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

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3 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) expressionA → ( 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
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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
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one year ago
```							Dear Student,Please find below the solution to your problem.Denote byα : the coefficient of linear expansionβ : the coefficient of surface expansionγ : the coefficient of volumetric expansionThen a length increases asL → L ( 1 + α ΔT)But this means that for isotropic (same in every direction) expansion a surface (length x length) increases asA → 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) expressionA → ( 1 + βΔT) , we see the relationβ = 2α .LikewiseV → 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.Henceγ = 3 αThanks and Regards
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4 months ago
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