A uniform wire of resistance R and length L is cut into four equal parts , each of lengths L/4, which are then connected in parallel combination. The effective resistance of the combination. The effective resistance of the combination will be

Options

A. R
B. 4R
C. `R/4`
D. `R/16`

Answer

To find the effective resistance of a uniform wire of resistance \( R \) and length \( L \) that is cut into four equal parts and connected in parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Resistance of Each Segment**:
The original wire has a resistance \( R \) and length \( L \). When the wire is cut into four equal parts, each part has a length of \( \frac{L}{4} \). The resistance of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where \( \rho \) is the resistivity of the material, \( L \) is the length, and \( A \) is the cross-sectional area. Since the resistivity and cross-sectional area remain constant, the resistance of each segment (let's call it \( R' \)) can be calculated as:
\[ R' = \rho \frac{L/4}{A} = \frac{R}{4} \]

2. **Set Up the Parallel Resistance Formula**:
When these four segments are connected in parallel, the effective resistance \( R_{eq} \) can be calculated using the formula for resistances in parallel:
\[ \frac{1}{R_{eq}} = \frac{1}{R'} + \frac{1}{R'} + \frac{1}{R'} + \frac{1}{R'} \]
Since all four resistances are equal, this simplifies to:
\[ \frac{1}{R_{eq}} = 4 \cdot \frac{1}{R'} \]

3. **Substitute the Value of \( R' \)**:
Substitute \( R' = \frac{R}{4} \) into the equation:
\[ \frac{1}{R_{eq}} = 4 \cdot \frac{1}{\frac{R}{4}} = 4 \cdot \frac{4}{R} = \frac{16}{R} \]

4. **Calculate the Effective Resistance**:
Taking the reciprocal to find \( R_{eq} \):
\[ R_{eq} = \frac{R}{16} \]

### Final Answer:
The effective resistance of the combination is:
\[ R_{eq} = \frac{R}{16} \]