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Prove that beta is equal to 2 alpha. And gamma is equal to 3 alpha

Prove that beta is equal to 2 alpha. And gamma is equal to 3 alpha

Grade:12th pass

3 Answers

Arun
25757 Points
5 years ago
Denote by 
α : the coefficient of linear expansion 
β : the coefficient of surface expansion 
γ : the coefficient of volumetric expansion 

Then a length increases as 

L → L ( 1 + α ΔT) 

But this means that for isotropic (same in every direction) expansion a surface (length x length) increases as 

A → A ( 1 + α ΔT)( 1 + α ΔT) ≈ A (1 +2 α ΔT) 
where we have neglected the (usually very small) square term (α ΔT)² . 

Comparing with the (definition of β) expression

A → ( 1 + βΔT) , we see the relation 

β = 2α . 

Likewise 

V → V ( 1 + γ ΔT) from the definition of volumetric expansion coefficient. 
But also we can approximate (volume = length x length x length) 

V → V ( 1 + α ΔT)³ ≈ V ( 1 + 3 α ΔT) , neglecting higher powers of α ΔT. 

Hence 
γ = 3 α
mary
13 Points
3 years ago
Using solvings show that one beta particle is equal to two alpha particle and one gamma particle is equal to one particle,simultaneously
 
Rishi Sharma
askIITians Faculty 646 Points
2 years ago
Dear Student,
Please find below the solution to your problem.

Denote by
α : the coefficient of linear expansion
β : the coefficient of surface expansion
γ : the coefficient of volumetric expansion
Then a length increases as
L → L ( 1 + α ΔT)
But this means that for isotropic (same in every direction) expansion a surface (length x length) increases as
A → A ( 1 + α ΔT)( 1 + α ΔT) ≈ A (1 +2 α ΔT)
where we have neglected the (usually very small) square term (α ΔT)² .
Comparing with the (definition of β) expression
A → ( 1 + βΔT) , we see the relation
β = 2α .

Likewise
V → V ( 1 + γ ΔT) from the definition of volumetric expansion coefficient.
But also we can approximate (volume = length x length x length)
V → V ( 1 + α ΔT)³ ≈ V ( 1 + 3 α ΔT) , neglecting higher powers of α ΔT.
Hence
γ = 3 α

Thanks and Regards

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