Question

Two metal wires of identical dimensions 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. $\frac{\sigma_1\sigma_2}{\sigma_1+\sigma_2}$
• C. $\frac{2\sigma_1\sigma_2}{\sigma_1+\sigma_2}$
• D. $\frac{\sigma_1+\sigma_2}{2\sigma_1\sigma_2}$

Answer 1

Answer: (C)

here, $R _{\text{eff }}= R _{1}+ R _{2}$
or $\frac{( L + L )}{\sigma_{\text{eff }} A }=\frac{ L }{\sigma_{1} A }+\frac{ L }{\sigma_{2} A }$
(both wires are identical so have same cross section $A$ )
Or $\frac{2}{\sigma_{\text{eff }}}=\frac{1}{\sigma_{1}}+\frac{1}{\sigma_{2}}$
Or $\sigma_{\text{eff }}=\frac{2 \sigma_{1} \sigma_{2}}{\sigma_{1}+\sigma_{2}}$