Estimating copper resistivity at elevated temperature (700 K) A copper specimen has resistivity ρ ≈ 1.8 × 10^-8 Ω·m at room temperature (~293 K). Using a linear temperature coefficient model with α ≈ 3.9 × 10^-3 K^-1, estimate the resistivity at 700 K. Choose the closest value.

Introduction / Context:
Metallic resistivity increases with temperature due to enhanced electron–phonon scattering. For engineering estimates far from phase transitions, a linear temperature coefficient model is widely used. This question checks your ability to apply ρ(T) = ρ0 [1 + α (T − T0)].

Given Data / Assumptions:

• Base resistivity: ρ0 = 1.8 × 10^-8 Ω·m at T0 ≈ 293 K.
• Target temperature: T = 700 K.
• Temperature coefficient: α ≈ 3.9 × 10^-3 K^-1 (typical for copper near room temperature).
• Linear model valid over this range for an estimate.

Concept / Approach:

Use the linear relation ρ(T) = ρ0 [1 + α (T − T0)]. Compute the temperature rise and scale ρ0 accordingly to get the elevated-temperature resistivity. Select the closest option to the computed value.

Step-by-Step Solution:

Verification / Alternative check:

Using α in the range 3.8–4.1 × 10^-3 K^-1 yields ρ(T) ≈ 4.6–4.8 × 10^-8 Ω·m, consistent with the selected value.

Why Other Options Are Wrong:

2.0 and 1.6 × 10^-8 Ω·m are near or below room-temperature values; 3.0 × 10^-8 Ω·m underestimates the increase; 6.0 × 10^-8 Ω·m overestimates typical linear scaling to 700 K.

Common Pitfalls:

Forgetting to convert temperature to Kelvin for ΔT or using α with the wrong sign; assuming constant α far outside the fitted range without sanity checks.

Final Answer:

4.7 × 10^-8 Ω·m