Question

What happens to the resistance of the wire when its length is increased to twice its original length?

Answer 1

This gets to the core of how resistance works.

The short and direct answer is:

The resistance of the wire will double.

Here's a more detailed explanation of why.

1. The Simple Explanation (Analogy)

Think of the wire as a narrow hallway and the electric current as people trying to get through it.

• Resistance is the difficulty the people have moving through the hallway (bumping into walls, other people, etc.).
• If you make the hallway twice as long, the people have to travel twice the distance and will face twice the overall difficulty.

Similarly, when electrons travel through a longer wire, they encounter more atoms to bump into along their path, which increases the total opposition (resistance) to their flow.

2. The Physics Explanation (The Formula)

The resistance ($R$) of a wire is determined by the following formula:

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

Where:
$R$ = Resistance (in Ohms, Ω)
$\rho$ (rho) = Resistivity. This is a property of the material itself (e.g., copper has very low resistivity, while rubber has very high resistivity).
$L$ = Length of the wire.
$A$ = Cross-sectional area of the wire (how thick it is).

From this formula, you can see that resistance ($R$) is directly proportional to the length ($L$). If the resistivity ($\rho$) and area ($A$) stay the same, whatever you do to the length will happen to the resistance.

Let's do the math:

• Original Resistance: $R_{original} = \rho \frac{L}{A}$
• New Length: $L_{new} = 2 \times L$
• New Resistance: $R{new} = \rho \frac{L{new}}{A} = \rho \frac{2L}{A}$

If you compare the two, you can see that:

$R{new} = 2 \times \left( \rho \frac{L}{A} \right) = 2 \times R{original}$

So, doubling the length doubles the resistance.

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An Important Distinction: Stretching the Wire

There is a common "trick" version of this question in physics problems:

Question: What happens to the resistance if a wire is stretched to twice its original length?

Answer: The resistance will quadruple (increase by 4 times).

Why? When you stretch a wire, its volume must stay the same. To make it twice as long, it must also get thinner.

1. Length Doubles: $L_{new} = 2L$
2. Area Halves: To keep the volume constant (Volume = Length × Area), the cross-sectional area must become half of what it was. $A_{new} = A/2$

Now, let's put these new values into the resistance formula:

$R{new} = \rho \frac{L{new}}{A{new}} = \rho \frac{2L}{A/2} = \rho \frac{4L}{A} = 4 \times \left( \rho \frac{L}{A} \right) = 4 \times R{original}$

Summary

| Scenario | Change in Length | Change in Area | Change in Resistance |
| -------------------------------------------- | --------------------- | ----------------- | -------------------------- |
| Adding length (e.g., connecting a second identical wire) | Doubles (x2) | Stays the same | Doubles (x2) |
| Stretching the original wire | Doubles (x2) | Halves (x0.5) | Quadruples (x4) |