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Two metal wires of identical dimensions are connected in series if ${\sigma _1}$ and ${\sigma _2}$ are the conductivity of the metal wires respectively, the conductivity of the combination is$\left( a \right){\text{ }}\dfrac{{2{\sigma _1}{\sigma _2}}}{{{\sigma _1} + {\sigma _2}}}$$\left( b \right){\text{ }}\dfrac{{{\sigma _1} + {\sigma _2}}}{{2{\sigma _1}{\sigma _2}}}$$\left( c \right){\text{ }}\dfrac{{{\sigma _1} + {\sigma _2}}}{{2{\sigma _1}{\sigma _2}}}$$\left( d \right){\text{ }}\dfrac{{{\sigma _1}{\sigma _2}}}{{{\sigma _1} + {\sigma _2}}}$

Last updated date: 20th Jun 2024
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Hint Just take the word. What will resistance mean? It’s the refusal to just accept or allow one thing. Within the same method, once the current is passed through a device or just through a wire, it shows resistance to permit it to flow through it. This is often known as the “resistance” of that device/wire. The property of the material by which it's this resistance is termed “Resistivity”. It primarily shows however robust a tool will resist the current.

Formula used
Conductivity is equal to the reciprocal of resistivity,
Mathematically we can write it as
$\sigma = \dfrac{1}{\rho }$

Since the conductivity is given for the individual and we have to calculate the conductivity in the parallel combination. So let’s solve this.
For the resistivity rho, the net resistance of a metal wire will be
We have,
${R_1} = {\rho _1}\dfrac{L}{A}$
Similarly
${R_2} = {\rho _2}\dfrac{L}{A}$
Therefore for the two metal wires, their net effective resistance will be,
${R_{eq}} = {R_1} + {R_2}$
$\Rightarrow \rho \dfrac{{2L}}{A} = {\rho _{_1}}\dfrac{L}{A} + {\rho _2}\dfrac{L}{A}$
$\Rightarrow 2\rho = {\rho _1} + {\rho _2}$
As we know
Conductivity is equal to:
$\sigma = \dfrac{1}{\rho }$
We have,
$\Rightarrow \dfrac{2}{\sigma } = \dfrac{{{\sigma _1} + {\sigma _2}}}{{{\sigma _1}{\sigma _2}}}$
So the for the combined wires, the effective conductivity will be,
$\Rightarrow \sigma = \dfrac{{{\sigma _1} + {\sigma _2}}}{{2{\sigma _1}{\sigma _2}}}$

Note Resistivity could be a constant term that is the same for a selected material. Every completely different material has different resistivity. However, if the material is the same, the resistivity is the same. For instance, all copper has the same ohmic resistance. We can change its resistance by ever-changing the length & diameter. We tend to cannot change the ohmic resistance of any copper. It’s the same for all copper. To vary resistivity, we've to vary the material. So, for a selected material resistivity could be a constant, however, the resistance isn't as a result of we can change it by ever-changing the length & width of the material.