Question

Two metal wires of identical dimensions are connected in series. If ${\delta }_1$ and ${\delta }_2$ are the conductivities of the metal wires respectively, the effective conductivity of the combination is

• A. $\frac{2{\delta }_1{\delta }_2}{{\delta }_1+{\delta }_2}$
• B. $\frac{{\delta }_1+{\delta }_2}{2{\delta }_1+{\delta }_2}$
• C. $\frac{2{\delta }_1+{\delta }_2}{{\delta }_1+{\delta }_2}$
• D. $\frac{{\delta }_1{\delta }_2}{{\delta }_1+{\delta }_2}$

Answer 1

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


$R_{eff}=R_1+R_2$

$R=\rho \frac{l}{A}$

${\rho }_{eff}\frac{2l}{A}=\frac{{\rho }_{1}l}{A}+\frac{{\rho }_{2}l}{A}$

$2{\rho }_{eff}={\rho }_1+{\rho }_2$
$\because (\sigma =\frac{1}{\rho })$

$\frac{2}{{\sigma }_{eff}}=\frac{1}{{\sigma }_1}+\frac{1}{{\sigma }_2}$

$\frac{2}{{\sigma }_{eff}}=\frac{{\sigma }_2+{\sigma }_1}{{\sigma }_1{\sigma }_2}$

${\sigma }_{eff}=\frac{2{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$