Question

Two wires of same dimensions but different resistivities ${\rho }_1$ and ${\rho }_2$ are connected in series. The equivalent resistivity of the combination is

• A. $\sqrt{{\rho }_1{\rho }_2}$
• B. ${\rho }_1+{\rho }_2$
• C. $\frac{{\rho }_1+{\rho }_2}{2}$
• D. None of the above

Answer 1

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

Let $A$ and $l$ be the area and length of each wire respectively.
Then equivalent resistance of the wires in series, $\begin{array}{}{R}_{eq}={\rho }_{eq}\frac{2l}{A}& \text{(1)}\end{array}$
Resistance of wire having resistivity ${\rho }_1$, $\begin{array}{}{R}_1={\rho }_1\frac{l}{A}& \text{(2)}\end{array}$
Resistance of wire having resistivity ${\rho }_2$, $\begin{array}{}{R}_2={\rho }_2\frac{l}{A}& \text{(3)}\end{array}$
Since the wires are in series their equivalent resistance, $R_{eq}=R_1+R_2$
Substituting the values from (1), (2) and (3),
${\rho }_{eq}\frac{2l}{A}={\rho }_1\frac{l}{A}+{\rho }_2\frac{l}{A}$
$\Rightarrow {\rho }_{eq}=\frac{{\rho }_1+{\rho }_2}{2}$