What happens to resistance if wire is stretched?
2 Answers
When a wire is stretched, its resistance increases. This relationship can be understood through the fundamental properties of resistance and how they change with physical alterations to the wire. Here’s a detailed explanation:
### Key Concepts
1. **Resistance Formula**: The resistance \( R \) of a conductor is given by the formula:
\[
R = \frac{\rho L}{A}
\]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material (a property that depends on the material itself and is constant for a given material at a constant temperature),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
2. **Effect of Stretching**:
- **Length Increase**: When a wire is stretched, its length \( L \) increases. According to the formula, if the length increases while the resistivity remains constant, the resistance \( R \) will increase.
- **Cross-Sectional Area Decrease**: Stretching the wire also decreases its cross-sectional area \( A \). Since resistance is inversely proportional to the area, a decrease in area will further increase the resistance.
### Mathematical Perspective
Let's denote the initial length and area of the wire as \( L_0 \) and \( A_0 \), respectively. When the wire is stretched to a new length \( L = kL_0 \) (where \( k > 1 \)), the new cross-sectional area can be related to the initial area by the principle of conservation of volume (assuming the volume of the wire remains constant):
\[
A = \frac{A_0 L_0}{L} = \frac{A_0 L_0}{kL_0} = \frac{A_0}{k}
\]
Now, substituting these new values into the resistance formula:
\[
R = \frac{\rho (kL_0)}{\frac{A_0}{k}} = \frac{\rho k^2 L_0}{A_0}
\]
This shows that the new resistance \( R \) increases by a factor of \( k^2 \), since both the length increases linearly (proportional to \( k \)) and the area decreases inversely (also proportional to \( k \)).
### Conclusion
In summary, when a wire is stretched:
- Its **length** increases, contributing to a higher resistance.
- Its **cross-sectional area** decreases, which further increases resistance.
The overall effect is that the resistance of the wire increases significantly when stretched. This principle is important in applications like electrical engineering, where understanding how materials behave under stress is crucial for designing circuits and components.
### Key Concepts
1. **Resistance Formula**: The resistance \( R \) of a conductor is given by the formula:
\[
R = \frac{\rho L}{A}
\]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material (a property that depends on the material itself and is constant for a given material at a constant temperature),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
2. **Effect of Stretching**:
- **Length Increase**: When a wire is stretched, its length \( L \) increases. According to the formula, if the length increases while the resistivity remains constant, the resistance \( R \) will increase.
- **Cross-Sectional Area Decrease**: Stretching the wire also decreases its cross-sectional area \( A \). Since resistance is inversely proportional to the area, a decrease in area will further increase the resistance.
### Mathematical Perspective
Let's denote the initial length and area of the wire as \( L_0 \) and \( A_0 \), respectively. When the wire is stretched to a new length \( L = kL_0 \) (where \( k > 1 \)), the new cross-sectional area can be related to the initial area by the principle of conservation of volume (assuming the volume of the wire remains constant):
\[
A = \frac{A_0 L_0}{L} = \frac{A_0 L_0}{kL_0} = \frac{A_0}{k}
\]
Now, substituting these new values into the resistance formula:
\[
R = \frac{\rho (kL_0)}{\frac{A_0}{k}} = \frac{\rho k^2 L_0}{A_0}
\]
This shows that the new resistance \( R \) increases by a factor of \( k^2 \), since both the length increases linearly (proportional to \( k \)) and the area decreases inversely (also proportional to \( k \)).
### Conclusion
In summary, when a wire is stretched:
- Its **length** increases, contributing to a higher resistance.
- Its **cross-sectional area** decreases, which further increases resistance.
The overall effect is that the resistance of the wire increases significantly when stretched. This principle is important in applications like electrical engineering, where understanding how materials behave under stress is crucial for designing circuits and components.
When a wire is stretched, its physical dimensions change, and as a result, its electrical resistance increases. This is because the resistance of a wire depends on its material properties as well as its geometry, such as its length and cross-sectional area. Let’s break down the effect in detail using the following principles:
### 1. **Ohm's Law & Resistance Formula**
The resistance \( R \) of a wire is given by the formula:
\[
R = \rho \frac{L}{A}
\]
Where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material (a constant that depends on the material),
- \( L \) is the length of the wire, and
- \( A \) is the cross-sectional area of the wire.
### 2. **Effect of Stretching the Wire**
When a wire is stretched, its length \( L \) increases and its cross-sectional area \( A \) decreases. These changes both affect the resistance:
#### **Increase in Length:**
- Stretching the wire increases its length. Since resistance is directly proportional to the length \( L \), an increase in length will cause an increase in the resistance.
For example, if you stretch the wire to double its length, the resistance would also double (assuming the cross-sectional area decreases uniformly).
#### **Decrease in Cross-Sectional Area:**
- As the wire is stretched, its volume remains constant (assuming no material is lost), which means the cross-sectional area \( A \) must decrease as the length increases.
- Resistance is inversely proportional to the cross-sectional area \( A \), so as the area decreases, the resistance increases.
