Two different wires having lengths L1 and L2 and respective temperature coefficient of linear expansion α1 and α2, are joined end–to–end. then the effective temperature coefficient of linear expansion is :

Engineering

Physics

Thermal Expansion

Question

Two different wires having lengths L₁ and L₂ and respective temperature coefficient of linear expansion α₁ and α₂, are joined end–to–end. then the effective temperature coefficient of linear expansion is :

$4 \frac{α_{1} α_{2}}{α_{1} + α_{2}} \frac{L_{1} L_{2}}{{(L_{2} + L_{1})}^{2}}$

$\frac{α_{1} + α_{2}}{2}$

$2 \sqrt{α_{1} α_{2}}$

$\frac{α_{1} L_{1} + α_{2} L_{2}}{L_{1} + L_{2}}$

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At t°C ; ℓeq = ℓ1 + ℓ2

At t + $∆$ t°C; ℓ_eq' = ℓ₁' + ℓ₂'

ℓ_eq (1 + α_eq $∆$ t) = ℓ₁ (1 + α₁ $∆$ t) + ℓ₂ (1 + α₂ $∆$ t)

(ℓ₁ + ℓ₂) (1 + α_eq $∆$ t) = ℓ₁ + ℓ₂ + ℓ₁α₁ $∆$ t + ℓ₂α₂ $∆$ t

$∴ α_{eq} = \frac{α_{1} ℓ_{1} + α_{2} ℓ_{2}}{ℓ_{1} + ℓ_{2}}$