α₁+α₂/2, α₁+α₂/2
α₁+α₂/2, α₁+α₂
α₁+α₂, α₁+α₂/2
α₁+α₂, α₁α₂/(α₁+α₂)
Let R₀ be the initial resistance of both conductors.
At temperature θ their resistance will be,
R₁ = R₀(1 + α₁θ) and R₂ = R₀(1 + α₂θ)
for series combination,
Rₛ = R₁ + R₂
Rₛ₀(1 + αₛθ) = R₀(1 + α₁θ) + R₀(1 + α₂θ)
where Rₛ₀ = R₀ + R₀ = 2R₀.
2R₀(1 + αₛθ) = 2R₀ + R₀θ(α₁ + α₂)
or αₛ = (α₁ + α₂)/2
for parallel combination,
Rₚ = R₁R₂/(R₁ + R₂)
Rₚ₀(1 + αₚθ) = R₀(1 + α₁θ)R₀(1 + α₂θ)/[R₀(1 + α₁θ) + R₀(1 + α₂θ)]
where, Rₚ₀ = R₀R₀/(R₀ + R₀) = R₀/2
R₀/2(1 + αₚθ) = R₀²(1 + α₁θ + α₂θ + α₁α₂θ²)/R₀(2 + α₁θ + α₂θ)
as α₁ and α₂ are small quantities α₁α₂ is negligible
or αₚ = (α₁ + α₂)/2 + (α₁ + α₂)θ = (α₁ + α₂)/2[1 - (α₁ + α₂)θ]
as (α₁ + α₂)² is negligible.
αₚ = (α₁ + α₂)/2