When the length of wire is stretched, n times its length?
2 Answers
When a wire is stretched to \( n \) times its original length, its physical properties change in a way that affects its resistance. Here’s a detailed explanation:
### 1. **Change in Cross-Sectional Area**
When a wire is stretched, its length increases, but its volume remains constant (assuming the wire is incompressible). As the length increases, the cross-sectional area of the wire decreases to maintain constant volume. The relationship between the original and stretched dimensions can be used to determine the change in cross-sectional area.
**Original Length**: \( L \)
**Original Cross-Sectional Area**: \( A \)
**Stretched Length**: \( L' = n \cdot L \)
To find the new cross-sectional area \( A' \), use the volume conservation principle:
\[ \text{Volume} = \text{Original Volume} = \text{Stretched Volume} \]
\[ A \cdot L = A' \cdot L' \]
Substitute \( L' = n \cdot L \):
\[ A \cdot L = A' \cdot (n \cdot L) \]
Solve for \( A' \):
\[ A' = \frac{A}{n} \]
### 2. **Change in Resistance**
Resistance \( R \) of a wire is given by:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) is the resistivity of the material
- \( L \) is the length of the wire
- \( A \) is the cross-sectional area of the wire
For the stretched wire:
- New Length \( L' = n \cdot L \)
- New Cross-Sectional Area \( A' = \frac{A}{n} \)
Substitute these into the resistance formula:
\[ R' = \rho \frac{L'}{A'} = \rho \frac{n \cdot L}{\frac{A}{n}} \]
\[ R' = \rho \frac{n^2 \cdot L}{A} \]
Since the original resistance was \( R = \rho \frac{L}{A} \), the resistance of the stretched wire is:
\[ R' = n^2 \cdot R \]
### Summary
When a wire is stretched to \( n \) times its original length:
- The cross-sectional area decreases to \( \frac{A}{n} \).
- The resistance increases to \( n^2 \) times the original resistance.
This quadratic relationship shows that the resistance increases significantly with stretching, due to the reduction in cross-sectional area.
### 1. **Change in Cross-Sectional Area**
When a wire is stretched, its length increases, but its volume remains constant (assuming the wire is incompressible). As the length increases, the cross-sectional area of the wire decreases to maintain constant volume. The relationship between the original and stretched dimensions can be used to determine the change in cross-sectional area.
**Original Length**: \( L \)
**Original Cross-Sectional Area**: \( A \)
**Stretched Length**: \( L' = n \cdot L \)
To find the new cross-sectional area \( A' \), use the volume conservation principle:
\[ \text{Volume} = \text{Original Volume} = \text{Stretched Volume} \]
\[ A \cdot L = A' \cdot L' \]
Substitute \( L' = n \cdot L \):
\[ A \cdot L = A' \cdot (n \cdot L) \]
Solve for \( A' \):
\[ A' = \frac{A}{n} \]
### 2. **Change in Resistance**
Resistance \( R \) of a wire is given by:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) is the resistivity of the material
- \( L \) is the length of the wire
- \( A \) is the cross-sectional area of the wire
For the stretched wire:
- New Length \( L' = n \cdot L \)
- New Cross-Sectional Area \( A' = \frac{A}{n} \)
Substitute these into the resistance formula:
\[ R' = \rho \frac{L'}{A'} = \rho \frac{n \cdot L}{\frac{A}{n}} \]
\[ R' = \rho \frac{n^2 \cdot L}{A} \]
Since the original resistance was \( R = \rho \frac{L}{A} \), the resistance of the stretched wire is:
\[ R' = n^2 \cdot R \]
### Summary
When a wire is stretched to \( n \) times its original length:
- The cross-sectional area decreases to \( \frac{A}{n} \).
- The resistance increases to \( n^2 \) times the original resistance.
This quadratic relationship shows that the resistance increases significantly with stretching, due to the reduction in cross-sectional area.
When a wire is stretched to \(N\) times its original length, several key changes occur in its properties, including its resistance, cross-sectional area, and mechanical stress. Here’s a detailed look at these changes:
### 1. **Change in Resistance**
The electrical resistance \( R \) of a wire is determined by its length \( L \), cross-sectional area \( A \), and the material’s resistivity \( \rho \). The relationship is given by:
\[ R = \rho \frac{L}{A} \]
When the wire is stretched to \(N\) times its original length, the new length \( L' \) becomes:
\[ L' = N \cdot L \]
Assuming the wire is stretched uniformly, the volume of the wire remains constant. Therefore, the cross-sectional area \( A' \) of the wire will decrease as the length increases. The original volume \( V \) is:
\[ V = L \cdot A \]
After stretching, the volume \( V' \) is:
\[ V' = L' \cdot A' = N \cdot L \cdot A' \]
Since volume is conserved:
\[ L \cdot A = N \cdot L \cdot A' \]
Thus:
\[ A' = \frac{A}{N} \]
The new resistance \( R' \) can be calculated using the new length \( L' \) and new cross-sectional area \( A' \):
\[ R' = \rho \frac{L'}{A'} = \rho \frac{N \cdot L}{\frac{A}{N}} = \rho \frac{N^2 \cdot L}{A} = N^2 \cdot R \]
So, when a wire is stretched to \(N\) times its original length, its resistance increases by a factor of \(N^2\).
