Chapter 25: Problem 14
A wire 6.50 m long with diameter of 2.05 mm has a resistance of 0.0290
Short Answer
Expert verified
The wire is most likely made of copper.
Step by step solution
01
Convert Diameter to Radius
To find the cross-sectional area, we first need to convert the diameter of the wire to a radius. The conversion is straightforward: radius is half the diameter. Thus, the radius is given by mm, which is mm or meters.
02
Calculate Cross-Sectional Area
Using the radius, we calculate the cross-sectional area of the wire. The formula for the area of a circle is . Substituting the value of , . Calculating this gives square meters.
03
Use Resistance Formula
We use the resistance formula to solve for the resistivity . We know that , meters, and square meters. Rearranging for , we get .
04
Calculate Resistivity
Substitute the known values into the rearranged formula to calculate the resistivity: . Evaluating this expression gives .
05
Identify the Material
The calculated resistivity is close to the known resistivity of copper, which is approximately . Therefore, the material of the wire is most likely copper.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross-Sectional Area
To comprehend how to determine the cross-sectional area of a wire, let’s start by understanding its basic geometry. The cross-section of a wire, much like a circle, requires us to find its radius to calculate the area.
Given the diameter of the wire is 2.05 mm, the radius is simply half of that. So, you divide 2.05 by 2 to get 1.025 mm, or 1.025 x 10⁻³ meters to keep the units consistent with standard calculations.
To find the area, use the formula for the area of a circle, which is . Substitute the radius into the formula: . This calculation is crucial as it will impact the value of resistance found in subsequent steps.
Given the diameter of the wire is 2.05 mm, the radius is simply half of that. So, you divide 2.05 by 2 to get 1.025 mm, or 1.025 x 10⁻³ meters to keep the units consistent with standard calculations.
To find the area, use the formula for the area of a circle, which is
Resistance Formula
The resistance of a wire is determined by several factors, including the material it is made from, its length, and its cross-sectional area. In physics, these relationships are expressed in the resistance formula:
, where: . By substituting the computed area and the given resistance and length of the wire, you can find the unknown resistivity, which leads us to the final goal of identifying the material.
is the resistance in ohms ( ). is the resistivity of the material in ohm meters. is the length of the wire in meters. is the cross-sectional area in square meters.
Material Identification
After calculating the resistivity of the wire using the measures given and applying the resistance formula, we end up with a value of approximately .
With this resistivity value in hand, we can compare it to known resistivities of various materials. This process is akin to comparing a fingerprint to a database to identify a person.
Copper, commonly used in electrical applications due to its excellent conductive properties, has a known resistivity of roughly . Given the close match, we deduce that the wire is most likely composed of copper, confirming its use in many practical situations.
With this resistivity value in hand, we can compare it to known resistivities of various materials. This process is akin to comparing a fingerprint to a database to identify a person.
Copper, commonly used in electrical applications due to its excellent conductive properties, has a known resistivity of roughly
Wire Properties
Understanding the properties of a wire involves examining several key aspects that affect its function and application. The primary properties include:
- Length: Longer wires have higher resistance because electrons encounter more obstacles over longer distances. In our case, the wire is 6.50 meters long.
- Diameter: The wire’s diameter directly influences its cross-sectional area, which in turn affects resistance; a larger area reduces resistance.
- Material: Different materials conduct electricity differently; metals like copper have low resistivity, making them excellent conductors.