The linear expansivity, $\alpha$, and cubic expansivity, $\gamma$, of a material are related by the equation?

Analyze the question: It asks for the relationship between linear expansivity (α) and cubic (volume) expansivity (γ) for a solid material.

Concept: When a solid is heated, each of its dimensions expands. Linear expansivity α is the fractional increase in length per unit rise in temperature. Cubic expansivity γ is the fractional increase in volume per unit rise in temperature.

Derivation:

Take a cube of side L₀ at the starting temperature, so its volume is V₀ = L₀³.

After a temperature rise Δθ, the new side is:

L = L₀(1 + αΔθ)

The new volume is:

V = L³ = L₀³(1 + αΔθ)³

Expanding with the binomial theorem:

V = V₀(1 + 3αΔθ + 3α²Δθ² + α³Δθ³)

Because α is very small (about 10⁻⁵ per K), the terms α² and α³ are negligible, so:

V ≈ V₀(1 + 3αΔθ)

Compare this with the definition V = V₀(1 + γΔθ):

γ = 3α

Checking the options:

• A. γ = α is wrong.
• B. γ = 3α is correct.
• C. γ = 2α would be the areal expansivity relation (β = 2α), so it is wrong.
• D. γ = α/3 is wrong.

Tip: Remember the pattern α : β : γ = 1 : 2 : 3. Linear is 1D, area is 2D, volume is 3D, so the multiplier equals the number of dimensions.