The equivalent resistance of series combination of four equal resistors is S. If they are joined in parallel, the total resistance is P. The relation between S and P is given by S = nP. Then the minimum possible value of n is

Options

A. 12
B. 14
C. 16
D. 10

Answer

To solve the problem, we need to find the minimum possible value of \( n \) in the relation \( S = nP \), where \( S \) is the equivalent resistance of four equal resistors in series and \( P \) is the equivalent resistance of the same resistors in parallel.

### Step-by-Step Solution:

1. **Define the Resistance of Each Resistor**:
Let the resistance of each of the four equal resistors be \( R \).

2. **Calculate the Equivalent Resistance in Series**:
When resistors are connected in series, the total resistance \( S \) is given by:
\[ S = R_1 + R_2 + R_3 + R_4 = R + R + R + R = 4R \]

3. **Calculate the Equivalent Resistance in Parallel**:
When the same resistors are connected in parallel, the total resistance \( P \) is given by:
\[ \frac{1}{P} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \frac{1}{R_4} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{4}{R} \]
Therefore, rearranging gives:
\[ P = \frac{R}{4} \]

4. **Express \( R \) in Terms of \( S \) and \( P \)**:
From the series resistance, we have:
\[ R = \frac{S}{4} \quad \text{(Equation 1)} \]
From the parallel resistance, we have:
\[ R = 4P \quad \text{(Equation 2)} \]

5. **Equate the Two Expressions for \( R \)**:
Set Equation 1 equal to Equation 2:
\[ \frac{S}{4} = 4P \]

6. **Solve for \( S \)**:
Multiply both sides by 4:
\[ S = 16P \]

7. **Relate \( S \) and \( P \)**:
From the relation \( S = nP \), we can substitute \( S \):
\[ 16P = nP \]

8. **Divide Both Sides by \( P \)** (assuming \( P \neq 0 \)):
\[ n = 16 \]

### Conclusion:
The minimum possible value of \( n \) is \( 16 \).