Electrical resistivity (\( \rho \)) is a measure of how strongly a material opposes the flow of electric current. It is the reciprocal of conductivity (\( \sigma \)). This article explains how to find resistivity when conductivity is known, using the formula \( \rho = \dfrac{1}{\sigma} \). We’ll provide five practical examples to illustrate the calculations.
Formula to Determine Electrical Resistivity
Electrical resistivity (\( \rho \)) can be calculated using the formula:
\[ \rho = \dfrac{1}{\sigma} \]
where:
- \( \rho \) is the resistivity (in ohm meters, \( \Omega \cdot \text{m} \)),
- \( \sigma \) is the conductivity (in siemens per meter, S/m).
Example 1: Resistivity of Copper
Scenario: The conductivity of copper is \( 5.95 \times 10^7 \, \text{S/m} \). What is the resistivity?
Step-by-Step Calculation:
1. Given:
\[ \sigma = 5.95 \times 10^7 \, \text{S/m} \]
2. Substitute Values into the Resistivity Formula:
\[ \rho = \dfrac{1}{\sigma} \]
\[ \rho = \dfrac{1}{5.95 \times 10^7} \]
3. Perform the Calculation:
\[ \rho \approx 1.68 \times 10^{-8} \, \Omega \cdot \text{m} \]
Final Value
The resistivity of copper is:
\[ \rho \approx 1.68 \times 10^{-8} \, \Omega \cdot \text{m} \]
Example 2: Resistivity of Aluminum
Scenario: The conductivity of aluminum is \( 3.77 \times 10^7 \, \text{S/m} \). Find the resistivity.
Step-by-Step Calculation:
1. Given:
\[ \sigma = 3.77 \times 10^7 \, \text{S/m} \]
2. Substitute Values into the Resistivity Formula:
\[ \rho = \dfrac{1}{\sigma} \]
\[ \rho = \dfrac{1}{3.77 \times 10^7} \]
3. Perform the Calculation:
\[ \rho \approx 2.65 \times 10^{-8} \, \Omega \cdot \text{m} \]
Final Value
The resistivity of aluminum is:
\[ \rho \approx 2.65 \times 10^{-8} \, \Omega \cdot \text{m} \]
Example 3: Resistivity of Silver
Scenario: The conductivity of silver is \( 6.29 \times 10^7 \, \text{S/m} \). Calculate the resistivity.
Step-by-Step Calculation:
1. Given:
\[ \sigma = 6.29 \times 10^7 \, \text{S/m} \]
2. Substitute Values into the Resistivity Formula:
\[ \rho = \dfrac{1}{\sigma} \]
\[ \rho = \dfrac{1}{6.29 \times 10^7} \]
3. Perform the Calculation:
\[ \rho \approx 1.59 \times 10^{-8} \, \Omega \cdot \text{m} \]
Final Value
The resistivity of silver is:
\[ \rho \approx 1.59 \times 10^{-8} \, \Omega \cdot \text{m} \]
Example 4: Resistivity of Iron
Scenario: The conductivity of iron is \( 1.03 \times 10^7 \, \text{S/m} \). What is the resistivity?
Step-by-Step Calculation:
1. Given:
\[ \sigma = 1.03 \times 10^7 \, \text{S/m} \]
2. Substitute Values into the Resistivity Formula:
\[ \rho = \dfrac{1}{\sigma} \]
\[ \rho = \dfrac{1}{1.03 \times 10^7} \]
3. Perform the Calculation:
\[ \rho \approx 9.71 \times 10^{-8} \, \Omega \cdot \text{m} \]
Final Value
The resistivity of iron is:
\[ \rho \approx 9.71 \times 10^{-8} \, \Omega \cdot \text{m} \]
Example 5: Resistivity of Gold
Scenario: The conductivity of gold is \( 4.10 \times 10^7 \, \text{S/m} \). Find the resistivity.
Step-by-Step Calculation:
1. Given:
\[ \sigma = 4.10 \times 10^7 \, \text{S/m} \]
2. Substitute Values into the Resistivity Formula:
\[ \rho = \dfrac{1}{\sigma} \]
\[ \rho = \dfrac{1}{4.10 \times 10^7} \]
3. Perform the Calculation:
\[ \rho \approx 2.44 \times 10^{-8} \, \Omega \cdot \text{m} \]
Final Value
The resistivity of gold is:
\[ \rho \approx 2.44 \times 10^{-8} \, \Omega \cdot \text{m} \]
Summary
To find the resistivity (\( \rho \)) given the conductivity (\( \sigma \)), use the formula:
\[ \rho = \dfrac{1}{\sigma} \]
In the examples provided:
1. Copper: \( \rho \approx 1.68 \times 10^{-8} \, \Omega \cdot \text{m} \)
2. Aluminum: \( \rho \approx 2.65 \times 10^{-8} \, \Omega \cdot \text{m} \)
3. Silver: \( \rho \approx 1.59 \times 10^{-8} \, \Omega \cdot \text{m} \)
4. Iron: \( \rho \approx 9.71 \times 10^{-8} \, \Omega \cdot \text{m} \)
5. Gold: \( \rho \approx 2.44 \times 10^{-8} \, \Omega \cdot \text{m} \)
