Question

When wire of resistance R is stretched to N times its previous length, new resistance of stretched wire is R after stretching.?

Answer 1

When a wire is stretched, its resistance changes due to changes in both its length and its cross-sectional area. To understand the new resistance after stretching, let’s go through the process step by step.

### Initial Situation
- **Initial Length (L):** \( L \)
- **Initial Cross-Sectional Area (A):** \( A \)
- **Initial Resistance (R):** \( R \)

The resistance \( R \) of the wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where \( \rho \) is the resistivity of the material.

### After Stretching
When the wire is stretched to \( N \) times its original length:
- **New Length (L'):** \( N \times L \)
  
Since the volume of the wire remains constant during stretching, we can express the relationship between the original and new cross-sectional area. The volume \( V \) of the wire is given by:
\[ V = L \times A \]

After stretching, the volume remains the same:
\[ V' = L' \times A' = (N \times L) \times A' \]

Setting the volumes equal:
\[ L \times A = (N \times L) \times A' \]
\[ A' = \frac{A}{N} \]

### New Resistance
The new resistance \( R' \) can be calculated using the new length and new cross-sectional area:
\[ R' = \rho \frac{L'}{A'} \]
Substitute \( L' = N \times L \) and \( A' = \frac{A}{N} \):
\[ R' = \rho \frac{N \times L}{\frac{A}{N}} \]
\[ R' = \rho \frac{N^2 \times L}{A} \]

Since the original resistance \( R \) was \( \rho \frac{L}{A} \):
\[ R' = N^2 \times R \]

### Conclusion
After stretching the wire to \( N \) times its original length, the new resistance of the wire will be \( N^2 \) times the original resistance. So, if the initial resistance was \( R \), the new resistance \( R' \) is \( N^2 \times R \).

Answer 2

When a wire of resistance \( R \) is stretched to \( N \) times its original length, the new resistance of the wire can be found using the relationship between resistance, length, and cross-sectional area of the wire.

**Understanding the Relationship:**

1. **Initial Resistance \( R \):**
   - Let the original length of the wire be \( L \).
   - Let the original cross-sectional area be \( A \).
   - The resistance \( R \) of the wire is given by:
     \[
     R = \rho \frac{L}{A}
     \]
     where \( \rho \) is the resistivity of the material.

2. **After Stretching:**
   - The new length of the wire is \( L' = N \cdot L \).
   - The volume of the wire remains constant before and after stretching. Therefore:
     \[
     \text{Original volume} = L \cdot A = \text{New volume} = L' \cdot A'
     \]
     \[
     L \cdot A = (N \cdot L) \cdot A'
     \]
     \[
     A' = \frac{A}{N}
     \]
   - The new cross-sectional area \( A' \) is \( \frac{A}{N} \).

3. **New Resistance \( R' \):**
   - The resistance after stretching is:
     \[
     R' = \rho \frac{L'}{A'}
     \]
     \[
     R' = \rho \frac{N \cdot L}{\frac{A}{N}}
     \]
     \[
     R' = \rho \frac{N^2 \cdot L}{A}
     \]
     \[
     R' = N^2 \cdot \left( \rho \frac{L}{A} \right)
     \]
     \[
     R' = N^2 \cdot R
     \]

**Conclusion:**

The new resistance \( R' \) of the wire after it has been stretched to \( N \) times its original length is \( N^2 \cdot R \).