Question

Each side of a regular hexagon has resistance R. The effective resistance between the two opposite vertices of the hexagon is:

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

Answer 1

The Correct Option is C

A regular hexagon has 6 sides, each with resistance R. We need to find the equivalent resistance between two opposite vertices.

The hexagon can be divided into two parallel paths between the opposite vertices:

1. One direct path with resistance 2R (two sides in series)
2. Two parallel paths each with resistance 2R (four sides forming two parallel branches of 2R each)

The three parallel resistances are:

• First path: 2R
• Second path: R + R = 2R
• Third path: R + R = 2R

Equivalent resistance for three 2R resistors in parallel:

\[ \frac{1}{R_{eq}} = \frac{1}{2R} + \frac{1}{2R} + \frac{1}{2R} = \frac{3}{2R} \]

\[ R_{eq} = \frac{2R}{3} \]

Answer 2

Effective Resistance Between Opposite Vertices of a Regular Hexagon

To determine the effective resistance between two opposite vertices of a regular hexagon where each side has a resistance \( R \), we can leverage the symmetry of the hexagon. The hexagon can be divided into three parallel paths connecting the two opposite vertices. Each path consists of two resistors in series, so the resistance of each path is \( 2R \). Since there are three such paths in parallel, we use the formula for parallel resistances:

\[ \frac{1}{R_{eff}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]

Here, \( R_1 = R_2 = R_3 = 2R \). Thus,

\[ \frac{1}{R_{eff}} = \frac{1}{2R} + \frac{1}{2R} + \frac{1}{2R} = \frac{3}{2R}. \]

Inverting this equation gives us the effective resistance:

\[ R_{eff} = \frac{2R}{3}. \]

Step-by-Step Solution

Identify that the hexagon has 6 sides, each with resistance \( R \).

Recognize that there are three parallel paths between the two opposite vertices.

Calculate the resistance of each path: each path has two resistors in series, so the resistance is \( 2R \).

Use the formula for parallel resistances:

\[ \frac{1}{R_{eff}} = \frac{1}{2R} + \frac{1}{2R} + \frac{1}{2R}. \]

Solve for the effective resistance:

\[ R_{eff} = \frac{2R}{3}. \]

Final Answer

The effective resistance between the two opposite vertices of the hexagon is

\[ \boxed{\frac{2R}{3}} \]