First slide

Temperature dependence

Question

Two wires of resistance R₁ and R₂ have temperature coefficient of resistance $α_{1}$ and $α_{2}$ respectively. These are joined in series. The effective temperature coefficient of resistance is

Easy

A

$\frac{α_{1} + α_{2}}{2}$
B

$\sqrt{α_{1} α_{2}}$
C

$\frac{α_{1} R_{1} + α_{2} R_{2}}{R_{1} + R_{2}}$
D

$\frac{\sqrt{R_{1} R_{2} + α_{1} α_{2}}}{\sqrt{R_{1}^{2} + R_{2}^{2}}}$

Solution

$\begin{array}{l} R_{t_{1}} = R_{1} (1 + α_{1} t) and R_{t_{2}} = R_{2} (1 + α_{2} t) \\ Also R_{eq .} = R_{t_{1}} = R_{t_{2}} \Rightarrow R_{eq .} = R_{1} + R_{2} + (R_{1} α_{1} + R_{2} α_{2}) t \end{array}$
$\Rightarrow R_{eq} = (R_{1} + R_{2}) \{1 + (\frac{R_{1} α_{1} + R_{2} α_{2}}{R_{1} + R_{2}}) . t\}$
$So α_{eff} = \frac{R_{1} α_{1} + R_{2} α_{2}}{R_{1} + R_{2}}$

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