Problem 14 - University Physics with Modern Physics

Question

A wire 6.50 \(\mathrm{m}\) long with diameter of 2.05 \(\mathrm{mm}\) has a resistance of 0.0290\(\Omega .\) What material is the wire most likely made of?

Step-by-Step Solution

Answer

The wire is most likely made of copper.

Step 1: Convert Units

First, convert the diameter from millimeters to meters. We have a diameter of 2.05 mm, which is 0.00205 m when converted. The length of the wire is already in meters, so no conversion is needed for the length.

Step 2: Calculate the Cross-sectional Area

The cross-sectional area of the wire is circular, so we use the area formula for a circle: \[A = \pi \left(\frac{d}{2}\right)^2 = \pi \left(\frac{0.00205}{2}\right)^2 = \pi (0.001025)^2 \approx 3.30 \times 10^{-6} \mathrm{m}^2.\] This is the cross-sectional area of the wire.

Step 3: Use Resistance Formula to Find Resistivity

The formula for resistance is given by:\[R = \rho \frac{L}{A}\]where \( R = 0.0290 \Omega \), \( L = 6.50 \mathrm{m} \), and \( A = 3.30 \times 10^{-6} \mathrm{m}^2 \). We need to solve for \( \rho \) (resistivity).Rearrange the formula to solve for \( \rho \):\[\rho = R \frac{A}{L} = 0.0290 \times \frac{3.30 \times 10^{-6}}{6.50} \approx 1.47 \times 10^{-8} \Omega \cdot \mathrm{m}.\] This is the resistivity of the material the wire is made from.

Step 4: Compare Resistivity to Known Values

Compare the calculated resistivity to known resistivity values for various materials. Common materials and their resistivities include:- Copper: approx. \(1.68 \times 10^{-8} \Omega \cdot \mathrm{m}\)- Aluminum: approx. \(2.82 \times 10^{-8} \Omega \cdot \mathrm{m}\)- Gold: approx. \(2.44 \times 10^{-8} \Omega \cdot \mathrm{m}\)The resistivity of \(1.47 \times 10^{-8} \Omega \cdot \mathrm{m}\) is closest to that of copper, suggesting the wire is likely made of copper.

Key Concepts

Understanding Resistance in Wires

Resistance is a measure of how much a material opposes the flow of electric current. It essentially tells us how hard it is for the electrical charge to pass through a conductor, like a wire. The resistance of a wire is influenced by several factors:

• The length of the wire: Longer wires have more resistance.
• The cross-sectional area: Thinner wires have more resistance.
• The material of the wire: Different materials have different intrinsic resistances.

Resistance is often represented using the symbol \( R \) and is measured in Ohms \( \Omega \). The formula to calculate resistance is \( R = \rho \frac{L}{A} \),where \( \rho \) is the resistivity of the material, \( L \) is the length of the wire, and \( A \) is the cross-sectional area.

Exploring Cross-sectional Area

The cross-sectional area of a wire is crucial in determining its resistance. The formula used to calculate the area depends on the shape. For a wire, which is circular in cross-section, the area is calculated using the formula for the area of a circle:\(A = \pi \left(\frac{d}{2}\right)^2 \)where \( d \) is the diameter of the wire. The area directly affects resistance: a larger area means lower resistance because more space is available for the current to flow through.To visualize, imagine water flowing through a pipe. A larger diameter pipe allows more water to flow easily, just like a wire with a larger cross-sectional area allows more electric current. This concept is fundamental in wiring and designing circuits to ensure efficiency and safety.

Understanding Unit Conversion

Unit conversion is essential when working with measurements to ensure all units are consistent, making calculations accurate and reliable. In the exercise, the diameter of the wire was given in millimeters and had to be converted to meters to match the unit of length (meters) used throughout the other calculations. Here's a simple guide for unit conversion:

• Identify the units you need to convert from and to. For example, millimeters to meters.
• Use the conversion factor. Here it's 1 meter = 1000 millimeters.
• Calculate by dividing the millimeter measurement by 1000.

By ensuring all units are compatible, you avoid costly errors in calculations, which could lead to incorrect results. Being meticulous with unit conversion is a skill that is invaluable in science and engineering.

Materials Identification Based on Resistivity

Identifying materials based on their resistivity is like detective work in physics. Every material has a characteristic resistivity, a property indicating how strongly it opposes the flow of electric current. By calculating the resistivity from given measurements, it is possible to deduce the material of a wire.Here's how you can identify a material:

• Measure the resistance, length, and cross-sectional area of the wire.
• Use the resistance formula \( R = \rho \frac{L}{A} \) rearranged to find resistivity: \( \rho = R \frac{A}{L} \).
• Compare the calculated resistivity with known resistivity values of materials like copper, aluminum, and others.

In the exercise, the computed resistivity was close to that of copper, suggesting the wire is likely copper. This process is important in industries and fields where specific materials must be used for their electrical properties. Understanding how to identify materials through resistivity helps ensure the right material is chosen for the job, optimizing performance and safety.