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Calculate the resistance of a copper wire with a length of 1000 m and a cross-sectional area of 2 mm². The resistivity of copper is [tex]1.6 \times 10^{-8} \ \Omega \cdot \text{m}[/tex].

Sagot :

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.