Question

What will be the new resistance when a wire of resistance R is stretched to four times its initial length?

Answer 1

When a wire of resistance \( R \) is stretched to four times its initial length, its resistance changes due to alterations in both its length and its cross-sectional area. Here's a step-by-step explanation of how to determine the new resistance:

1. **Understand the Relationship Between Resistance, Length, and Cross-sectional Area**:
   The resistance \( R \) of a wire is given by the formula:
   \[
   R = \rho \frac{L}{A}
   \]
   where:
   - \( \rho \) is the resistivity of the material (a constant for a given material),
   - \( L \) is the length of the wire,
   - \( A \) is the cross-sectional area of the wire.

2. **Effect of Stretching the Wire**:
   When the wire is stretched, its length \( L \) increases, and its cross-sectional area \( A \) decreases. Specifically, if the wire is stretched to four times its original length, the new length \( L' \) is:
   \[
   L' = 4L
   \]

3. **Volume Conservation**:
   The volume of the wire remains constant before and after stretching. Hence:
   \[
   \text{Initial Volume} = \text{Final Volume}
   \]
   Mathematically:
   \[
   L \times A = L' \times A'
   \]
   Substituting \( L' = 4L \):
   \[
   L \times A = 4L \times A'
   \]
   Solving for the new cross-sectional area \( A' \):
   \[
   A' = \frac{A}{4}
   \]

4. **Calculate the New Resistance**:
   Using the resistance formula for the new length \( L' \) and the new area \( A' \):
   \[
   R' = \rho \frac{L'}{A'}
   \]
   Substitute \( L' = 4L \) and \( A' = \frac{A}{4} \):
   \[
   R' = \rho \frac{4L}{\frac{A}{4}}
   \]
   Simplify the expression:
   \[
   R' = \rho \frac{4L \times 4}{A} = 16 \frac{\rho L}{A}
   \]
   Therefore:
   \[
   R' = 16R
   \]

So, when a wire with initial resistance \( R \) is stretched to four times its original length, the new resistance of the wire becomes \( 16R \).

Answer 2

When a wire is stretched to four times its initial length, its resistance changes according to specific principles of physics. Let's break it down step by step to understand how this happens:

### Understanding Resistance and Stretching

1. **Resistance and Length Relationship**:
   The resistance \( R \) of a wire is given by the formula:
   \[
   R = \rho \frac{L}{A}
   \]
   where:
   - \( \rho \) is the resistivity of the material (a property that does not change with length),
   - \( L \) is the length of the wire,
   - \( A \) is the cross-sectional area of the wire.

2. **Effect of Stretching on Length**:
   When the wire is stretched to four times its initial length, the new length \( L' \) is:
   \[
   L' = 4L
   \]

3. **Effect of Stretching on Cross-Sectional Area**:
   Stretching a wire not only increases its length but also decreases its cross-sectional area. Assuming the volume of the wire remains constant during stretching, the volume \( V \) before and after stretching is:
   \[
   V = L \times A = L' \times A'
   \]
   Here, \( A' \) is the new cross-sectional area after stretching. Substituting \( L' = 4L \):
   \[
   L \times A = 4L \times A'
   \]
   Solving for \( A' \):
   \[
   A' = \frac{A}{4}
   \]

4. **New Resistance Calculation**:
   Using the formula for resistance with the new length \( L' \) and new area \( A' \):
   \[
   R' = \rho \frac{L'}{A'}
   \]
   Substituting \( L' = 4L \) and \( A' = \frac{A}{4} \):
   \[
   R' = \rho \frac{4L}{\frac{A}{4}}
   \]
   Simplify the expression:
   \[
   R' = \rho \frac{4L \times 4}{A} = 16 \rho \frac{L}{A}
   \]
   Therefore:
   \[
   R' = 16R
   \]

### Conclusion

When a wire with an initial resistance \( R \) is stretched to four times its initial length, the new resistance becomes \( 16R \). The increase in resistance is due to both the increase in length and the decrease in cross-sectional area of the wire.