Equivalent resistance between the adjacent corners of a regular n–sided polygon of uniform wire of resistance` R` would be :

Options

A. `((n-1)R)/(2n-1)`
B. `((n-1)R)/(n)`
C. `(n^2-R)/(n-1)`
D. `((n-1)R)/(n^2)`

Answer

To find the equivalent resistance between the adjacent corners of a regular n-sided polygon made of a uniform wire of resistance R, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have a regular n-sided polygon, and we need to find the equivalent resistance between two adjacent corners (let's call them A and B). The total resistance of the wire forming the polygon is R.

2. **Resistance of Each Edge:**
Since the polygon has n sides, the resistance of each edge (R_edge) can be calculated as:
\[ R_{\text{edge}} = \frac{R}{n} \]

3. **Identifying the Configuration:**
When looking at the polygon, the two adjacent corners A and B are connected by two edges:
- One edge directly between A and B.
- The other edge connects A to the next corner (let's call it C) and then C to B.

4. **Resistance Values:**
- The resistance of the direct edge AB is \( R_{\text{edge}} = \frac{R}{n} \).
- The resistance of the path A to C to B consists of two edges, which is:
\[ R_{AC} + R_{CB} = \frac{R}{n} + \frac{R}{n} = \frac{2R}{n} \]

5. **Equivalent Resistance Calculation:**
Now, we have two resistances in parallel:
- \( R_1 = \frac{R}{n} \) (direct edge AB)
- \( R_2 = \frac{2R}{n} \) (path ACB)

The formula for equivalent resistance \( R_{\text{eq}} \) for two resistors in parallel is:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} \]

Substituting the values:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{\frac{R}{n}} + \frac{1}{\frac{2R}{n}} = \frac{n}{R} + \frac{n}{2R} = \frac{n}{R} + \frac{n}{2R} = \frac{2n + n}{2R} = \frac{3n}{2R} \]

Therefore, the equivalent resistance \( R_{\text{eq}} \) is:
\[ R_{\text{eq}} = \frac{2R}{3n} \]

6. **Final Result:**
The equivalent resistance between the adjacent corners of a regular n-sided polygon of uniform wire of resistance R is:
\[ R_{\text{eq}} = \frac{2R}{3n} \]