Question

5 equal resistors are combined suitably to get maximum and minimum effective resistances. Their ratio is

• A. 5 : 1
• B. 125 : 1
• C. 1 : 5
• D. 1 : 25
• E. 25 : 1

Hint:

The ratio of maximum to minimum resistance for \(n\) identical resistors is always \(n^2\). Here \(5^2 = 25\). This shortcut works for both resistors and capacitors (though for capacitors, the parallel/series roles are reversed).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
For a set of equal resistors, the maximum resistance is obtained when they are connected in series, and the minimum resistance is obtained when they are connected in parallel.

Step 2: Key Formula or Approach:
For \(n\) equal resistors of resistance \(R\):
1. Series resistance: \(R_{max} = nR\)
2. Parallel resistance: \(R_{min} = \frac{R}{n}\)

Step 3: Detailed Explanation:
Given \(n = 5\) equal resistors. Let each resistor have resistance \(R\).
Maximum resistance (Series):
\[ R_{max} = R + R + R + R + R = 5R \]
Minimum resistance (Parallel):
\[ \frac{1}{R_{min}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{5}{R} \implies R_{min} = \frac{R}{5} \]
Calculating the ratio:
\[ \text{Ratio} = \frac{R_{max}}{R_{min}} = \frac{5R}{R/5} = 25 : 1 \]

Step 4: Final Answer:
The ratio is 25 : 1.