Question

'n' conducting wires of same dimensions but having resistivity 1,2,3,..... n are connected in series the equivalent resistivity of the combination is ____

Answer 1

Hint: Resistivity is a measurement of a material's resistance to electrical conduction at a given size. Although these terms are less often used, resistivity may often be referred to as specific electrical resistance or volume resistivity.

Complete answer:
As current flows sequentially through the resistors, they are assumed to be in sequence. The current through and resistor is the same so there is only one direction for the charges to pass through. The algebraic number of the individual resistances equals the corresponding resistance of a group of resistors in a sequence relation.
Each resistor receives the same amount of current from the voltage source, so the current through each resistor is the same. The voltage provided by the voltage source and the resistance of the resistors determine the current flowing through the circuit. If a current passes through each resistor, a potential reduction occurs that is equal to the loss of electric potential energy.
Electrical resistance of a conductor with a unit cross-sectional area and length is called resistivity. Resistivity is a property of each material that can be used to compare the ability of different materials to conduct electric currents. Weak conductors have a high resistivity.
We know that
\[{\text{R}} = \dfrac{{\rho L}}{A}\]
When the lengths of 'n' conducting wires are the same
but having resistivities 1,2,3…n are connected in series.
\[{\text{R}} = {{\text{R}}_1} + {{\text{R}}_2} + \ldots \ldots \ldots {{\text{R}}_n}\]
\[ \Rightarrow \dfrac{{\rho L}}{A} = \dfrac{L}{A}(1 + 2 + 3 + \ldots ..n)\]
\[ \Rightarrow \dfrac{{\rho L}}{A} = \dfrac{L}{A} \times \dfrac{{n(n + 1)}}{2}\]
Hence, equivalent resistivity \[\rho = \dfrac{{n(n + 1)}}{2}\]

Note: Conductivity is the reciprocal of resistivity, and it, like resistivity, characterises materials by how well electric current passes through them. mho per metre, or ampere per volt-metre, is the metre-kilogram-second unit of conductivity. High conductivities and low resistivities are characteristics of good electrical conductors. High resistivities and low conductivities are characteristics of good insulators, or dielectrics.