Question

You are given three resistors of each of resistance $4ohm.$ How many effective resistances you can obtain using all three resistors. Draw equivalent circuits in each case.

Answer 1

Step 1: Given data:
$R_A=R_B=R_C=4ohms$

There is three effective resistance that can be obtained by the combination of a given set of resistors.

Step 2:- Case (a): When three resistors are in parallel.

Effective resistances in case of the parallel combination

$$\begin{gathered} \frac{1}{R_{eq}}=\frac{1}{R_A}+\frac{1}{R_B}+\frac{1}{R_C} \\ \frac{1}{R_{eq}}=\frac{1}{4}+\frac{1}{4}+\frac{1}{4} \\ {R}_{eq}=\frac{4}{3}\Omega \end{gathered}$$

Step 3:- Case (b) When three resistors are in series.

Effective resistances in case of the series combination

$$\begin{gathered} R_{eq}=R_A+R_B+R_C \\ {R}_{eq}=4+4+4 \\ {R}_{eq}=12\Omega \end{gathered}$$

Step 4:- Case (c): When two resistors are parallel and one resistor is in series of them.

Resistances $AandB$ are connected in parallel.

So, the equivalent resistance is

$$\begin{gathered} \frac{1}{R_p}=\frac{1}{R_A}+\frac{1}{R_B} \\ \frac{1}{R_p}=\frac{1}{4}+\frac{1}{4} \\ {R}_p=2\Omega \end{gathered}$$

Now, resistance $R_p=2\Omega$ and $R_C=4\Omega$ are in series

So, the net equivalent resistance is

$$\begin{gathered} R_{eq}=R_p+R_C \\ {R}_{eq}=2+4 \\ {R}_{eq}=6\Omega \end{gathered}$$

Hence, there is three effective resistance that can be obtained.