Resistance (\( R \)) is a measure of how much a material opposes the flow of electric current. It can be determined using the formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
where:
- \( R \) is the resistance (in ohms, \( \Omega \)),
- \( \rho \) is the resistivity (in ohm meters, \( \Omega \cdot \text{m} \)),
- \( L \) is the length of the material (in meters, m),
- \( A \) is the cross-sectional area (in square meters, \( \text{m}^2 \)).
We’ll illustrate how to find resistance with five practical examples.
Example 1: Resistance of a Copper Wire
Scenario: A copper wire has a length of \( 2 \, \text{m} \), a cross-sectional area of \( 1 \, \text{mm}^2 \) (\( 1 \times 10^{-6} \, \text{m}^2 \)), and a resistivity of \( 1.68 \times 10^{-8} \, \Omega \cdot \text{m} \). What is the resistance?
Step-by-Step Calculation:
1. Given:
\[ \rho = 1.68 \times 10^{-8} \, \Omega \cdot \text{m} \]
\[ L = 2 \, \text{m} \]
\[ A = 1 \, \text{mm}^2 = 1 \times 10^{-6} \, \text{m}^2 \]
2. Substitute Values into the Resistance Formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
\[ R = \dfrac{1.68 \times 10^{-8} \cdot 2}{1 \times 10^{-6}} \]
3. Perform the Calculation:
\[ R = 3.36 \times 10^{-2} \, \Omega \]
Final Value
The resistance of the copper wire is:
\[ R = 0.0336 \, \Omega \]
Example 2: Resistance of an Aluminum Rod
Scenario: An aluminum rod has a length of \( 5 \, \text{m} \), a cross-sectional area of \( 2 \, \text{mm}^2 \) (\( 2 \times 10^{-6} \, \text{m}^2 \)), and a resistivity of \( 2.65 \times 10^{-8} \, \Omega \cdot \text{m} \). Calculate the resistance.
Step-by-Step Calculation:
1. Given:
\[ \rho = 2.65 \times 10^{-8} \, \Omega \cdot \text{m} \]
\[ L = 5 \, \text{m} \]
\[ A = 2 \times 10^{-6} \, \text{m}^2 \]
2. Substitute Values into the Resistance Formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
\[ R = \dfrac{2.65 \times 10^{-8} \cdot 5}{2 \times 10^{-6}} \]
3. Perform the Calculation:
\[ R = 6.625 \times 10^{-2} \, \Omega \]
Final Value
The resistance of the aluminum rod is:
\[ R = 0.06625 \, \Omega \]
Example 3: Resistance of a Silver Cable
Scenario: A silver cable has a length of \( 10 \, \text{m} \), a cross-sectional area of \( 0.5 \, \text{mm}^2 \) (\( 0.5 \times 10^{-6} \, \text{m}^2 \)), and a resistivity of \( 1.59 \times 10^{-8} \, \Omega \cdot \text{m} \). What is the resistance?
Step-by-Step Calculation:
1. Given:
\[ \rho = 1.59 \times 10^{-8} \, \Omega \cdot \text{m} \]
\[ L = 10 \, \text{m} \]
\[ A = 0.5 \times 10^{-6} \, \text{m}^2 \]
2. Substitute Values into the Resistance Formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
\[ R = \dfrac{1.59 \times 10^{-8} \cdot 10}{0.5 \times 10^{-6}} \]
3. Perform the Calculation:
\[ R = 3.18 \times 10^{-1} \, \Omega \]
Final Value
The resistance of the silver cable is:
\[ R = 0.318 \, \Omega \]
Example 4: Resistance of an Iron Bar
Scenario: An iron bar has a length of \( 3 \, \text{m} \), a cross-sectional area of \( 3 \, \text{mm}^2 \) (\( 3 \times 10^{-6} \, \text{m}^2 \)), and a resistivity of \( 9.71 \times 10^{-8} \, \Omega \cdot \text{m} \). Calculate the resistance.
Step-by-Step Calculation:
1. Given:
\[ \rho = 9.71 \times 10^{-8} \, \Omega \cdot \text{m} \]
\[ L = 3 \, \text{m} \]
\[ A = 3 \times 10^{-6} \, \text{m}^2 \]
2. Substitute Values into the Resistance Formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
\[ R = \dfrac{9.71 \times 10^{-8} \cdot 3}{3 \times 10^{-6}} \]
3. Perform the Calculation:
\[ R = 9.71 \times 10^{-2} \, \Omega \]
Final Value
The resistance of the iron bar is:
\[ R = 0.0971 \, \Omega \]
Example 5: Resistance of a Gold Wire
Scenario: A gold wire has a length of \( 0.5 \, \text{m} \), a cross-sectional area of \( 0.1 \, \text{mm}^2 \) (\( 0.1 \times 10^{-6} \, \text{m}^2 \)), and a resistivity of \( 2.44 \times 10^{-8} \, \Omega \cdot \text{m} \). What is the resistance?
Step-by-Step Calculation:
1. Given:
\[ \rho = 2.44 \times 10^{-8} \, \Omega \cdot \text{m} \]
\[ L = 0.5 \, \text{m} \]
\[ A = 0.1 \times 10^{-6} \, \text{m}^2 \]
2. Substitute Values into the Resistance Formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
\[ R = \dfrac{2.44 \times 10^{-8} \cdot 0.5}{0.1 \times 10^{-6}} \]
3. Perform the Calculation:
\[ R = 1.22 \times 10^{-1} \, \Omega \]
Final Value
The resistance of the gold wire is:
\[ R = 0.122 \, \Omega \]
Summary
To find the resistance (\( R \)) given the resistivity (\( \rho \)), length (\( L \)), and cross-sectional area (\( A \)), use the formula:
\[ R = \dfrac{\rho \cdot L}{A} \]
In the examples provided:
1. Copper wire: \( R = 0.0336 \, \Omega \)
2. Aluminum rod: \( R = 0.06625 \, \Omega \)
3. Silver cable: \( R = 0.318 \, \Omega \)
4. Iron bar: \( R = 0.0971 \, \Omega \)
5. Gold wire: \( R = 0.122 \, \Omega \)
