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

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

Ans.
(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

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

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

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