A wire of resistance R is cut into three equal parts. If these three parts are then joined in parallel, calculate the total resistance of the combination so formed.

Given,

• Resistance of the wire = ($\text R$)
• Number of parts cut of the wire = 3

As, resistance of a wire is directly proportional to its length then on increasing the length resistance increases and vice versa.

As all part are of equal length then resistance of each part is given by,

$\text R' = \dfrac{\text R}{3}$

When three parts are connected in parallel combination then the total resistance is given by,

$\dfrac{1}{\text R$\text P} = \dfrac{1}{\text R'} + \dfrac{1}{\text R'} + \dfrac{1}{\text R'} \\[1em] = \dfrac{3}{\text R'} \\[1em] = \dfrac{3}{\dfrac{\text R}{3}} \\[1em] = \dfrac{3\times 3}{\text R} \\[1em] \Rightarrow \dfrac{1}{\text R\text P} = \dfrac{9}{\text R} \\[1em] \Rightarrow \text R_\text P = \dfrac{\text R}{9}

Hence, the total resistance of the combination so formed $\dfrac{\textbf R}{\bold 9}$.