Question

Two wires of same dimensions but 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 the length of each wire will be $l$ and $A$ be the area of cross-section of each wire.

Let us consider a single rod of same dimension (i.e. length $2l$, Area $A$) as of composite rod. If resistance of the single rod is the same as that of the composite rod will become equivalent resistivity of the composite rod.

Therefore,

$R=R_1+R_2$

$\Rightarrow \frac{{\rho }_{eq}\left(2l\right)}{A}=\frac{{\rho }_{1}l}{A}+\frac{{\rho }_{2}l}{A}$

$\Rightarrow \rho =\frac{{\rho }_1+{\rho }_2}{2}$

Hence, option $\left(c\right)$ is correct.

Why this question? If the resistance of the single rod is same as that of the composite rod, then the resistivity of the rod will become equivalent resistivity of the composite rod.