Relation Between alpha beta and gamma
We have already studied about alpha beta and gamma in thermal expansion. In this article we are going to discuss that, what is the relation between alpha beta and gamma in physics?
Relation Between alpha beta and gamma for isotropic solids
Suppose that we have a cube of some isotropic material of side L, having :-
- Coefficient of linear expansion (α)
- Coefficient of areal expansion (β) and
- Coefficient of volumetric expansion (γ)
Let the cube be heated by a small range of temperature ΔT, due to which its dimensions changes.
New side length L = L0 (1 + αΔT ) ……….(1)
New surface area of one face A = A0 (1 + βΔT ) ……….(2)
New volume V = V0 (1 + γΔT ) ……….(3)
Now, let us derive the relation between alpha beta and gamma.
New surface area of one face can also be written as
A0+ΔA = (L)2 = (L0 + ΔL)2
A0+βA0ΔT = (L0 + αL0ΔT)2
A0 (1 + βΔT) = (L0)2(1 + αΔT)2
A0 (1 + βΔT) = (L0)2(1 + αΔT)2
because A0 = (L0)2
⇒ 1 + βΔT = (1 + αΔT)2
⇒ 1 + βΔT = 1 + α2 ΔT2 + 2αΔT
⇒ βΔT = α2 ΔT2 + 2αΔT
The value of α is of the order of 10-5 so its higher powers can be neglected i.e., α2 ΔT2 ≈ 0
This gives us
βΔT = 2αΔT
βΔT = 2αΔT
⇒ β = 2α or α = β/2 ……….(4)
Again new volume V can be written as
V0 + ΔV = (L)3 = (L0 + ΔL)3
V0 + γV0ΔT = (L0 + αL0ΔT)3
V0(1 + γΔT) = (L0)3(1 + αΔT)3
because V0 = (L0)3
V0(1 + γΔT) = (L0)3(1 + αΔT)3
1 + γΔT = (1 + αΔT)3
1 + γΔT = 1 + α3ΔT3 +3αΔT + 3α2 ΔT2
again neglecting the terms containing higher powers of α, i.e., α3ΔT3 ≈ 0 and 3α2 ΔT2 ≈ 0 , we get
1 + γΔT = 1 + 3αΔT
⇒ γΔT = 3αΔT
⇒ γ = 3α or α = γ/3 ……….(5)
From equations (4) and (5),
α = β/2 = γ/3
or
6α = 3β = 2γ
This is the required relation between alpha beta and gamma.
Also α : β : γ = 1 : 2 : 3
Relation Between alpha beta and gamma for non-isotropic solid
For non-isotropic solid
β = α1 + α2 and γ = α1 + α2 + α3
Here α1 , α2 and α3 are coefficient of linear expansion in X, Y and Z direction.
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