Four wires of equal length and of resistance 5ohm each are connected in the form of a square. The equivalent resistance between the diagonally opposite corners of the square is

Options

A. `5 Omega`
B. `10 Omega`
C. `20 Omega`
D. `5/4 Omega`

Answer

To find the equivalent resistance between the diagonally opposite corners of a square formed by four resistors of equal resistance (5 ohms each), we can follow these steps:

### Step 1: Understand the Configuration
We have four resistors, each of 5 ohms, arranged in a square. Let's label the corners of the square as A, B, C, and D. We need to find the equivalent resistance between points B and D (the diagonally opposite corners).

### Step 2: Identify the Paths
When looking for the equivalent resistance between B and D, we can see that there are two paths:
1. From B to A to D (through resistors R1 and R2).
2. From B to C to D (through resistors R3 and R4).

### Step 3: Calculate the Resistance for Each Path
Each path consists of two resistors in series:
- For path B to A to D:
- R1 = 5 ohms (B to A)
- R2 = 5 ohms (A to D)
- Total resistance for this path (R_ABD) = R1 + R2 = 5 + 5 = 10 ohms.

- For path B to C to D:
- R3 = 5 ohms (B to C)
- R4 = 5 ohms (C to D)
- Total resistance for this path (R_BCD) = R3 + R4 = 5 + 5 = 10 ohms.

### Step 4: Combine the Two Paths
Now, we have two resistances (10 ohms each) in parallel between points B and D. The formula for the equivalent resistance (R_eq) of two resistors in parallel (R1 and R2) is given by:

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

Substituting the values:

\[ \frac{1}{R_{eq}} = \frac{1}{10} + \frac{1}{10} \]

\[ \frac{1}{R_{eq}} = \frac{2}{10} = \frac{1}{5} \]

### Step 5: Calculate the Equivalent Resistance
Taking the reciprocal gives us the equivalent resistance:

\[ R_{eq} = 5 \text{ ohms} \]

### Final Answer
The equivalent resistance between the diagonally opposite corners of the square is **5 ohms**.

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