A blacksmith heated a metal whose cubic expansivity  is 3.9 x 10$^{-6}$K$^{-1}$. Calculate the area expansivity.

Analyze the question: We are given the cubic (volume) expansivity, γ = 3.9 × 10⁻⁶ K⁻¹, and asked for the area (superficial) expansivity, β.

Concept: For an isotropic solid, the three expansivities are related to the linear expansivity α:

• Linear: α
• Area: β = 2α
• Cubic: γ = 3α

So the ratio is α : β : γ = 1 : 2 : 3.

Solution:

1. Find the linear expansivity: α = γ / 3 = (3.9 × 10⁻⁶) / 3 = 1.3 × 10⁻⁶ K⁻¹

2. Find the area expansivity: β = 2α = 2 × 1.3 × 10⁻⁶ = 2.6 × 10⁻⁶ K⁻¹

Alternatively, β = (2/3)γ = (2/3)(3.9 × 10⁻⁶) = 2.6 × 10⁻⁶ K⁻¹.

Answer: The area expansivity is 2.6 × 10⁻⁶ K⁻¹, which is option B.

Note that option C (1.3 × 10⁻⁶) is the linear expansivity, a common trap if you stop after the first step.

Tip: Remember "1-2-3": linear, area, cubic. To go from cubic to area, multiply by 2/3. To go from linear to area, multiply by 2.