A uniform rectangular plate of length ℓ and width ω has a coefficient of linear expansion α. Show that, if we neglect very small quantities, the change in area of the plate due to a temperature change ∆T is ∆A = 2αℓω ∆T. See Fig. 17–21.

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Start by recalling the formula for linear expansion: the change in length of an object due to a temperature change is given by Δℓ = αℓΔT, where α is the coefficient of linear expansion, ℓ is the original length, and ΔT is the temperature change.

For a rectangular plate, the area A is given by A = ℓω, where ℓ is the length and ω is the width. When the temperature changes, both the length and width expand linearly.

The new length of the plate after expansion is ℓ + Δℓ = ℓ + αℓΔT, and the new width is ω + Δω = ω + αωΔT. The new area of the plate is therefore (ℓ + αℓΔT)(ω + αωΔT).

Expand the expression for the new area: (ℓ + αℓΔT)(ω + αωΔT) = ℓω + ℓ(αωΔT) + ω(αℓΔT) + (αℓΔT)(αωΔT). The term (αℓΔT)(αωΔT) is very small compared to the other terms and can be neglected.

Simplify the expression: The change in area ΔA = (new area) - (original area) = [ℓω + ℓ(αωΔT) + ω(αℓΔT)] - ℓω = 2αℓωΔT. This shows that the change in area is proportional to the coefficient of linear expansion, the original area, and the temperature change.

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Coefficient of Linear Expansion

The coefficient of linear expansion (α) quantifies how much a material expands per unit length for each degree of temperature increase. It is a crucial property in understanding how solids respond to temperature changes, as it directly influences the change in dimensions of an object, such as length and width.

Area Expansion

Area expansion refers to the increase in surface area of a two-dimensional object when subjected to a temperature change. For a rectangular plate, the change in area (∆A) can be derived from the linear expansions of its length and width, leading to the formula ∆A = 2αℓω∆T, which accounts for the contributions from both dimensions.

Temperature Change (∆T)

Temperature change (∆T) is the difference in temperature that causes materials to expand or contract. In the context of thermal expansion, it is essential to determine how much a material will change in size, as the extent of expansion is directly proportional to this temperature difference, influencing the overall change in area of the plate.