### 2. **Change in Mechanical Stress**
Mechanical stress \( \sigma \) in the wire is given by:
\[ \sigma = \frac{F}{A} \]
where \( F \) is the applied force and \( A \) is the cross-sectional area. As the wire is stretched, its cross-sectional area decreases, which increases the stress for the same force.
If the wire is stretched to \(N\) times its original length, the cross-sectional area reduces to \( \frac{A}{N} \), hence the new stress \( \sigma' \) becomes:
\[ \sigma' = \frac{F}{A'} = \frac{F}{\frac{A}{N}} = N \cdot \frac{F}{A} = N \cdot \sigma \]
Thus, the mechanical stress increases linearly with the factor \(N\) of the length stretching.
### 3. **Strain in the Wire**
Strain \( \epsilon \) is defined as the ratio of the change in length to the original length:
\[ \epsilon = \frac{\Delta L}{L} \]
In this case, if the wire is stretched to \(N\) times its original length, the strain is:
\[ \epsilon = \frac{L' - L}{L} = \frac{N \cdot L - L}{L} = N - 1 \]
### Summary
- **Resistance**: Increases by a factor of \(N^2\).
- **Mechanical Stress**: Increases by a factor of \(N\).
- **Strain**: Is \(N - 1\).
These effects illustrate how stretching a wire affects its electrical and mechanical properties, and understanding these changes is crucial in applications requiring precise control over the material's behavior under stress.
### 1. **Change in Resistance**
The electrical resistance \( R \) of a wire is determined by its length \( L \), cross-sectional area \( A \), and the material’s resistivity \( \rho \). The relationship is given by:
\[ R = \rho \frac{L}{A} \]
When the wire is stretched to \(N\) times its original length, the new length \( L' \) becomes:
\[ L' = N \cdot L \]
Assuming the wire is stretched uniformly, the volume of the wire remains constant. Therefore, the cross-sectional area \( A' \) of the wire will decrease as the length increases. The original volume \( V \) is:
\[ V = L \cdot A \]
After stretching, the volume \( V' \) is:
\[ V' = L' \cdot A' = N \cdot L \cdot A' \]
Since volume is conserved:
\[ L \cdot A = N \cdot L \cdot A' \]
Thus:
\[ A' = \frac{A}{N} \]
The new resistance \( R' \) can be calculated using the new length \( L' \) and new cross-sectional area \( A' \):
\[ R' = \rho \frac{L'}{A'} = \rho \frac{N \cdot L}{\frac{A}{N}} = \rho \frac{N^2 \cdot L}{A} = N^2 \cdot R \]
So, when a wire is stretched to \(N\) times its original length, its resistance increases by a factor of \(N^2\).
### 2. **Change in Mechanical Stress**
Mechanical stress \( \sigma \) in the wire is given by:
\[ \sigma = \frac{F}{A} \]
where \( F \) is the applied force and \( A \) is the cross-sectional area. As the wire is stretched, its cross-sectional area decreases, which increases the stress for the same force.
If the wire is stretched to \(N\) times its original length, the cross-sectional area reduces to \( \frac{A}{N} \), hence the new stress \( \sigma' \) becomes:
\[ \sigma' = \frac{F}{A'} = \frac{F}{\frac{A}{N}} = N \cdot \frac{F}{A} = N \cdot \sigma \]
Thus, the mechanical stress increases linearly with the factor \(N\) of the length stretching.
### 3. **Strain in the Wire**
Strain \( \epsilon \) is defined as the ratio of the change in length to the original length:
\[ \epsilon = \frac{\Delta L}{L} \]
In this case, if the wire is stretched to \(N\) times its original length, the strain is:
\[ \epsilon = \frac{L' - L}{L} = \frac{N \cdot L - L}{L} = N - 1 \]
### Summary
- **Resistance**: Increases by a factor of \(N^2\).
- **Mechanical Stress**: Increases by a factor of \(N\).
- **Strain**: Is \(N - 1\).
These effects illustrate how stretching a wire affects its electrical and mechanical properties, and understanding these changes is crucial in applications requiring precise control over the material's behavior under stress.