Question

A square shaped wire with resistance of each side \( 3 \, \Omega \) is bent to form a complete circle. The resistance between two diametrically opposite points of the circle in unit of \( \Omega \) will be \(\dots\dots\dots\).

Hint:

For a uniform ring of total resistance \( R \), the resistance between points subtending an angle \( \theta \) at the center is \( R' = \frac{R \theta (2\pi - \theta)}{4\pi^2} \). For diametrically opposite points, \( \theta = \pi \), simplifying to \( R/4 \).

Answer 1

Correct Answer: 3

Step 1: Understanding the Concept:
The total resistance of a wire is proportional to its length. When a wire is bent into a circle, the resistance between diametrically opposite points is equivalent to two halves of the wire connected in parallel.
Step 2: Key Formula or Approach:
1. Total resistance: \( R_{total} = n \times r_{side} \).
2. Parallel resistance: \( R_{eq} = \frac{R_{half} \times R_{half}}{R_{half} + R_{half}} \).
Step 3: Detailed Explanation:
1. Total Resistance of the wire:
The wire has 4 sides, each of \( 3 \, \Omega \).
\[ R_{total} = 4 \times 3 = 12 \, \Omega \]
2. Resistance between diametrically opposite points:
When the \( 12 \, \Omega \) wire is bent into a circle, diametrically opposite points divide the circle into two equal lengths.
Each half-circle has a resistance:
\[ R_{half} = \frac{12 \, \Omega}{2} = 6 \, \Omega \]
These two halves are in parallel between the measurement points.
3. Equivalent Resistance:
\[ R_{eq} = \frac{6 \times 6}{6 + 6} = \frac{36}{12} = 3 \, \Omega \]
Step 4: Final Answer:
The resistance is 3 \( \Omega \).