To calculate the resistance of a copper wire, we can use the formula for electrical resistance:
[tex]\[ R = \rho \times \frac{L}{A} \][/tex]
where:
- [tex]\( R \)[/tex] is the resistance,
- [tex]\( \rho \)[/tex] is the resistivity of the material,
- [tex]\( L \)[/tex] is the length of the wire,
- [tex]\( A \)[/tex] is the area of cross-section.
Let's go through the solution step by step:
1. Given Values:
- Length of the wire ([tex]\( L \)[/tex]) = 1000 meters.
- Area of cross-section ([tex]\( A \)[/tex]) = 2 mm².
- Resistivity of copper ([tex]\( \rho \)[/tex]) = [tex]\( 1.6 \times 10^{-8} \)[/tex] ohm meters (Ω·m).
2. Convert Area to Square Meters:
The given area of cross-section is in square millimeters (mm²). To use it in the formula, we need to convert it to square meters (m²).
[tex]\[
1 \text{ mm}^2 = 1 \times 10^{-6} \text{ m}^2
\][/tex]
Therefore,
[tex]\[
2 \text{ mm}^2 = 2 \times 10^{-6} \text{ m}^2
\][/tex]
3. Apply the Values to the Formula:
Now we substitute the given values into the formula for resistance:
[tex]\[
R = \rho \times \frac{L}{A} = (1.6 \times 10^{-8} \, \Omega \cdot \text{m}) \times \frac{1000 \text{ m}}{2 \times 10^{-6} \text{ m}^2}
\][/tex]
4. Simplify the Expression:
[tex]\[
R = 1.6 \times 10^{-8} \, \Omega \cdot \text{m} \times \frac{1000}{2 \times 10^{-6} \text{ m}^2}
\][/tex]
Simplify the fraction inside:
[tex]\[
\frac{1000}{2 \times 10^{-6}} = \frac{1000}{0.000002} = 500000000
\][/tex]
Hence, the calculation becomes:
[tex]\[
R = 1.6 \times 10^{-8} \, \Omega \cdot \text{m} \times 500000000
\][/tex]
5. Calculate the Product:
[tex]\[
R = 8.0 \, \Omega
\][/tex]
Therefore, the resistance of the copper wire with the given dimensions and resistivity is [tex]\( 8.0 \)[/tex] ohms.
