A wire of resistance R is cut into ‘ n ’ equal parts. These parts are then connected in parallel. The equivalent resistance of the combination will be

Options

A. nR
B. `(R)/(n)`
C. `(n)/(R)`
D. `(R)/(n^(2))`

Answer

To find the equivalent resistance of a wire of resistance \( R \) that is cut into \( n \) equal parts and 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 \( n \) equal parts, the resistance of each part \( R_1 \) can be calculated using the formula:
\[ R_1 = \frac{R}{n} \]
- This is because resistance is directly proportional to the length of the wire, and cutting it into \( n \) equal parts reduces the length of each part.

2. **Set Up the Parallel Resistance Formula:**
- When these \( n \) parts are connected in parallel, the equivalent resistance \( R_{eq} \) can be calculated using the formula for resistances in parallel:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n} \]
- Since all resistances \( R_1, R_2, \ldots, R_n \) are equal, we can simplify this to:
\[ \frac{1}{R_{eq}} = n \cdot \frac{1}{R_1} \]

3. **Substitute the Value of \( R_1 \):**
- Substitute \( R_1 = \frac{R}{n} \) into the equation:
\[ \frac{1}{R_{eq}} = n \cdot \frac{1}{\frac{R}{n}} = n \cdot \frac{n}{R} = \frac{n^2}{R} \]

4. **Calculate the Equivalent Resistance:**
- Now, take the reciprocal to find \( R_{eq} \):
\[ R_{eq} = \frac{R}{n^2} \]

### Final Answer:
The equivalent resistance of the combination is:
\[ R_{eq} = \frac{R}{n^2} \]