Question

A wire of resistance $ R $ is bent into a triangular pyramid as shown in the figure, with each segment having the same length. The resistance between points $ A $ and $ B $ is $ \frac{R}{n} $. The value of $ n $ is:

• A. 16
• B. 14
• C. 10
• D. 12

Hint:

In complex circuits involving parallel and series combinations, it’s useful to break down the network into simpler parts and use symmetry to simplify the calculation.

Answer 1

The Correct Option is D

A triangular pyramid constructed from a wire of total resistance \( R \), where each segment possesses equal length, is analyzed. The resistance between points \( A \) and \( B \) is sought. Given that this resistance is expressed as \( \frac{R}{n} \), an examination of the resistance pathways within the pyramid is required. The resistance between any two points in such a circuit is determined by the series and parallel combination of individual segment resistances. Recognizing the pyramid's symmetry, where each leg represents a resistance path, the value of \( n \) can be derived by exploiting this symmetry and the principles of series and parallel resistances. Upon solving the circuit, the value of \( n \) is found to be \( 12 \). Consequently, the resistance between points \( A \) and \( B \) is \( \frac{R}{12} \). %
Final Answer n = 12