A copper wire has a length L = 2 m, a cross-sectional area A = 0.5 mm², and resistivity ρ = 1.7 × 10^-8 Ωm. Calculate the resistance of another wire made of the same material whose length is twice the length of the wire but has the same cross-sectional area.

Given,

• Initial length (L) = 2 m
• Initial area (A) = 0.5 mm² = 0.5 x 10^-6 m²
• Resistivity (ρ) = 1.7 × 10^-8 Ωm
• Final length (L') = 2 x L = 2 x 2 m = 4 m
• Final area (A') = A = 0.5 mm² = 0.5 x 10^-6 m²

Resistance when L = 2 m is given by,

$\text R = \text ρ \dfrac{\text L}{\text A} \\[1em] = 1.7\times 10^{-8} \times \dfrac{2}{0.5\times 10^{-6}} \\[1em] = 1.7\times 10^{-8} \times \dfrac{20}{5\times 10^{-6}} \\[1em] = 1.7\times 10^{-8} \times 4\times 10^6 \\[1em] = 1.7\times 4\times 10^{-8 + 6} \\[1em] = 1.7\times 4\times 10^{-2} \\[1em] = 6.8\times 10^{-2} \\[1em] \Rightarrow \text R = 0.068\ \text Ω$

Resistance when L' = 4 m is given by,

$\text R' = \text ρ \dfrac{\text L'}{\text A'} \\[1em] = \text ρ \dfrac{2\text L}{\text A} \\[1em] = 2\text ρ \dfrac{\text L}{\text A} \\[1em] = 2\text R \\[1em] = 2\times 0.068 \\[1em] \Rightarrow \text R' = 0.136\ \text Ω$

Hence, the resistance of the wire when the length is double is 0.136 Ω.