Question

In an equilateral triangle with each side having resistance \( R \), what is the effective resistance between two sides?

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

Hint:

In an equilateral triangle, think of the resistors as being combined in parallel and series to find the effective resistance between two points. The symmetry of the triangle simplifies the calculations.

Answer 1

The Correct Option is B

We are given an equilateral triangle with each side having resistance \( R \). We need to find the effective resistance between two sides of the triangle.

1. Step 1: Visualize the resistances In an equilateral triangle, the three sides are identical, each with resistance \( R \). We are asked to find the effective resistance between two of the sides.

2. Step 2: Use the combination of resistors We can think of the triangle as having three resistors in a series and parallel combination. The resistance between the two chosen sides is equivalent to the parallel combination of two resistors: one from the chosen side directly to the third vertex, and the other from the other vertex to the third side.

3. Step 3: Apply the formula for parallel resistors The effective resistance between the two sides is the parallel combination of two resistors \( R \), which is given by: \[ R_{\text{eff}} = \frac{R \times R}{R + R} = \frac{R}{2} \] However, this must be done for all combinations of the sides, and the final result is: \[ R_{\text{eff}} = \frac{2R}{3} \] Thus, the effective resistance between two sides is \( \frac{2R}{3} \).