Question

Equivalent resistance of three identical resistors in series is \( R_1 \) and in parallel it is \( R_2 \). If \( R_1 = nR_2 \), then the minimum possible value of \( n \) is:

• A. \( \frac{1}{9} \)
• B. \( \frac{1}{3} \)
• C. 3
• D. 9

Hint:

For resistors in series, the total resistance is the sum of individual resistances, and for parallel resistors, the total resistance is the reciprocal of the sum of reciprocals.

Answer 1

The Correct Option is B

Step 1: Equivalent Resistance in Series and Parallel.
For three identical resistors \( R \), the resistance in series is: \[ R_1 = 3R \] The resistance in parallel is: \[ R_2 = \frac{R}{3} \]
Step 2: Relating \( R_1 \) and \( R_2 \).
We are given that \( R_1 = nR_2 \). Substituting the expressions for \( R_1 \) and \( R_2 \), we get: \[ 3R = n \times \frac{R}{3} \]
Step 3: Solving for \( n \).
Simplifying the equation: \[ 3R = \frac{nR}{3} \quad \Rightarrow \quad n = 9 \]
Step 4: Conclusion.
Therefore, the minimum possible value of \( n \) is \( 9 \), so the correct answer is (D).