Resistance formula for wires

Steps:

• Resistance R = ρ L / A is the same for both wires P and Q.
• Cross-sectional area A ∝ d², so with d_Q = 2 d_P, A_Q = 4 A_P.
• Resistivity ρ_Q = 8 ρ_P (eight times that of wire P).
• Thus, L_Q / L_P = (A_Q / A_P) × (ρ_P / ρ_Q) = 4 × (1/8) = 1/2.

Why C is correct:

• The ratio follows from rearranging the resistance formula R = ρ L / A to L ∝ A / ρ, yielding L_Q / L_P = 1/2 given the factors of 4 and 8.

Why the others are wrong:

• A. 1/8: Would require ρ_Q = 32 ρ_P to offset A_Q = 4 A_P.
• B. 1/4: Would require ρ_Q = 16 ρ_P to offset A_Q = 4 A_P.
• D. 1: Would require ρ_Q = 4 ρ_P to offset A_Q = 4 A_P.

Final answer: C