Four wires each of same length diameter and material are connected to each other to form a square. If the resistance of each wire is R then equivalent resistance across the opposite corners is

Options

A. R
B. R/2
C. R/4
D. none of these

Answer

To find the equivalent resistance across the opposite corners of a square formed by four wires, we can follow these steps:

### Step 1: Understand the Configuration
We have four wires connected to form a square. Each wire has the same resistance \( R \). We need to find the equivalent resistance between two opposite corners of the square.

### Step 2: Identify the Resistors in the Circuit
Label the corners of the square as A, B, C, and D. The resistances can be represented as follows:
- Resistance \( R_1 \) between A and B
- Resistance \( R_2 \) between B and C
- Resistance \( R_3 \) between C and D
- Resistance \( R_4 \) between D and A

### Step 3: Determine the Series and Parallel Combinations
When looking for the equivalent resistance between corners A and B:
- The resistances \( R_1 \) (A to B) and \( R_3 \) (C to D) are in series.
- The resistances \( R_2 \) (B to C) and \( R_4 \) (D to A) are also in series.

### Step 4: Calculate the Resistance in Series
For the resistors in series:
- The total resistance of \( R_1 \) and \( R_4 \) (A to B and D to A) is:
\[ R_{AB} = R + R = 2R \]
- The total resistance of \( R_2 \) and \( R_3 \) (B to C and C to D) is:
\[ R_{CD} = R + R = 2R \]

### Step 5: Combine the Series Resistances in Parallel
Now, we have two resistances \( R_{AB} = 2R \) and \( R_{CD} = 2R \) in parallel between A and B:
\[ \frac{1}{R_{eq}} = \frac{1}{R_{AB}} + \frac{1}{R_{CD}} = \frac{1}{2R} + \frac{1}{2R} \]

### Step 6: Simplify the Parallel Resistance
Calculating the above expression:
\[ \frac{1}{R_{eq}} = \frac{1 + 1}{2R} = \frac{2}{2R} = \frac{1}{R} \]
Thus, the equivalent resistance \( R_{eq} \) is:
\[ R_{eq} = R \]

### Final Answer
The equivalent resistance across the opposite corners A and B is \( R \).

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