Problem 113 - Physics

Question

A wire has a resistance of \(21.0 \Omega\). It is melted down, and from the same volume of metal a new wire is made that is three times longer than the original wire. What is the resistance of the new wire?

Step-by-Step Solution

Answer

The new resistance is \(189.0 \Omega\).

Step 1: Understand the relationship between resistance, length, and area

Resistance \(R\) of a wire is given by \(R = \rho \frac{L}{A}\), where \(\rho\) is resistivity, \(L\) is length, and \(A\) is the cross-sectional area. When the wire is stretched, while the volume remains constant, \(L\) and \(A\) change.

Step 2: Relate original and new wire dimensions

When the length of the wire becomes three times longer (\(L' = 3L\)), and since the volume \(V = AL\) remains constant, the new cross-sectional area \(A'\) will be \(A' = \frac{A}{3}\).

Step 3: Calculate the resistance of the new wire

Using the relation \(R' = \rho \frac{L'}{A'}\) and substituting \(L' = 3L\) and \(A' = \frac{A}{3}\), we get \(R' = \rho \frac{3L}{A/3} = 9\rho \frac{L}{A} = 9R\). Since the original resistance \(R\) is \(21.0 \Omega\), the new resistance \(R'\) will be \(9 \times 21.0 \Omega = 189.0 \Omega\).

Key Concepts

Resistivity

Resistivity is a fundamental property of materials that affects how they conduct electricity. It is represented by the symbol \( \rho \) and is measured in ohm-meters.
The resistivity of a material indicates how strongly it opposes the flow of electric current.

• Materials with low resistivity, like copper, allow electricity to flow easily.
• Materials with high resistivity, like rubber, do not allow electricity to flow easily.

In the context of our wire problem, resistivity remains constant because the material is not changed even though the wire's dimensions are altered.
This means the resistance changes primarily due to changes in dimensions, not the material property itself.

Cross-sectional Area

The cross-sectional area \( A \) of a wire significantly impacts its resistance. It is the area of the face of the wire when you cut it perpendicular to its length.

• Larger cross-sectional areas mean less resistance, allowing more current to pass through.
• Smaller cross-sectional areas mean more resistance.

In the exercise, when the wire is stretched, the cross-sectional area reduces. Specifically, if the wire's length triples, then, due to volume conservation, the new cross-sectional area is one-third of the original. Understanding this relationship is crucial for determining changes in resistance.

Volume Conservation

Volume Conservation comes into play when the shape of a wire changes, such as when it is melted and reformed. The principle states that the volume must remain constant, even if length and area change.
For a wire, the volume \( V \) is given by the product of its cross-sectional area \( A \) and its length \( L \). Thus, \( V = A \times L \).
When our wire was stretched to three times its length, the area had to decrease proportionally, to keep the volume unchanged. This direct relationship ensures that changes in one dimension affect the others, maintaining the total volume.

Ohm's Law

Ohm's Law is a fundamental principle in electrical circuits, defined by the equation \( V = IR \), where \( V \) is voltage, \( I \) is current, and \( R \) is resistance.
Although Ohm's Law isn't directly used to calculate the new resistance of the wire in this exercise, it helps understand how resistance affects current flow.

• When resistance increases (as with a longer, thinner wire), for the same voltage, less current flows.
• Conversely, decreasing resistance allows more current to flow for the same voltage.

Thus, understanding resistance, and its dependence on dimensions and material (resistivity), is critical for applying Ohm's Law successfully in practical scenarios.