A wire of length 'l' is stretched to double its length. Compare the resistance of the original and stretched wire.
Understand the Problem
The question asks to compare the electrical resistance of a wire when stretched to double its length against its original state, indicating a need to apply concepts of resistance in relation to length and possibly area.
Answer
The resistance of the stretched wire is $4 R_{\text{original}}$.
Answer for screen readers
The resistance of the stretched wire is four times greater than the resistance of the original wire:
$$ R_{\text{stretched}} = 4 R_{\text{original}} $$
Steps to Solve
- Understand the Formula for Resistance
The resistance $R$ of a wire is given by the formula:
$$ R = \frac{\rho L}{A} $$
where $\rho$ is the resistivity of the material, $L$ is the length of the wire, and $A$ is the cross-sectional area.
- Determine Resistance of Original Wire
For the original wire of length $L$ and area $A$, the resistance is:
$$ R_{\text{original}} = \frac{\rho L}{A} $$
- Analyze the Effects of Stretching
When the wire is stretched to double its length, the new length becomes $L' = 2L$.
Since volume remains constant, the relationship between area and length can be expressed as follows:
$$ A' = \frac{A}{2} $$
- Calculate Resistance of Stretched Wire
Using the new length and area, the resistance of the stretched wire is:
$$ R_{\text{stretched}} = \frac{\rho L'}{A'} = \frac{\rho (2L)}{\frac{A}{2}} = \frac{4\rho L}{A} $$
- Compare the Two Resistances
Now we can find how the resistance of the stretched wire compares to the original wire's resistance:
$$ R_{\text{stretched}} = 4 R_{\text{original}} $$
This indicates that the resistance of the stretched wire is four times that of the original wire.
The resistance of the stretched wire is four times greater than the resistance of the original wire:
$$ R_{\text{stretched}} = 4 R_{\text{original}} $$
More Information
When a wire is stretched to double its length, its cross-sectional area is reduced, which significantly increases its resistance due to the relationship between length, area, and resistivity. This concept is often encountered in physics and engineering.
Tips
- Not considering the change in cross-sectional area when calculating the resistance of the stretched wire.
- Forgetting that the resistivity is constant for a given material and does not change in this context.
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