Question

A uniform wire is cut into \[10\] equal parts and all the equal parts are connected in parallel. The effective resistance is
A. decreases \[10\] times
B. increases \[10\] times
C. increases \[100\] times
D. decreases \[100\] times

Answer 1

Hint:Try finding out the resistance of the ten individual wires and then find the effective resistance by connecting them in the parallel combination.Resistance in simple words is a measure of how much the current is slowed down. The bigger the resistance, the smaller the current. It’s S.I unit is ohms\[\left( \Omega \right)\].

Complete step by step answer:
The resistance of a wire depends on its length, its cross-sectional area and the resistivity of the material.Resistivity is the resistance of a material of unit length and unit cross-sectional area. It is the characteristic property of the material and is independent of its length and area of cross section. Mathematically, Resistance of a wire is given by:
\[R=\rho \dfrac{l}{A}\]
where $R$ is the resistance, $ρ$ is the resistivity, l is the length, $A$ is the cross sectional area.

Let us assume that the length of the wire is L metre and resistance R \[\Omega \] . So, when wire is cut into \[10\] equal parts, each part will have its length \[\dfrac{l}{10}\] . As resistance is directly proportional to its length, each wire will have its resistance \[\dfrac{R}{10}\Omega \](as the length of each wire got divided by \[10\],resistance will also get divided by 10).

When these wires are connected in parallel, the effective resistance will become
\[\dfrac{1}{{{R}_{eff}}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}+........+\dfrac{1}{{{R}_{10}}}\]
Substituting \[{{R}_{1}},{{R}_{2}},.........{{R}_{10}}=\dfrac{R}{10}\]
Hence, we get,
\[\dfrac{1}{{{R}_{eff}}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}+........+\dfrac{1}{{{R}_{10}}} \\
\Rightarrow \dfrac{1}{{{R}_{eff}}}=\dfrac{10}{R}+\dfrac{10}{R}+.....+\dfrac{10}{R} \\
\Rightarrow \dfrac{1}{{{R}_{eff}}}=\dfrac{100}{R} \\
\therefore {{\operatorname{R}}_{eff}}=\dfrac{R}{100} \\ \]
Therefore, resistance got decreased by \[100\] times as the original resistance we assumed was \[R\,\Omega \].

Hence, the correct answer is option D.

Note:The S.I units of resistivity are ohm-metre(\[\Omega m\]) and it depends on:
-Nature of the material, meaning it is different for different materials.
-Temperature of the material.