The resistance of the series combination of two resistances is S. When they are joined in parallel the total resistance is P. If S= nP then the minimum possible value of n is

Options

A. 2
B. 3
C. 4
D. 1

Explanation

To solve the problem, we need to analyze the relationship between the resistances in series and parallel combinations. Let's denote the two resistances as \( R_1 \) and \( R_2 \).

### Step 1: Write the expressions for series and parallel resistances
1. **Series Resistance (S)**:
\[ S = R_1 + R_2 \]

2. **Parallel Resistance (P)**:
\[ P = \frac{R_1 R_2}{R_1 + R_2} \]

### Step 2: Relate S and P using the given condition
We are given that \( S = nP \). Substituting the expressions for \( S \) and \( P \):
\[ R_1 + R_2 = n \left( \frac{R_1 R_2}{R_1 + R_2} \right) \]

### Step 3: Rearranging the equation
Multiply both sides by \( R_1 + R_2 \):
\[ (R_1 + R_2)^2 = n R_1 R_2 \]

### Step 4: Express n in terms of R1 and R2
Rearranging gives:
\[ n = \frac{(R_1 + R_2)^2}{R_1 R_2} \]

### Step 5: Expand the numerator
Expanding the numerator:
\[ n = \frac{R_1^2 + 2R_1R_2 + R_2^2}{R_1 R_2} \]
This can be rewritten as:
\[ n = \frac{R_1^2}{R_1 R_2} + 2 + \frac{R_2^2}{R_1 R_2} \]
Thus,
\[ n = \frac{R_1}{R_2} + \frac{R_2}{R_1} + 2 \]

### Step 6: Apply the AM-GM inequality
Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{R_1}{R_2} + \frac{R_2}{R_1} \geq 2 \]
This implies:
\[ n \geq 2 + 2 = 4 \]

### Conclusion
The minimum possible value of \( n \) is 4.