Question

A piece of wire with resistance R is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is ${R^{\prime }}^{}$, then the ratio $\frac{R}{{R^{\prime }}^{}}$ is:

• A. (1/25)
• B. (1/5)
• C. 5
• D. 25

Answer 1

Answer: (D)

Step 1: Given data:

Resistance of a wire = R

The wire is divided in 5 equal parts, that is the resistance of each part is the same.

The equivalent resistance of the wire = $R^{\prime }$

Step 2: Formula used

The equivalent resistance of the combination of resistances connected in parallel is given by;

$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+..............+\frac{1}{R_n}$

Step 3: Finding the equivalent resistance

Let, the new resistance of each one of the five parts be $R_n$ then,

$R_n=\frac{R}{5}$

Therefore

Resistance of five parts connected in parallel is given by

$\frac{1}{R_{eq}}=\frac{1}{R_n}+\frac{1}{R_n}+\frac{1}{R_n}+\frac{1}{R_n}+\frac{1}{R_n}$

$\frac{1}{R^{\prime }}=\frac{5}{R}+\frac{5}{R}+\frac{5}{R}+\frac{5}{R}+\frac{5}{R}$

$\frac{1}{R^{\prime }}=5\times \frac{5}{R}$

$\frac{1}{R^{\prime }}=\frac{25}{R}$

$R^{\prime }=\frac{R}{25}$

Step 4: Finding the ratio of R and R'

$R^{\prime }=\frac{R}{25}$

$\frac{R^{\prime }}{R}=\frac{1}{25}$

$\frac{R}{R^{\prime }}=\frac{25}{1}$

Hence,

option D) $\frac{R}{R^{\prime }}=25$ is the correct option.