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

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,

\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.