Two metal wires of same length, same area of crosssection have conductivities of their material $\sigma_1$…

Two metal wires of same length, same area of crosssection have conductivities of their material $\sigma_1$ and $\sigma_2$. If they are connected in series, the effective conductivity is
  1. $\frac{\sigma_1 \sigma_2}{\sigma_1+\sigma_2}$
  2. $\frac{2 \sigma_1 \sigma_2}{\sigma_1+\sigma_2}$
  3. $\frac{\sigma_1 \sigma_2}{\sigma_1-\sigma_2}$
  4. $\frac{\sigma_1 \sigma_2}{\sigma_1-\sigma_2}$

Solution

If they are connected in series, then $\begin{aligned} & R_{e q}=R_1+R_2 \\ & \frac{L+L}{\sigma_{e q} A}=\frac{L}{\sigma_1 A}+\frac{L}{\sigma_2 A} \\ & \frac{2}{\sigma_{e q}}=\frac{1}{\sigma_1}+\frac{1}{\sigma_2} \Rightarrow \sigma_{e q}=\frac{2 \sigma_1 \sigma_2}{\sigma_1+\sigma_2} \end{aligned}$

Asked in: AP EAMCET 2023 (17 May Shift 1)

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