How To Calculate Cross-Sectional Area

To determine the cross-sectional area ($A$) when resistivity ($\rho$), resistance ($R$), and length ($L$) are known, use the formula:

$$A={\displaystyle \frac{\rho \cdot L}{R}}$$

where:

• $A$ is the cross-sectional area (in square meters, ${\text{m}}^2$),
• $\rho$ is the resistivity (in ohm meters, $\mathrm{\Omega }\cdot \text{m}$),
• $R$ is the resistance (in ohms, $\mathrm{\Omega }$),
• $L$ is the length (in meters, $\text{m}$).

Problem 1: Area of a Copper Conductor

Scenario: A copper conductor has a resistivity of $1.68\times {10}^{-8}\mathrm{\Omega }\cdot \text{m}$, a resistance of $0.2\mathrm{\Omega }$, and a length of $5\text{m}$. What is its cross-sectional area?

Calculation:

1. Given:

  $$\rho =1.68\times {10}^{-8}\mathrm{\Omega }\cdot \text{m}$$

  $$R=0.2\mathrm{\Omega }$$

  $$L=5\text{m}$$

2. Substitute into the Area Formula:

  $$A={\displaystyle \frac{\rho \cdot L}{R}}$$

  $$A={\displaystyle \frac{1.68\times {10}^{-8}\cdot 5}{0.2}}$$

3. Calculate:

  $$A={\displaystyle \frac{8.4\times {10}^{-8}}{0.2}}=4.2\times {10}^{-7}{\text{m}}^2$$

Answer: The cross-sectional area of the copper conductor is $4.2\times {10}^{-7}{\text{m}}^2$.

Problem 2: Area of a Gold Wire

Scenario: A gold wire has a resistivity of $2.44\times {10}^{-8}\mathrm{\Omega }\cdot \text{m}$, a resistance of $0.5\mathrm{\Omega }$, and a length of $3\text{m}$. Determine its cross-sectional area.

Calculation:

1. Given:

  $$\rho =2.44\times {10}^{-8}\mathrm{\Omega }\cdot \text{m}$$

  $$R=0.5\mathrm{\Omega }$$

  $$L=3\text{m}$$

2. Substitute into the Area Formula:

  $$A={\displaystyle \frac{\rho \cdot L}{R}}$$

  $$A={\displaystyle \frac{2.44\times {10}^{-8}\cdot 3}{0.5}}$$

3. Calculate:

  $$A={\displaystyle \frac{7.32\times {10}^{-8}}{0.5}}=1.464\times {10}^{-7}{\text{m}}^2$$

Answer: The cross-sectional area of the gold wire is $1.464\times {10}^{-7}{\text{m}}^2$.

Problem 3: Area of a Tungsten Filament

Scenario: A tungsten filament has a resistivity of $5.6\times {10}^{-8}\mathrm{\Omega }\cdot \text{m}$, a resistance of $1\mathrm{\Omega }$, and a length of $0.1\text{m}$. Find its cross-sectional area.

Calculation:

1. Given:

  $$\rho =5.6\times {10}^{-8}\mathrm{\Omega }\cdot \text{m}$$

  $$R=1\mathrm{\Omega }$$

  $$L=0.1\text{m}$$

2. Substitute into the Area Formula:

  $$A={\displaystyle \frac{\rho \cdot L}{R}}$$

  $$A={\displaystyle \frac{5.6\times {10}^{-8}\cdot 0.1}{1}}$$

3. Calculate:

  $$A={\displaystyle \frac{5.6\times {10}^{-9}}{1}}=5.6\times {10}^{-9}{\text{m}}^2$$

Answer: The cross-sectional area of the tungsten filament is $5.6\times {10}^{-9}{\text{m}}^2$.