Question

A wire of cross-sectional area $A$, length $L_{1}$, resistivity $\rho_{1}$ and temperature coefficient of resistivity $\alpha_{1}$ is connected in series to a second wire of length $L_{2}$, resistivity $\rho_{2}$, temperature coefficient of resistivity $\alpha_{2}$ 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)

• A. $\alpha_{1}=-\alpha_{2}$
• B. $\rho_{1} L_{1} \alpha_{1}+\rho_{2} L_{2} \alpha_{2}=0$
• C. $L_{1} \alpha_{1}+L_{2} \alpha_{2}=0$
• D. None

Answer 1

Answer: (B)

Let initial resistances of the wires are $R_{1}$ and $R_{2}$ respectively. Then $R_{1}'+R_{2}^{\prime}=R_{1}+R_{2}$
$\Rightarrow R_{1}\left(1+\alpha_{1} \Delta T\right)+R_{2}\left(1+\alpha_{2} \Delta T\right)=R_{1}+R_{2}$
$\Rightarrow R_{1} \alpha_{1}+R_{2} \alpha_{2}=0$
$\Rightarrow \frac{\rho_{1} L_{1}}{A} \alpha_{1}+\frac{\rho_{2} L_{2}}{A} \alpha_{2}=0$
$\Rightarrow \rho_{1} L_{1} \alpha_{1}+\rho_{2} L_{2} \alpha_{2}=0$