Question

Five equal resistances are connected in a network as shown in the figure. The net resistance between points $A$ and $B$ is:

• A. $\frac{3R}{2}$
• B. $\frac{R}{2}$
• C. $R$
• D. $2R$

Answer 1

The correct option is C

$R$

Step 1: Given data

Five equal resistance are connected in a network,

Let, the resistance between $AD$ be $R_{AD}=R$,

The resistance between $DB$ be $R_{DB}=R$,

The resistance between $BE$ be $R_{BE}=R$,

The resistance between $EA$ be $R_{EA}=R$,

The resistance between $DE$ be $R_{DE}$,

Step 2. Find that if $R_{DE}$ is redundant,

It is a Wheatstone bridge so we know that

$\frac{R_{AD}}{R_{AE}}=\frac{R_{DB}}{R_{BE}}$

If $\left(\frac{R_{AD}}{R_{AE}}=\frac{R_{DB}}{R_{BE}}\right)$ condition satisfies then $R_{DE}$ become redundant,

$$\begin{gathered} \frac{R_{AD}}{R_{AE}}=\frac{R_{DB}}{R_{BE}} \\ \Rightarrow \frac{R}{R}=\frac{R}{R} \\ \Rightarrow 1=1 \end{gathered}$$

And here it satisfies the equations .Thus $R_{DE}$ is redundant .

Step 3: To find net resistance between point $A$and $B$,

By using the equivalent resistance formula for parallel connected resistance,

$$\begin{gathered} \frac{1}{R_{eq}}=\frac{1}{2R}+\frac{1}{2R} \\ \Rightarrow \frac{1}{R_{eq}}=\frac{2R+2R}{\left(2R\times 2R\right)} \\ \Rightarrow R_{eq}=\frac{4R^2}{4R} \\ \Rightarrow R_{eq}=R \end{gathered}$$

Hence the net resistance is $R$.

Therefore the correct answer is Option C.