Question

A metal wire of resistance $R$is cut into two pieces of equal length. The two pieces are connected together side by side. What is the resistance of the two connected wires?

Short answer

The resistance of the two connected wires is $R_{eq}=\frac{R}{4}$.

Step-by-step solution

Given Information

We have to find the resistance of the two connected wires.

Simplify

The resistance through the wire depends on the area and the resistivity $\rho$of the wire and the formula is given as

localid="1648650084630" $R=\frac{\rho L}{A}...\left(1\right)$

Where, $L$is the length of the filament, $A$is the area and $\rho$is the resistivity of the material. As shown by equation $\left(1\right)$, the resistance is directly proportional to the length

$R\alpha L$

If the wire has length $L$and it is divided into two pieces, the length of each wire is $L/2$. The resistance will be halved as the length is halved, so it will be $R/2$for each wire.

As when both new wires are connected side by side, so they will be in parallel and this Equation shows the equivalent resistance for the parallel connection in the form

localid="1648650104413" $\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_N}...\left(2\right)$

The two wires and each wire has resistance $R/2$, so we can get the equivalent resistance for the combination by using equation $\left(2\right)$in the form

$$\begin{gathered} \frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2} \\ \frac{1}{R_{eq}}=\frac{1}{R/2}+\frac{1}{R/2} \\ {R}_{eq}={\left(\frac{4}{R}\right)}^{-1} \end{gathered}$$

The equivalent resistance is $R_{eq}=\frac{R}{4}$