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Question

If α,β and γ are the coefficient of linear, areal and volume expansion respectively, then which of the following options correctly represent the relation between them?
  1. γ=3α
  2. α=3γ
  3. β=3α
  4. γ=3β


Solution

The correct option is A γ=3α
Let us consider a three dimensional solid having a volume expansion coefficient (γ). Let the original volume of the solid be (V).
Here we assume, Volume V=L3.
Then by heating the solid by changing the temperature (ΔT), the new volume of the solid can be written as,
V+ΔV=(L+ΔL)3    ............(1)
where (L+ΔL) is the new length of the solid along each direction in 3-dimensional space.
Using binomial expansion on (1), we get
V+ΔV=(L3+(ΔL)3+3L2ΔL+3L(ΔL)2)
From the definition of linear and volume expansions
ΔL=Lα ΔT and ΔV=Vγ ΔT
V+ΔV=L3+(LαΔT)3+3L2(LαΔT)+3L(LαΔT)2
V+Vγ ΔT=L3+L3(α3ΔT3+3α2ΔT2+3α ΔT)
[Neglecting α2 and α3 terms as α is very small and using V=L3]
VγΔT=L3×3αΔT
we get, γ =3α
Hence, option (a) is the correct answer.

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