What happens to resistance when A wire is stretched?
2 Answers
When a wire is stretched, several physical changes occur that directly affect its resistance. To understand what happens, we need to review some key factors in the relationship between resistance, length, and cross-sectional area of a wire.
### Resistance and its 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 for each material).
- \( L \) is the length of the wire.
- \( A \) is the cross-sectional area of the wire.
### How Stretching Affects Resistance
When a wire is stretched, both its length and its cross-sectional area are altered. Let's break down what happens:
#### 1. **Length Increases**
- As the wire is stretched, its length \( L \) increases. Since resistance is directly proportional to the length of the wire (as per the formula above), increasing the length increases the resistance.
- For example, if you double the length of the wire, the resistance will also roughly double.
#### 2. **Cross-sectional Area Decreases**
- Stretching the wire causes it to become thinner, which means its cross-sectional area \( A \) decreases. Since resistance is inversely proportional to the area, reducing the cross-sectional area will increase the resistance.
- If the cross-sectional area is halved (for example), the resistance will double, because the current has a smaller "path" to flow through.
#### 3. **Combined Effect on Resistance**
- When you stretch the wire, both the length \( L \) increases and the cross-sectional area \( A \) decreases. These changes work together to significantly increase the overall resistance.
- If the wire is stretched uniformly, and assuming the volume of the wire remains constant (this is an approximation but generally holds for metals), the new resistance can be derived mathematically based on how much the wire is stretched.
### Approximate Formula for Stretched Wire
If the wire is stretched by a factor \( n \) (i.e., if the length becomes \( n \) times the original length), the new resistance \( R' \) can be found using:
\[
R' = n^2 R
\]
Where:
- \( R \) is the original resistance.
- \( n \) is the stretch factor (the ratio of the new length to the original length).
This equation arises because:
- The length increases by \( n \), so resistance increases proportionally.
- The cross-sectional area decreases by a factor of \( n \), which increases the resistance further by another factor of \( n \).
Thus, the resistance increases by a factor of \( n^2 \).
### Example:
Suppose you have a wire with an initial length of 1 meter and it has a resistance of 10 ohms. If you stretch this wire so that its length doubles (i.e., \( n = 2 \)), the new resistance will be:
\[
R' = 2^2 \times 10 = 4 \times 10 = 40 \, \text{ohms}
\]
The resistance has increased by a factor of 4 due to the stretching.
### Conclusion:
When a wire is stretched, its resistance increases because:
1. The length increases (which directly increases resistance).
2. The cross-sectional area decreases (which also increases resistance).
This results in a significant increase in resistance, often by a factor of the square of the stretch factor.
### Resistance and its 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 for each material).
- \( L \) is the length of the wire.
- \( A \) is the cross-sectional area of the wire.
### How Stretching Affects Resistance
When a wire is stretched, both its length and its cross-sectional area are altered. Let's break down what happens:
#### 1. **Length Increases**
- As the wire is stretched, its length \( L \) increases. Since resistance is directly proportional to the length of the wire (as per the formula above), increasing the length increases the resistance.
- For example, if you double the length of the wire, the resistance will also roughly double.
#### 2. **Cross-sectional Area Decreases**
- Stretching the wire causes it to become thinner, which means its cross-sectional area \( A \) decreases. Since resistance is inversely proportional to the area, reducing the cross-sectional area will increase the resistance.
- If the cross-sectional area is halved (for example), the resistance will double, because the current has a smaller "path" to flow through.
#### 3. **Combined Effect on Resistance**
- When you stretch the wire, both the length \( L \) increases and the cross-sectional area \( A \) decreases. These changes work together to significantly increase the overall resistance.
- If the wire is stretched uniformly, and assuming the volume of the wire remains constant (this is an approximation but generally holds for metals), the new resistance can be derived mathematically based on how much the wire is stretched.
### Approximate Formula for Stretched Wire
If the wire is stretched by a factor \( n \) (i.e., if the length becomes \( n \) times the original length), the new resistance \( R' \) can be found using:
\[
R' = n^2 R
\]
Where:
- \( R \) is the original resistance.
- \( n \) is the stretch factor (the ratio of the new length to the original length).
This equation arises because:
- The length increases by \( n \), so resistance increases proportionally.
- The cross-sectional area decreases by a factor of \( n \), which increases the resistance further by another factor of \( n \).
Thus, the resistance increases by a factor of \( n^2 \).
### Example:
Suppose you have a wire with an initial length of 1 meter and it has a resistance of 10 ohms. If you stretch this wire so that its length doubles (i.e., \( n = 2 \)), the new resistance will be:
\[
R' = 2^2 \times 10 = 4 \times 10 = 40 \, \text{ohms}
\]
The resistance has increased by a factor of 4 due to the stretching.
### Conclusion:
When a wire is stretched, its resistance increases because:
1. The length increases (which directly increases resistance).
2. The cross-sectional area decreases (which also increases resistance).
This results in a significant increase in resistance, often by a factor of the square of the stretch factor.
