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

Step 1: Understanding the Concept:
The problem involves finding the equivalent resistance of a circuit formed by a resistive wire bent into a geometric shape. The key is to correctly identify the series and parallel combinations when a voltage source is connected across two points.
Step 2: Calculating Resistance of Each Side:
The total resistance of the uniform wire is R. Since it is bent into an equilateral triangle, it has three equal sides. The resistance is distributed equally among the three sides.
Let the vertices be A, B, and C.
Resistance of each side (AB, BC, CA) = \(\frac{R}{3}\).
Step 3: Analyzing the Circuit Configuration:
We need to find the equivalent resistance between the ends of one side, for example, between points A and B.
Imagine connecting a battery across A and B. The current will split at point A.
One path is directly from A to B through the side AB. The resistance of this path is \(R_1 = R_{AB} = \frac{R}{3}\).
The other path is from A to C and then from C to B. The sides AC and CB are in series along this path. The total resistance of this path is \(R_2 = R_{AC} + R_{CB} = \frac{R}{3} + \frac{R}{3} = \frac{2R}{3}\).
These two paths, \(R_1\) and \(R_2\), are in parallel with each other between points A and B.
Step 4: Calculating Equivalent Resistance:
The equivalent resistance \(R_{eq}\) for a parallel combination is given by the formula: \[ R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2} \] Substituting the values of \(R_1\) and \(R_2\):
\[ R_{eq} = \frac{(\frac{R}{3}) \times (\frac{2R}{3})}{(\frac{R}{3}) + (\frac{2R}{3})} = \frac{\frac{2R^2}{9}}{\frac{3R}{3}} = \frac{\frac{2R^2}{9}}{R} \] \[ R_{eq} = \frac{2R^2}{9R} = \frac{2R}{9} \] Step 5: Final Answer:
The equivalent resistance between the ends of any side of the triangle is \(\frac{2R}{9}\).