Question

Two wires of equal length and equal diameter and having resistivites ${\rho }_1$ and ${\rho }_2$ are connected in series. The equivalent resistivity of the combination is _____

• A. $\frac{{\rho }_1+{\rho }_2}{2}$
• B. $({\rho }_1+{\rho }_2)$
• C. $\frac{{\rho }_1{\rho }_2}{{\rho }_1+{\rho }_2}$
• D. $\sqrt{{\rho }_1{\rho }_2}$

Answer 1

The correct option is A $\frac{{\rho }_1+{\rho }_2}{2}$

The resistances of two wires are
$R_1={\rho }_1\frac{l}{A}$
and $R_2={\rho }_2\frac{l}{A}$
Now, equivalent resistance of series connection of wire
$R_{eq}=R_1+R_2$
$={\rho }_1=\frac{l}{A}+{\rho }_2\frac{l}{A}=\frac{l}{A}({\rho }_1+{\rho }_2)$
The resistance $R_{eq}$ can be given by
$R_{eq}={\rho }_{eq}\frac{2l}{A}$
Thus, ${\rho }_{eq}\frac{2l}{A}=\frac{l}{A}({\rho }_1+{\rho }_2)={\rho }_{eq}=\frac{{\rho }_1+{\rho }_2}{2}$.