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Prove that beta is equal to 2 alpha. And gamma is equal to 3 alpha
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
Using solvings show that one beta particle is equal to two alpha particle and one gamma particle is equal to one particle,simultaneously
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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