If the wire is stretched thin, the cross-sectional area becomes smaller, leading to a more significant increase in resistance.
### 3. **Quantitative Relationship (Assuming Constant Volume)**
If the volume of the wire remains constant during stretching (which is often assumed in ideal cases), we can relate the original and stretched dimensions of the wire:
- Initial volume: \( V_0 = L_0 \times A_0 \),
- Final volume: \( V = L \times A \).
Since the volume remains constant:
\[
L_0 \times A_0 = L \times A
\]
From this, we can see that as the length increases, the cross-sectional area decreases proportionally.
### 4. **Combined Effect on Resistance**
The increase in resistance due to stretching comes from two factors:
- **Longer length**: Resistance increases because of the longer path for the electrons.
- **Smaller cross-sectional area**: Resistance increases because the thinner wire offers less space for electrons to flow through.
Both these factors act together to **increase the resistance significantly**.
### Example:
Suppose you stretch the wire to twice its original length (\( L = 2L_0 \)). Using the constant volume assumption, the new cross-sectional area would be:
\[
A = \frac{A_0}{2}
\]
The new resistance \( R \) becomes:
\[
R = \rho \frac{2L_0}{\frac{A_0}{2}} = 4 \times \rho \frac{L_0}{A_0} = 4R_0
\]
In this case, the resistance increases fourfold if the wire is stretched to twice its length.
### 5. **Practical Considerations**
- **Material Limits**: In reality, stretching a wire too much can lead to structural failure, as most materials have a limit to how much they can be stretched (known as the breaking point).
- **Temperature**: Stretching the wire can also change its temperature due to internal friction, which might further affect the resistance since resistivity \( \rho \) itself depends on temperature.
### Conclusion:
When a wire is stretched, its resistance increases due to the increase in length and the decrease in cross-sectional area. The exact increase depends on how much the wire is stretched and the assumption that the wire's volume remains constant. Therefore, stretching a wire results in a significant increase in its electrical resistance.
### 1. **Ohm's Law & Resistance Formula**
The resistance \( R \) of a wire is given by the formula:
\[
R = \rho \frac{L}{A}
\]
Where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material (a constant that depends on the material),
- \( L \) is the length of the wire, and
- \( A \) is the cross-sectional area of the wire.
### 2. **Effect of Stretching the Wire**
When a wire is stretched, its length \( L \) increases and its cross-sectional area \( A \) decreases. These changes both affect the resistance:
#### **Increase in Length:**
- Stretching the wire increases its length. Since resistance is directly proportional to the length \( L \), an increase in length will cause an increase in the resistance.
For example, if you stretch the wire to double its length, the resistance would also double (assuming the cross-sectional area decreases uniformly).
#### **Decrease in Cross-Sectional Area:**
- As the wire is stretched, its volume remains constant (assuming no material is lost), which means the cross-sectional area \( A \) must decrease as the length increases.
- Resistance is inversely proportional to the cross-sectional area \( A \), so as the area decreases, the resistance increases.
If the wire is stretched thin, the cross-sectional area becomes smaller, leading to a more significant increase in resistance.
### 3. **Quantitative Relationship (Assuming Constant Volume)**
If the volume of the wire remains constant during stretching (which is often assumed in ideal cases), we can relate the original and stretched dimensions of the wire:
- Initial volume: \( V_0 = L_0 \times A_0 \),
- Final volume: \( V = L \times A \).
Since the volume remains constant:
\[
L_0 \times A_0 = L \times A
\]
From this, we can see that as the length increases, the cross-sectional area decreases proportionally.
### 4. **Combined Effect on Resistance**
The increase in resistance due to stretching comes from two factors:
- **Longer length**: Resistance increases because of the longer path for the electrons.
- **Smaller cross-sectional area**: Resistance increases because the thinner wire offers less space for electrons to flow through.
Both these factors act together to **increase the resistance significantly**.
### Example:
Suppose you stretch the wire to twice its original length (\( L = 2L_0 \)). Using the constant volume assumption, the new cross-sectional area would be:
\[
A = \frac{A_0}{2}
\]
The new resistance \( R \) becomes:
\[
R = \rho \frac{2L_0}{\frac{A_0}{2}} = 4 \times \rho \frac{L_0}{A_0} = 4R_0
\]
In this case, the resistance increases fourfold if the wire is stretched to twice its length.
### 5. **Practical Considerations**
- **Material Limits**: In reality, stretching a wire too much can lead to structural failure, as most materials have a limit to how much they can be stretched (known as the breaking point).
- **Temperature**: Stretching the wire can also change its temperature due to internal friction, which might further affect the resistance since resistivity \( \rho \) itself depends on temperature.
### Conclusion:
When a wire is stretched, its resistance increases due to the increase in length and the decrease in cross-sectional area. The exact increase depends on how much the wire is stretched and the assumption that the wire's volume remains constant. Therefore, stretching a wire results in a significant increase in its electrical resistance.