Question

A piece of wire of resistance $\mathrm{R}$ is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is $\mathrm{R}\text{'}$, then the ratio $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$ is _____.

• A. $\frac{1}{25}$
• B. $\frac{1}{5}$
• C. $5$
• D. $25$

Answer 1

The correct option is D

$25$

Step 1. Given data

It is given that a piece of wire of resistance $\mathrm{R}$ is cut into five equal parts. These parts are then connected in parallel.

The equivalent resistance of the combination is $\mathrm{R}\text{'}$.

Step 2: Formula used

Resistance is the opposition that an electrical device has to the flow of electrical current. It is represented by symbol $\mathrm{R}$.

The formula to find the resistance is,

$\mathrm{R}=\mathrm{\rho }\frac{\mathcal{l}}{\mathrm{A}}$

Here, $\mathrm{\rho }$ is the resistivity, $\mathcal{l}$ is the length of wire and $\mathrm{A}$ is the cross sectional area.

Step 3. Find the ratio of $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$.

The resistance $\mathrm{R}$ is directly proportional to length of the wire.

Thus, the resistance of each five parts after cutting the wire is,

${\mathrm{R}}_1={\mathrm{R}}_2={\mathrm{R}}_3={\mathrm{R}}_4={\mathrm{R}}_5=\frac{\mathrm{R}}{5}$

When all the resistances are connected in parallel,

Let us consider, $\mathrm{R}\text{'}$ is the equivalent resistance.

Therefore,

$\frac{1}{\mathrm{R}\text{'}}=\frac{1}{{\mathrm{R}}_1}+\frac{1}{{\mathrm{R}}_2}+\frac{1}{{\mathrm{R}}_3}+\frac{1}{{\mathrm{R}}_4}+\frac{1}{{\mathrm{R}}_5}$

$=\frac{1}{{\displaystyle \frac{\mathrm{R}}{5}}}+\frac{1}{{\displaystyle \frac{\mathrm{R}}{5}}}+\frac{1}{{\displaystyle \frac{\mathrm{R}}{5}}}+\frac{1}{{\displaystyle \frac{\mathrm{R}}{5}}}+\frac{1}{{\displaystyle \frac{\mathrm{R}}{5}}}$

$=\frac{5}{\mathrm{R}}+\frac{5}{\mathrm{R}}+\frac{5}{\mathrm{R}}+\frac{5}{\mathrm{R}}+\frac{5}{\mathrm{R}}$

$R^{\text{'}}=\frac{\mathrm{R}}{25}$

Now, we will find the ratio of $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$.

We will find the ratio of $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$.

$\Rightarrow$ $\frac{\mathrm{R}}{\mathrm{R}\text{'}}=\frac{\mathrm{R}}{{\displaystyle \frac{\mathrm{R}}{25}}}$

$=\frac{25\cancel{\mathrm{R}}}{\cancel{\mathrm{R}}}$

$=25$

The ratio of $\frac{\mathrm{R}}{\mathrm{R}\text{'}}$ is $25$.

Hence, option $\mathrm{D}$ is correct answer.