**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.

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:

\(\frac{dA}{dT}={{l}_{1}}\frac{d{{l}_{2}}}{dT}+{{l}_{2}}\frac{d{{l}_{1}}}{dT}\).

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.

Where,

A₀ = Original Area,

ΔT = Temperature Change,

ΔA = βA₀ΔT

Where,

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.