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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
25763 Points
4 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
2 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
one year 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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