Superficial Expansion

Superficial Expansion

When the temperature of a two – dimensional object is changed, its area changes, then the expansion is called superficial expansion. A two – dimensional object, such as a thin metallic plate, changes area when its temperature is raised or lowered.

Superficial Expansion

For an isotope material the area expansion may be expressed in terms of the linear expansion coefficient α.Consider a rectangular area of dimension l₁ and l₂.

The area (A₀) = l₁ x l₂.

Differentiating both sides with respect to T, we get:


dA/ dT = l₁ (αl₂) + l₂(αl₁) = αl₁l₂ + αl₁l₂

dA/ dT = 2αl₁l₂

dA = 2αl₁l₂dT … (1)

If α is constant over the temperature range of under consideration, then equation (1) can be integrated.

\(\int\limits_{{{A}_{i}}}^{{{A}_{f}}}{dA}=2\alpha {{l}_{1}}{{l}_{2}}\int\limits_{{{T}_{i}}}^{{{T}_{f}}}{dT}\).

ΔA = 2αA₀ΔT.


A₀ = Original Area,

ΔT = Temperature Change,

ΔA = βA₀ΔT


β = Coefficient of superficial expansion, it is twice the coefficient of linear expansion α.

Final area of the plate, A = A₀ (1 + βΔT)

The average coefficient of linear expansion, β = ΔA/ A₀ΔT.

The unit of β is °C⁻¹ (or) K⁻¹. Its dimension is θ⁻¹.