A uniform wire os resistance `12 Omega` is bent to form a circle. Effective resistance across two diametrically opposite points is

Options

A. `12 Omega `
B. ` 2 4 Omega `
C. ` 3 Omega`
D. ` 6 Omega `

Answer

To solve the problem of finding the effective resistance across two diametrically opposite points of a uniform wire bent into a circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Resistance of the Wire**:
The total resistance of the uniform wire is given as \( R = 12 \, \Omega \).

2. **Divide the Wire into Two Equal Parts**:
When the wire is bent into a circle, it is divided into two equal halves. Since the wire is uniform, the resistance of each half will be equal. Therefore, the resistance of each half is:
\[ R_1 = R_2 = \frac{R}{2} = \frac{12 \, \Omega}{2} = 6 \, \Omega \]

3. **Understand the Configuration**:
The two halves of the wire (each with resistance \( 6 \, \Omega \)) are connected in parallel when considering the effective resistance between the two diametrically opposite points.

4. **Use the Formula for Parallel Resistance**:
The formula for the equivalent resistance \( R_{eq} \) of two resistors \( R_1 \) and \( R_2 \) in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Substituting the values:
\[ \frac{1}{R_{eq}} = \frac{1}{6 \, \Omega} + \frac{1}{6 \, \Omega} \]

5. **Calculate the Equivalent Resistance**:
Simplifying the equation:
\[ \frac{1}{R_{eq}} = \frac{1 + 1}{6} = \frac{2}{6} = \frac{1}{3} \]
Therefore, the equivalent resistance is:
\[ R_{eq} = 3 \, \Omega \]

6. **Conclusion**:
The effective resistance across the two diametrically opposite points is:
\[ R_{eq} = 3 \, \Omega \]