Final answer:
The length of a copper wire with a diameter of 0.5 mm, resistivity of 1.6 × 10⁻⁸ Ω\u00b7m, and resistance of 100 Ω is calculated to be approximately 200 meters using the relationship between resistance, resistivity, cross-sectional area, and length.
Step-by-step explanation:
The question involves calculating the length of a copper wire based on its diameter, resistivity, and resistance. To find the wire length when the wire's resistance is 100 Ω, we use the formula R = ρL/A, where R is resistance, ρ (rho) is resistivity, L is length, and A is the cross-sectional area of the wire. The area (A) is obtained using A = πr², where r is the radius of the wire. Since the diameter is given as 0.5 mm, the radius is 0.25 mm or 0.00025 meters.
First, calculate the cross-sectional area, A = π * (0.00025 m)² = 1.9635 × 10⁻⁷ m². Then, rearrange the original formula to find the length (L), L = R * A / ρ. Substituting the values, we get L = 100 Ω * 1.9635 × 10⁻⁷ m² / (1.6 × 10⁻⁸ Ω\u00b7m). After performing the calculation, we find that the length of the copper wire must be approximately 200 meters to achieve a resistance of 100 Ω.