Relationship between Linear, Area and Cubic Expansivity

Derivation of The Relationship between Linear Expansivity and Area Expansivity:

Let us consider a sheet of metal that has been heated through a temperature change of θ. Let the initial length and breadth be L₁ and B₁ respectively and the final length and breadth be L₂ and B₂ respectively.

Let the initial area of the sheet, \( \scriptsize A_1 = L_1 \: \times \: B_1 \)

Let the final area of the sheet, \( \scriptsize A_2 = L_2 \: \times \: B_2 \)

From the formula of linear expansivity,

L₂ = L₁( 1 + α Δθ)

B₂ = B₁(1 + α Δθ)

where Δθ = θ₂ – θ₁

Hence \( \scriptsize A_2 = L_2 \: \times \: B_2 \\ = \scriptsize L_1 (1 \: + \: \alpha \Delta \theta) \: \times \: B_1(1 \: + \: \alpha \Delta \theta) \\ \scriptsize = L_1B_1(1 \: + \: \alpha \Delta \theta)^2 \\ \scriptsize = L_1B_1( 1 \: + \: 2 \alpha \Delta \theta \: + \: \alpha^2 \Delta \theta^2) \)

We can neglect α² because α is a very small quantity (e.g. 0.00002). Therefore, α² is approximately zero. I.e. α² = 0.

Hence we have;