Heat and Thermal ExpansionHard
Question
Two conductors have the same resistance at 0oC but their temperature coefficients of resistance are α1 and α2. The respective temperature coefficients of their series and parallel combinations are nearly
Options
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Solution
Let R0 be the initial resistance of both conductors
∴ At temperature q their resistance will be,
R1 = R0 (1 + α1θ) and R2 = R0 (1 + α2θ)
for, series combination, Rs = R1 + R2
R0s (1 + αsθ) = R0(1 + α1θ) + R0(1 + α2θ)
where Rs0 = R0 + R0 = 2R0
∴ 2R0 (1 + αsθ) = 2R0 + R0 θ (α1 + α2)
or
for parallel combination, Rp =
Rp0 (1 + αp θ) =
Where, Rp0
∴
as α1 and α2 are small quantities
∴ α1 α2 is negligible
or
[1 - (α1 + α2)θ]
as (α1 + α2)2 is negligible
∴
∴ At temperature q their resistance will be,
R1 = R0 (1 + α1θ) and R2 = R0 (1 + α2θ)
for, series combination, Rs = R1 + R2
R0s (1 + αsθ) = R0(1 + α1θ) + R0(1 + α2θ)
where Rs0 = R0 + R0 = 2R0
∴ 2R0 (1 + αsθ) = 2R0 + R0 θ (α1 + α2)
or
for parallel combination, Rp =
Rp0 (1 + αp θ) =
Where, Rp0
∴
as α1 and α2 are small quantities
∴ α1 α2 is negligible
or
[1 - (α1 + α2)θ]as (α1 + α2)2 is negligible
∴
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