Question

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

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

Answer 1

The correct option is B $\frac{2{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$

here, $R_{eff}=R_1+R_2$
or $\frac{(L+L)}{{\sigma }_{eff}A}=\frac{L}{{\sigma }_{1}A}+\frac{L}{{\sigma }_{2}A}$ (both wires are identical so have same cross section A )
or $\frac{2}{{\sigma }_{eff}}=\frac{1}{{\sigma }_1}+\frac{1}{{\sigma }_2}$
or ${\sigma }_{eff}=\frac{2{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$