Which of the following equations shows how how the linear expansivity ($\alpha$) and the cubic expansivity ($\gamma$) of a material are related?

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

Concept:

• Linear expansivity α: the fractional increase in length per unit original length per degree rise in temperature. L = L₀(1 + αθ)
• Cubic expansivity γ: the fractional increase in volume per unit original volume per degree rise in temperature. V = V₀(1 + γθ)

Solution: Consider a cube of side L₀. After heating by θ, each side becomes L₀(1 + αθ).

New volume: V = L₀³(1 + αθ)³ = V₀(1 + 3αθ + 3α²θ² + α³θ³)

Since α is very small (about 10⁻⁵ per K), α²θ² and α³θ³ are negligible, so:

V ≈ V₀(1 + 3αθ)

Comparing with V = V₀(1 + γθ):

γ = 3α

(Similarly, area expansivity β = 2α, so the pattern is α : β : γ = 1 : 2 : 3.)

Answer: Option D.

Tip: Remember the pattern 1-2-3: linear = α, area = 2α, volume = 3α. The number matches the number of dimensions.