \times 10^{-3} \, \Omega \]
Answer: The resistance of the silver cable is \( 4.77 \times 10^{-3} \, \Omega \).
Problem 3: Resistance of a Carbon Rod
Scenario: A carbon rod has a resistivity of \( 3.5 \times 10^{-5} \, \Omega \cdot \text{m} \), a length of \( 2 \, \text{m} \), and a cross-sectional area of \( 0.5 \times 10^{-4} \, \text{m}^2 \). Calculate its resistance.
Calculation:
1. Given:
\[ \rho = 3.5 \times 10^{-5} \, \Omega \cdot \text{m} \]
\[ L =
2 \, \text{m} \]
\[ A = 0.5 \times 10^{-4} \, \text{m}^2 \]
2. Substitute into the Resistance Formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
\[ R = \dfrac{3.5 \times 10^{-5} \cdot 2}{0.5 \times 10^{-4}} \]
3. Calculate:
\[ R = \dfrac{7 \times 10^{-5}}{0.5 \times 10^{-4}} = 1.4 \, \Omega \]
Answer: The resistance of the carbon rod is \( 1.4 \, \Omega \).
