Question

If an equilateral triangle is made of a uniform wire of resistance \( R \), then the equivalent resistance between the ends of a side is

• A. \( \frac{2R}{3} \)
• B. \( \frac{R}{3} \)
• C. \( \frac{R}{9} \)
• D. \( \frac{2R}{9} \)
• E. \( \frac{R}{6} \)

Hint:

Break network into simple series-parallel paths.

Answer 1

The Correct Option is D

Concept: Each side has resistance \( \frac{R}{3} \).

Step 1: Between two vertices.
One direct branch: \( \frac{R}{3} \) Other path: two sides → \( \frac{2R}{3} \)

Step 2: Parallel combination.
\[ R_{eq} = \frac{\frac{R}{3} \cdot \frac{2R}{3}}{\frac{R}{3} + \frac{2R}{3}} = \frac{2R^2/9}{R} = \frac{2R}{9} \]