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