Question

How to connect three resistors each of resistance 8 $\Omega$, so that the equivalent resistance of the combination is 12 $\Omega$? Draw diagram of the combination and justify your answer.

Hint:

Two equal resistors in parallel → Half resistance, then add series resistor.

Answer 1

To Obtain Equivalent Resistance of 12 Ω Using Three 8 Ω Resistors

We are given three resistors, each of resistance 8 Ω. We have to connect them in such a way that the equivalent resistance of the combination becomes 12 Ω.

Step 1: Try Possible Combinations

(A) All in Series:
R_eq = 8 + 8 + 8 = 24 Ω (Not required)

(B) All in Parallel:
1/R_eq = 1/8 + 1/8 + 1/8 = 3/8
R_eq = 8/3 Ω (Not required)

(C) Two in Parallel, Then in Series with Third:

First connect two 8 Ω resistors in parallel:

1/R = 1/8 + 1/8 = 2/8 = 1/4
R = 4 Ω

Now connect this 4 Ω in series with the third 8 Ω resistor:

R_eq = 4 + 8 = 12 Ω

Hence, the required equivalent resistance is obtained.

---

Diagram of the Combination:

        8Ω
   ----/\/\/\----
        |      |
        |      |
        8Ω     |
   ----/\/\/\---- 
              |
              |
             8Ω
          ----/\/\/\----
  

In the above diagram:
– The first two 8 Ω resistors are connected in parallel.
– Their equivalent (4 Ω) is connected in series with the third 8 Ω resistor.

Conclusion:
To obtain 12 Ω, connect two 8 Ω resistors in parallel (giving 4 Ω) and then connect this combination in series with the third 8 Ω resistor. The final equivalent resistance becomes 12 Ω.