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Step-by-Step Solution
Step 1: Understand the Given Problem
We have a square-shaped wire, where each side has a resistance of 3\,\Omega . The four sides are then bent to form a complete circle, so the total perimeter remains the same but is now in a circular shape. We need to find the equivalent resistance between two diametrically opposite points on this circle.
Step 2: Determine the Total Resistance of the Circular Wire
The total resistance of the wire is the sum of the resistances of all four sides of the original square. Since each side is 3\,\Omega , the total resistance in the circle is:
R_{\text{total}} = 3\,\Omega + 3\,\Omega + 3\,\Omega + 3\,\Omega = 12\,\Omega.
Step 3: Identify the Two Halves of the Circle
When taking two diametrically opposite points on the circle, the circle is split into two equal semicircles (arcs). Each arc will have half the total resistance, so each arc’s resistance is:
R_{\text{arc}} = \frac{12\,\Omega}{2} = 6\,\Omega.
Step 4: Calculate the Equivalent Resistance of the Two Semicircles
These two semicircles, each of 6\,\Omega , are connected in parallel between the diametrically opposite points. The formula for two resistors R_1 and R_2 in parallel is:
R_{\text{eq}} = \frac{R_1 \times R_2}{R_1 + R_2}.
Here R_1 = 6\,\Omega and R_2 = 6\,\Omega , so we get:
R_{\text{eq}} = \frac{6 \times 6}{6 + 6} = \frac{36}{12} = 3\,\Omega.
Step 5: State the Final Result
The resistance between two diametrically opposite points on the circular wire is 3\,\Omega .
