When a wire of resistance r is cut into two equal parts?
2 Answers
When you cut a wire of resistance \( r \) into two equal parts, each part will have its own resistance. To understand what happens to the resistance of each part, let's go through the process step-by-step.
### 1. Initial Resistance of the Wire
Let's denote the original wire's resistance as \( R \).
### 2. Cutting the Wire
When you cut the wire into two equal lengths, each part will have half the length of the original wire. The resistance of a wire is directly proportional to its length. Therefore, if you cut the wire into two equal parts, each part will have half the length of the original wire.
### 3. Calculating the Resistance of Each Part
The resistance \( R \) of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
If the original wire has length \( L \), then the resistance of the wire can be expressed as:
\[ R = \rho \frac{L}{A} \]
When you cut the wire into two equal lengths, each new length is \( \frac{L}{2} \). The resistance \( R_{\text{new}} \) of each part is:
\[ R_{\text{new}} = \rho \frac{\frac{L}{2}}{A} \]
Simplify this expression:
\[ R_{\text{new}} = \frac{1}{2} \rho \frac{L}{A} \]
Since \( \rho \frac{L}{A} \) is the resistance of the original wire \( R \):
\[ R_{\text{new}} = \frac{1}{2} R \]
### Conclusion
Each of the two equal-length parts of the wire will have a resistance that is **half** the resistance of the original wire. If the original wire had resistance \( r \), then each of the two cut pieces will have a resistance of \( \frac{r}{2} \).
### 1. Initial Resistance of the Wire
Let's denote the original wire's resistance as \( R \).
### 2. Cutting the Wire
When you cut the wire into two equal lengths, each part will have half the length of the original wire. The resistance of a wire is directly proportional to its length. Therefore, if you cut the wire into two equal parts, each part will have half the length of the original wire.
### 3. Calculating the Resistance of Each Part
The resistance \( R \) of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
If the original wire has length \( L \), then the resistance of the wire can be expressed as:
\[ R = \rho \frac{L}{A} \]
When you cut the wire into two equal lengths, each new length is \( \frac{L}{2} \). The resistance \( R_{\text{new}} \) of each part is:
\[ R_{\text{new}} = \rho \frac{\frac{L}{2}}{A} \]
Simplify this expression:
\[ R_{\text{new}} = \frac{1}{2} \rho \frac{L}{A} \]
Since \( \rho \frac{L}{A} \) is the resistance of the original wire \( R \):
\[ R_{\text{new}} = \frac{1}{2} R \]
### Conclusion
Each of the two equal-length parts of the wire will have a resistance that is **half** the resistance of the original wire. If the original wire had resistance \( r \), then each of the two cut pieces will have a resistance of \( \frac{r}{2} \).
When a wire of resistance \( R \) is cut into two equal parts, the resistance of each part is affected by the length and cross-sectional area of the wire. Here’s a detailed explanation of what happens:
### Original Wire Resistance
The resistance \( R \) of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
### Cutting the Wire
When the wire is cut into two equal parts:
1. **Length of Each Part**: The length of each new wire is \( \frac{L}{2} \), where \( L \) is the original length of the wire.
2. **Cross-Sectional Area**: The cross-sectional area \( A \) of each part remains the same because cutting the wire does not change its width.
### Resistance of Each Part
Let’s find the resistance \( R' \) of each of the two new wires. Using the resistance formula for each part:
\[ R' = \rho \frac{L/2}{A} \]
Since \( \frac{L/2}{A} = \frac{1}{2} \cdot \frac{L}{A} \), we have:
\[ R' = \frac{1}{2} \cdot \frac{R}{1} \]
\[ R' = \frac{R}{2} \]
### Summary
Each of the two new wires will have a resistance that is **half** of the original wire's resistance. If the original wire had a resistance \( R \), then each of the cut pieces will have a resistance of \( \frac{R}{2} \).
### Original Wire Resistance
The resistance \( R \) of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
### Cutting the Wire
When the wire is cut into two equal parts:
1. **Length of Each Part**: The length of each new wire is \( \frac{L}{2} \), where \( L \) is the original length of the wire.
2. **Cross-Sectional Area**: The cross-sectional area \( A \) of each part remains the same because cutting the wire does not change its width.
### Resistance of Each Part
Let’s find the resistance \( R' \) of each of the two new wires. Using the resistance formula for each part:
\[ R' = \rho \frac{L/2}{A} \]
Since \( \frac{L/2}{A} = \frac{1}{2} \cdot \frac{L}{A} \), we have:
\[ R' = \frac{1}{2} \cdot \frac{R}{1} \]
\[ R' = \frac{R}{2} \]
### Summary
Each of the two new wires will have a resistance that is **half** of the original wire's resistance. If the original wire had a resistance \( R \), then each of the cut pieces will have a resistance of \( \frac{R}{2} \).