When a wire is stretched, its resistance changes due to modifications in its physical dimensions. To understand this, let's break it down step-by-step using the formula for resistance:
### Formula for Resistance
The resistance \( R \) of a wire is given by the equation:
\[
R = \frac{\rho L}{A}
\]
Where:
- \( R \) is the resistance.
- \( \rho \) is the resistivity of the material (a property of the material, which doesn't change when the wire is stretched).
- \( L \) is the length of the wire.
- \( A \) is the cross-sectional area of the wire.
### What Happens When the Wire is Stretched?
When a wire is stretched, two key dimensions change:
1. **The Length (L):** When you stretch a wire, its length increases.
2. **The Cross-sectional Area (A):** As the wire is stretched, the material is pulled thinner, which decreases the cross-sectional area.
Both these changes affect the resistance of the wire.
### Impact of Length
As the wire is stretched, the length \( L \) increases. Since resistance is directly proportional to the length of the wire (\( R \propto L \)), an increase in length will cause the resistance to **increase**.
### Impact of Cross-sectional Area
The cross-sectional area \( A \) decreases as the wire becomes thinner. Since resistance is inversely proportional to the cross-sectional area (\( R \propto \frac{1}{A} \)), a decrease in the area will also cause the resistance to **increase**.
### Combined Effect
When the wire is stretched, both the increase in length and the decrease in cross-sectional area contribute to an overall **increase in resistance**.
### Mathematical Explanation
If a wire is stretched uniformly and its volume stays constant, we can express the relationship between the original length \( L_0 \), the stretched length \( L_1 \), and the original cross-sectional area \( A_0 \), and the new area \( A_1 \) as:
\[
L_0 A_0 = L_1 A_1
\]
If the wire is stretched to \( L_1 = nL_0 \) (where \( n \) is a stretch factor, i.e., how much the wire has been stretched), then the new area \( A_1 \) is given by:
\[
A_1 = \frac{A_0}{n}
\]
Substituting this into the resistance formula:
\[
R_1 = \frac{\rho L_1}{A_1} = \frac{\rho nL_0}{A_0 / n} = n^2 \cdot \frac{\rho L_0}{A_0} = n^2 R_0
\]
So, the resistance \( R_1 \) of the stretched wire is proportional to the square of the stretch factor \( n \). This shows that even a moderate stretch can lead to a significant increase in resistance.
### Summary
When a wire is stretched:
- **Length increases** → resistance increases.
- **Cross-sectional area decreases** → resistance increases.
- **Overall**, the resistance increases, and it does so proportionally to the square of the stretch factor, assuming volume remains constant.
This is a crucial concept in many engineering fields, particularly when dealing with materials under mechanical stress or strain.
### Formula for Resistance
The resistance \( R \) of a wire is given by the equation:
\[
R = \frac{\rho L}{A}
\]
Where:
- \( R \) is the resistance.
- \( \rho \) is the resistivity of the material (a property of the material, which doesn't change when the wire is stretched).
- \( L \) is the length of the wire.
- \( A \) is the cross-sectional area of the wire.
### What Happens When the Wire is Stretched?
When a wire is stretched, two key dimensions change:
1. **The Length (L):** When you stretch a wire, its length increases.
2. **The Cross-sectional Area (A):** As the wire is stretched, the material is pulled thinner, which decreases the cross-sectional area.
Both these changes affect the resistance of the wire.
### Impact of Length
As the wire is stretched, the length \( L \) increases. Since resistance is directly proportional to the length of the wire (\( R \propto L \)), an increase in length will cause the resistance to **increase**.
### Impact of Cross-sectional Area
The cross-sectional area \( A \) decreases as the wire becomes thinner. Since resistance is inversely proportional to the cross-sectional area (\( R \propto \frac{1}{A} \)), a decrease in the area will also cause the resistance to **increase**.
### Combined Effect
When the wire is stretched, both the increase in length and the decrease in cross-sectional area contribute to an overall **increase in resistance**.
### Mathematical Explanation
If a wire is stretched uniformly and its volume stays constant, we can express the relationship between the original length \( L_0 \), the stretched length \( L_1 \), and the original cross-sectional area \( A_0 \), and the new area \( A_1 \) as:
\[
L_0 A_0 = L_1 A_1
\]
If the wire is stretched to \( L_1 = nL_0 \) (where \( n \) is a stretch factor, i.e., how much the wire has been stretched), then the new area \( A_1 \) is given by:
\[
A_1 = \frac{A_0}{n}
\]
Substituting this into the resistance formula:
\[
R_1 = \frac{\rho L_1}{A_1} = \frac{\rho nL_0}{A_0 / n} = n^2 \cdot \frac{\rho L_0}{A_0} = n^2 R_0
\]
So, the resistance \( R_1 \) of the stretched wire is proportional to the square of the stretch factor \( n \). This shows that even a moderate stretch can lead to a significant increase in resistance.
### Summary
When a wire is stretched:
- **Length increases** → resistance increases.
- **Cross-sectional area decreases** → resistance increases.
- **Overall**, the resistance increases, and it does so proportionally to the square of the stretch factor, assuming volume remains constant.
This is a crucial concept in many engineering fields, particularly when dealing with materials under mechanical stress or strain.