The relationship between the coefficient of linear expansion $\alpha$ and volumetric expansion ($\gamma$) is-----------------

Analyze the question: You are asked for the relationship between the coefficient of linear expansion (α) and the coefficient of volume (cubic) expansion (γ) of a solid.

Concept: When a solid is heated, each of its dimensions expands. For a cube of side L₀:

• New side: L = L₀(1 + αΔT)
• Original volume: V₀ = L₀³
• New volume: V = L³ = L₀³(1 + αΔT)³

Solution:

Expand (1 + αΔT)³:

(1 + αΔT)³ = 1 + 3αΔT + 3α²ΔT² + α³ΔT³

Since α is very small (about 10⁻⁵ per °C for most solids), the terms α²ΔT² and α³ΔT³ are negligible. So:

V ≈ V₀(1 + 3αΔT)

By definition, volume expansion is V = V₀(1 + γΔT). Comparing the two:

γ = 3α

Checking the options:

• A. γ = α⁻³ is wrong (it gives the inverse cube).
• B. γ = α is wrong (it ignores the three dimensions).
• C. 3α is correct, though it is written without "γ =" in front. It clearly means γ = 3α.
• D. γ = α³ is wrong (this would make γ tiny, which is not what happens).

Tip: Remember the pattern 1 : 2 : 3, which is linear : area : volume.

• Linear expansion: α
• Area (superficial) expansion: β = 2α
• Volume (cubic) expansion: γ = 3α

So α : β : γ = 1 : 2 : 3. Each time you add a dimension, you multiply α by that number of dimensions.