Question

Calculate the equivalent resistance between two diametrically opposite points of a circular wire of total resistance \(12\,\Omega\).

• A. \(2\,\Omega\)
• B. \(3\,\Omega\)
• C. \(4\,\Omega\)
• D. \(6\,\Omega\)

Hint:

For circular wire resistance problems, remember that resistance is proportional to the length of the wire. Connecting diametrically opposite points divides the circle into two equal semicircles which act as parallel resistors.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A uniform circular wire with a total resistance of \(12\,\Omega\) is connected into a circuit at two diametrically opposite points.
This connection physically divides the circular wire into two equal semicircular arcs.
Step 2: Key Formula or Approach:
The resistance of a uniform wire is directly proportional to its length.
When divided into two halves, each half will have half the total resistance, and they will act as two resistors connected in parallel.
The equivalent resistance \(R_{eq}\) for two parallel resistors is given by:
\[ R_{eq} = \frac{R_1 R_2}{R_1 + R_2} \] Step 3: Detailed Solution:
Let the total resistance be \(R = 12\,\Omega\).
Since diametrically opposite points divide the circle into two equal semicircles, the resistance of each semicircle is:
\[ R_1 = R_2 = \frac{R}{2} = \frac{12}{2} = 6\,\Omega \] These two semicircular sections are connected across the same two points, making them parallel resistors.
Applying the parallel resistance formula:
\[ R_{eq} = \frac{6 \times 6}{6 + 6} \] \[ R_{eq} = \frac{36}{12} \] \[ R_{eq} = 3\,\Omega \] Step 4: Final Answer:
The equivalent resistance between the two diametrically opposite points is \(3\,\Omega\).