First slide

Temperature dependence

Question

A wire of cross-sectional are d A, length L₁, resistivity $ρ_{1}$ and temperature coefficient of resistivity $α_{1}$ is connected in series to a second wire of length L₂, resistivity $ρ_{2}$ , temperature coefficient of resistivity $α_{2}$ and the same area A, so that wires carry same current. Total resistance R is
independent of temperature for small temperature change if (Thermal expansion effect is negligible)

Moderate

A

$α_{1} = - α_{2}$
B

$ρ_{1} L_{1} α_{1} + ρ_{2} L_{2} α_{2} = 0$
C

$L_{1} α_{1} + L_{2} α_{2} = 0$
D

None

Solution

Let initial resistances of the wires are R₁ and R₂respectively. Then ${R^{'}}_{1} + {R^{'}}_{2} = R_{1} + R_{2}$
$\begin{array}{l} \Rightarrow R_{1} (1 + α_{1} ΔT) + R_{2} (1 + α_{2} ΔT) = R_{1} + R_{2} \\ \Rightarrow R_{1} α_{1} + R_{2} α_{2} = 0 \\ \Rightarrow \frac{ρ_{1} L_{1}}{A} α_{1} + \frac{ρ_{2} L_{2}}{A} α_{2} = 0 \\ ρ_{1} L_{1} α_{1} + ρ_{2} L_{2} α_{2} = 0 \end{array}$

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