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Step-by-Step Solution
Step 1: Understand the Problem
We have a wire of resistance $R$, uniform cross-section, and length $l$. This wire is bent to form a complete circle. We want to find the resistance between any two points on the circle that are diametrically opposite.
Step 2: Visualize the Circle Split into Two Equal Halves
When the wire is bent into a circle and you pick two diametrically opposite points, the circle essentially splits into two equal semicircles. Since the wire is uniform, each semicircle will have exactly half the total length. Hence, each semicircle has resistance $ \frac{R}{2} $.
Step 3: Recognize the Parallel Combination
These two semicircular segments of resistance $ \frac{R}{2} $ each are connected between the same two endpoints (the diametrically opposite points). Therefore, these two segments act like two resistors in parallel, each having resistance $ \frac{R}{2} $.
Step 4: Calculate the Equivalent Resistance
For two resistors in parallel, $R_{\mathrm{eq}}$ is given by:
$ \frac{1}{R_{\mathrm{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} $
Here, $ R_1 = \frac{R}{2} $ and $ R_2 = \frac{R}{2} $. Thus,
$ \frac{1}{R_{\mathrm{eq}}} = \frac{1}{\frac{R}{2}} + \frac{1}{\frac{R}{2}} = \frac{2}{R} + \frac{2}{R} = \frac{4}{R}. $
Hence,
$ R_{\mathrm{eq}} = \frac{R}{4}. $
Step 5: State the Final Answer
The equivalent resistance between the two diametrically opposite points on the wire is $ \frac{R}{4} $.
