First slide

Resistors

Question

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

Moderate

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

At temperature $θ$
$R_{1}^{1} = R_{1} [1 + α_{1} Δ θ], R_{2}^{1} = R_{2} [1 + α_{2} Δ θ]$
At $0 ° C$ , $R_{eq} = R_{1} {+R}_{2}$
At $θ ° C$ , $R_{e q}^{1} = R_{1}^{1} + R_{2}^{1} = R_{1} [1 + α_{1} θ] + R_{2} [1 + α_{2} θ]$
Let equivalent temperature coefficient is $α^{1}$
At $θ ° C$ , $R_{e q}^{1} = R_{e q} [1 + α^{1} θ]$
$(R_{1} + R_{2}) [1 + α^{1} θ] = R_{1} [1 + α_{1} θ] + R_{2} [1 + α_{2} θ]$
$α^{1} = \frac{R_{1} α_{1} + R_{2} α_{2}}{R_{1} + R_{2}}$

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