Question

Prove that coefficient of superficial expansion is double of coefficient of linear expansion.

​

Answer 1

Explanation:

Relationship between coefficient of superficial expansion and coefficient of linear expansion:

Let

ll = length of the surface

bb = breadth of the surface

Δll = change in length

Δbb = change in breadth

ΔT = change in temperature

Coefficient of linear expansion is given by

\alpha =α= Δll /ll ΔT

\alphaα ll ΔT = Δll

⇒ ll + Δll = l+\alpha ll+αl ΔT

⇒ ll + Δll = ll (1 + \alphaα ΔT) -----------------(1)

Similarly

⇒ bb + Δbb = b(1+\alphab(1+α ΔT) --------------(2)

Now, coefficient of areal expansion is given by

\betaβ = ΔA/AΔT ------------------(3)

A + ΔA = ll (1 + \alphaα ΔT)b(1+\alphab(1+α ΔT)

A + ΔA = lblb (1+\alpha(1+α ΔT)^2

A + ΔA = A(1 + 2\alphaα ΔT) [∵ \alphaα is very small]

A + ΔA = A + 2A\alphaα ΔT

ΔA = 2A\alphaα ΔT ---------------------(4)

Substituting (4) in (3), we get

\betaβ = 2A\alphaα ΔT/AΔT

∴ \beta =2\alphaβ=2α

Hence, the coefficient of superficial expansion is twice the coefficient of linear expansion.