Question

When A uniform wire of resistance r is cut into equal parts?

Answer 1

When a uniform wire with resistance \( r \) is cut into \( n \) equal parts, the resistance of each of these parts will change. To understand this, let's break it down:

1. **Initial Resistance**:
   - Suppose the original wire has a total resistance of \( r \).
   - The resistance \( r \) depends on the length \( L \), the cross-sectional area \( A \), and the resistivity \( \rho \) of the wire, and is given by:
     \[
     r = \frac{\rho L}{A}
     \]

2. **Cutting the Wire**:
   - If the wire is cut into \( n \) equal parts, each part will be a smaller piece of the original wire. Since the wire is uniform, each part will have the same cross-sectional area \( A \) and the same resistivity \( \rho \).

3. **Length of Each Piece**:
   - The length of each of these \( n \) pieces will be \( \frac{L}{n} \).

4. **Resistance of Each Piece**:
   - The resistance \( r' \) of each smaller piece can be calculated using the same formula:
     \[
     r' = \frac{\rho \left(\frac{L}{n}\right)}{A}
     \]
   - Simplify this to:
     \[
     r' = \frac{r}{n}
     \]

So, each of the \( n \) pieces of wire will have a resistance that is \( \frac{1}{n} \) of the original resistance \( r \).

### Example
- **Original Wire**: Suppose the resistance of the wire is \( 10 \, \Omega \).
- **Cut into 5 Pieces**: Each piece will have a resistance of:
  \[
  \frac{10 \, \Omega}{5} = 2 \, \Omega
  \]

### Summary
When a uniform wire of resistance \( r \) is cut into \( n \) equal parts, the resistance of each part will be \( \frac{r}{n} \). This relationship arises because resistance is directly proportional to the length of the wire, and cutting the wire into smaller lengths reduces its resistance accordingly.

Answer 2

When a uniform wire of resistance \( R \) is cut into \( n \) equal parts, the resistance of each part changes. Here’s a detailed explanation:

### Resistance Calculation:

1. **Initial Wire Resistance**:
   - The original wire has a resistance \( R \).

2. **Length of Each Part**:
   - When the wire is cut into \( n \) equal parts, each part will have a length of \( \frac{L}{n} \), where \( L \) is the original length of the wire.

3. **Resistance of Each Part**:
   - The resistance \( R \) of a wire is given by the formula:
     \[
     R = \rho \frac{L}{A}
     \]
     where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area.
   - If each part is \( \frac{L}{n} \) long, the resistance of each part \( R_{\text{part}} \) can be calculated as:
     \[
     R_{\text{part}} = \rho \frac{\frac{L}{n}}{A} = \frac{R}{n}
     \]
   - Therefore, the resistance of each of the \( n \) parts is \( \frac{R}{n} \).

### Summary:

When a uniform wire of resistance \( R \) is cut into \( n \) equal parts, the resistance of each part becomes \( \frac{R}{n} \).