Question

A uniform wire of resistance $50\mathrm{\Omega }$ is cut into five equal parts. These parts are now connected in parallel. Calculate the equivalent resistance of the combination.

Answer 1

Step 1: Given information

Resistance, $\mathrm{R}=50\mathrm{\Omega }$

Wire cuts in five equal parts.

Step 2: To find

We have to find the equivalent resistance of the combination.

Step 3: Calculate each resistance of the combination

We know the relation,

$\Rightarrow \mathrm{R}\propto \mathrm{l}$

Where,

$\mathrm{l}$ is the length of the conductor and is the resistance of the conductor.

Whenever the wire is split into five equal sections, the length of each portion decreases by one-fifth, as does the resistance of each part.

$$\begin{gathered} R=\frac{50}{5} \\ =10\mathrm{\Omega } \end{gathered}$$

Step 3: Calculate the equivalent resistance of the combination

Use the expression for equivalent resistors in parallel,

$$\begin{gathered} \frac{1}{{\mathrm{R}}_{\mathrm{eq}}}=\frac{1}{{\mathrm{R}}_1}+\frac{1}{{\mathrm{R}}_2}+\frac{1}{{\mathrm{R}}_3}+\frac{1}{{\mathrm{R}}_4}+\frac{1}{{\mathrm{R}}_5} \\ \frac{1}{{\mathrm{R}}_{\mathrm{eq}}}=\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}} \\ \frac{1}{{\mathrm{R}}_{\mathrm{eq}}}=\frac{5}{\mathrm{R}} \\ {\mathrm{R}}_{\mathrm{eq}}=\frac{\mathrm{R}}{5} \end{gathered}$$

Equivalent resistance, $R_{eq}=\frac{R}{5}$

$$\begin{gathered} =\frac{10}{5} \\ =2\mathrm{\Omega } \end{gathered}$$

Therefore, the equivalent resistance is $2\mathrm{\Omega }$.