A wire of resistance r is cut into m equal parts. These parts are then connected in parallel . The equivalent resistance of the combination will be

Options

A. mr
B. `r//m`
C. `m//r`
D. `r//m^(2)`

Answer

To find the equivalent resistance of a wire of resistance \( R \) that is cut into \( m \) equal parts and then connected in parallel, we can follow these steps:

### Step-by-Step Solution

1. **Determine the Resistance of Each Part**:
- When the wire of resistance \( R \) is cut into \( m \) equal parts, the resistance of each part, denoted as \( r' \), can be calculated using the formula:
\[ r' = \frac{R}{m} \]
- Here, \( r' \) is the resistance of each individual part after cutting.

2. **Set Up the Parallel Resistance Formula**:
- When resistors are connected in parallel, the equivalent resistance \( R_{eq} \) can be calculated using the formula:
\[ \frac{1}{R_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \ldots + \frac{1}{r_m} \]
- Since all parts have the same resistance \( r' \), we can simplify this to:
\[ \frac{1}{R_{eq}} = \frac{1}{r'} + \frac{1}{r'} + \ldots + \frac{1}{r'} \quad (m \text{ times}) \]

3. **Calculate the Total Contribution**:
- This means:
\[ \frac{1}{R_{eq}} = m \cdot \frac{1}{r'} \]
- Substituting \( r' = \frac{R}{m} \) into the equation gives:
\[ \frac{1}{R_{eq}} = m \cdot \frac{1}{\frac{R}{m}} = \frac{m^2}{R} \]

4. **Find the Equivalent Resistance**:
- To find \( R_{eq} \), take the reciprocal of both sides:
\[ R_{eq} = \frac{R}{m^2} \]

5. **Final Answer**:
- Thus, the equivalent resistance of the combination of the \( m \) parts connected in parallel is:
\[ R_{eq} = \frac{R}{m^2} \]