Question

If a wire is stretched to double its length. What will be its new resistivity?

Answer 1

The short answer is:

The new resistivity will be exactly the same.

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The Detailed Explanation

What is Resistivity?

Resistivity (ρ) is an intrinsic property of a material. This means it depends only on what the material is made of (e.g., copper, aluminum, rubber) and its temperature. It does not depend on the material's shape or size.

Think of it like density. If you have a block of aluminum and you stretch it or squash it, its density remains the same because it's still aluminum. Similarly, when you stretch a copper wire, it is still made of copper, so its resistivity does not change.

What You Might Be Thinking Of: Resistance

It's very common to confuse resistivity with resistance (R). Resistance does change when you stretch the wire.

Resistance is a property of a specific object that measures how much it opposes the flow of electric current. It depends on both the material's resistivity and its dimensions (length and cross-sectional area).

The formula for resistance is:

$R = \rho \frac{L}{A}$

where:
R = Resistance
ρ = Resistivity (this stays constant)
L = Length of the wire
A = Cross-sectional area of the wire

How Resistance Changes When the Wire is Stretched

Let's see what happens to the resistance when you stretch the wire to double its length.

1. Length (L): The new length ($L{new}$) is double the original length ($L{old}$).
   $L{new} = 2 \cdot L{old}$

2. Area (A): When you stretch a wire, it gets thinner. The key assumption is that the volume of the wire remains constant.
   Volume = Length × Area
   $V{old} = V{new}$
   $L{old} \cdot A{old} = L{new} \cdot A{new}$

   Now, substitute the new length into the equation:
   $L{old} \cdot A{old} = (2 \cdot L{old}) \cdot A{new}$

   You can cancel $L{old}$ from both sides, which gives you the relationship for the new area:
   $A{old} = 2 \cdot A{new}$
   or
   $A{new} = \frac{A_{old}}{2}$
   So, doubling the length halves the cross-sectional area.

3. Calculating the New Resistance ($R_{new}$):
   $R{new} = \rho \frac{L{new}}{A_{new}}$

   Now, substitute the new length and new area:
   $R{new} = \rho \frac{2 \cdot L{old}}{\frac{A_{old}}{2}}$

   Simplifying the fraction gives:
   $R{new} = 4 \cdot (\rho \frac{L{old}}{A_{old}})$

   Since the part in the parenthesis is the original resistance ($R{old}$), we get:
   $R{new} = 4 \cdot R_{old}$

Summary

| Property | How it Changes | Final Value |
| :---------- | :-------------------------------------------------------------------------------- | :---------------------------------------------- |
| Resistivity | Does NOT change. It's an intrinsic property of the material. | Remains the same. |
| Resistance | Does change. Length is doubled (factor of 2) and area is halved (factor of 1/2). | Becomes four times the original resistance. |