Two conductors have the same resistances at but their temperature coefficients of resistance are and . The respective temperature coefficients for their series and parallel combinations are

Question

Two conductors have the same resistances at $0^{o} C$ but their temperature coefficients of resistance are $α_{1}$ and $α_{2}$ . The respective temperature coefficients for their series and parallel combinations are [2024]

Smart Achievers · Accepted Answer

(2)

Series: $R_{e q} = R_{1} + R_{2}$

$2 R (1 + α_{e q} ∆ θ) = R (1 + α_{1} ∆ θ) + R (1 + α_{2} ∆ θ)$

$2 R (1 + α_{e q} ∆ θ) = 2 R + (α_{1} + α_{2}) R ∆ θ$

$\Rightarrow α_{e q} = \frac{α_{1} + α_{2}}{2}$

Parallel: $\frac{1}{R_{e q}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$

$\frac{1}{\frac{R}{2} (1 + α_{e q} ∆ θ)} = \frac{1}{R (1 + α_{1} ∆ θ)} + \frac{1}{R (1 + α_{2} ∆ θ)}$

$\frac{2}{1 + α_{e q} ∆ θ} = \frac{1}{1 + α_{1} ∆ θ} + \frac{1}{1 + α_{2} ∆ θ}$

$\frac{2}{1 + α_{e q} ∆ θ} = \frac{1 + α_{2} ∆ θ + 1 + α_{1} ∆ θ}{(1 + α_{1} ∆ θ) (1 + α_{2} ∆ θ)}$

$2 [(1 + α_{1} ∆ θ) (1 + α_{2} ∆ θ)]$

$= [2 + (α_{1} + α_{2}) ∆ θ] [1 + α_{e q} ∆ θ]$

$2 [1 + α_{1} ∆ θ + α_{2} ∆ θ + α_{1} α_{2} ∆ θ]$

$= 2 + 2 α_{e q} ∆ θ + (α_{1} + α_{2}) ∆ θ + α_{e q} (α_{1} + α_{2}) ∆ θ^{2}$

Neglecting small terms

$2 + 2 (α_{1} + α_{2}) ∆ θ = 2 + 2 α_{e q} ∆ θ + (α_{1} + α_{2}) ∆ θ$

$(α_{1} + α_{2}) ∆ θ = 2 α_{e q} ∆ θ \Rightarrow α_{e q} = \frac{α_{1} + α_{2}}{2}$

Solution

Q.
Two conductors have the same resistances at $0^{o} C$ but their temperature coefficients of resistance are $α_{1}$ and $α_{2}$ . The respective temperature coefficients for their series and parallel combinations are [2024]

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

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

3 $\frac{α_{1} + α_{2}}{2}, α_{1} + α_{2}$

4 $α_{1} + α_{2}, \frac{α_{1} α_{2}}{α_{1} + α_{2}}$

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Solution

Q. Two conductors have the same resistances at 0oC but their temperature coefficients of resistance are α1 and α2. The respective temperature coefficients for their series and parallel combinations are [2024]

1 α1+α2,α1+α22

2 α1+α22,α1+α22

3 α1+α22,α1+α2