Question

A piece of wire of resistance R is cut into three equal parts. These parts are then connected in parallel. If the equivalent resistance of this parallel combination is R₁, what is the value of the ratio R₁ : R?

Answer 1

The resistance of each part is ${\displaystyle \frac{\text{R}}{3}}$ Ω (as resistance is proportional to the length of the wire.)

${\displaystyle \frac{1}{{\text{R}}_1}=\frac{3}{\text{R}}+\frac{3}{\text{R}}+\frac{3}{\text{R}}}$

${\displaystyle \frac{1}{{\text{R}}_1}=\frac{3+3+3}{\text{R}}}$

${\displaystyle \frac{1}{{\text{R}}_1}=\frac{9}{\text{R}}}$

∴ ${\displaystyle {\text{R}}_1=\frac{\text{R}}{9}}$

∴ ${\displaystyle \frac{{\text{R}}_1}{\text{R}}=\frac{1}{9}}$

Answer 2

The resistance of each part is ${\displaystyle \frac{\text{R}}{3}}$ Ω (as resistance is proportional to the length of the wire.)

${\displaystyle \frac{1}{{\text{R}}_1}=\frac{3}{\text{R}}+\frac{3}{\text{R}}+\frac{3}{\text{R}}}$

${\displaystyle \frac{1}{{\text{R}}_1}=\frac{3+3+3}{\text{R}}}$

${\displaystyle \frac{1}{{\text{R}}_1}=\frac{9}{\text{R}}}$

∴ ${\displaystyle \frac{1}{{\text{R}}_1}=\frac{9}{\text{R}}}$

∴ ${\displaystyle \frac{\text{R}}{{\text{R}}_1}=9}$