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 $R_1^{\prime }+R_2^{\prime }=R_1+R_2$
$\Rightarrow R_1\left(1+{\alpha }_1\Delta T\right)+R_2\left(1+{\alpha }_2\Delta T\right)=R_1+R_2$
$\Rightarrow R_1{\alpha }_1+R_2{\alpha }_2=0$
$\Rightarrow \frac{{\rho }_1{L}_1}{A}{\alpha }_1+\frac{{\rho }_2{L}_2}{A}{\alpha }_2=0$
$\Rightarrow {\rho }_1{L}_1{\alpha }_1+{\rho }_2{L}_2{\alpha }_2=0$