JEE MAIN - Physics (2022 - 29th July Morning Shift - No. 12)

Two metallic wires of identical dimensions are connected in series. If $$\sigma_{1}$$ and $$\sigma_{2}$$ are the conductivities of the these wires respectively, the effective conductivity of the combination is :
$$ \frac{\sigma_{1} \sigma_{2}}{\sigma_{1}+\sigma_{2}} $$
$$ \frac{2 \sigma_{1} \sigma_{2}}{\sigma_{1}+\sigma_{2}} $$
$$ \frac{\sigma_{1}+\sigma_{2}}{2 \sigma_{1} \sigma_{2}} $$
$$ \frac{\sigma_{1}+\sigma_{2}}{\sigma_{1} \sigma_{2}} $$

Explanation

$$R = {R_1} + {R_2}$$

$$ \Rightarrow {{{l_1} + {l_2}} \over {\sigma A}} = {{{l_1}} \over {{\sigma _1}A}} + {{{l_2}} \over {{\sigma _2}A}}$$

$$ \Rightarrow {2 \over \sigma } = {1 \over {{\sigma _1}}} + {1 \over {{\sigma _2}}}$$

$$ \Rightarrow \sigma = {{2{\sigma _1}{\sigma _2}} \over {{\sigma _1} + {\sigma _2}}}$$

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