Problem 72 - Fundamentals of Physics

Question

A steel trolley-car rail has a cross-sectional area of \(56.0 \mathrm{~cm}^{2}\). What is the resistance of \(10.0 \mathrm{~km}\) of rail? The resistivity of the steel is \(3.00 \times 10^{-7} \Omega \cdot \mathrm{m}\)

Step-by-Step Solution

Answer

The resistance is approximately 0.536 ohms.

Step 1: Understanding the Problem

We need to calculate the electrical resistance of a steel rail. We have the length, cross-sectional area, and resistivity of the material, which are all needed to use the resistance formula.

Step 2: Recall the Formula for Resistance

The formula to calculate the resistance (R) of a conductor is given by: \( R = \frac{\rho \cdot L}{A} \), where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area of the conductor.

Step 3: Convert Units

Convert the given dimensions to appropriate SI units. The cross-sectional area is already in square centimeters, but it should be in square meters for this calculation: \( 56.0 \mathrm{~cm}^2 = 5.60 \times 10^{-3}\mathrm{~m}^2 \). The length needs to be converted from kilometers to meters: \( 10.0 \mathrm{~km} = 10,000 \mathrm{~m} \).

Step 4: Plug Values into the Formula

Insert the given values into the resistance formula: \( R = \frac{3.00 \times 10^{-7} \Omega \,\cdot\, \mathrm{m} \times 10,000 \mathrm{~m}}{5.60 \times 10^{-3} \mathrm{~m}^2} \).

Step 5: Perform the Calculation

Calculate the resistance using the plugged-in values: \( R = \frac{(3.00 \times 10^{-7}) \times 10,000}{5.60 \times 10^{-3}} \). This simplifies to \( R = \frac{3.00 \times 10^{-3}}{5.60 \times 10^{-3}} \) ohms.

Step 6: Compute the Result

Simplify the fraction to find the resistance: \( R = \frac{3.00}{5.60} \) ohms, which equals approximately \( 0.536 \) ohms.

Key Concepts

Resistivity

Resistivity is a fundamental property of materials that explains how much they oppose the flow of electric current. It's an inherent characteristic of the material itself. Unlike resistance, which depends on the shape and size of the object, resistivity is a constant for a given material at a specific temperature.
For instance, in the problem of the steel rail, the resistivity is given as \(3.00 \times 10^{-7} \, \Omega \cdot \mathrm{m}\). This means that a steel wire of this specific resistivity will offer this certain magnitude of resistance to the electric flow across one meter of its length if it has a unit cross-sectional area. This fixed value helps us characterize different materials and decide which one to use for specific applications.

• Low resistivity: Indicates the material is a good conductor (e.g., copper).
• High resistivity: Suggests the material is a good insulator (e.g., rubber).

The equation \( R = \frac{\rho \cdot L}{A} \) features resistivity \( \rho \), explaining how it plays a critical role in determining overall resistance.

Cross-sectional Area

The cross-sectional area of a conductor, such as the steel rail in our problem, is crucial in determining its electrical resistance. The larger the area, the less resistance the object will have because a larger area permits more electrons to flow through simultaneously.
In the case of the steel rail, the cross-sectional area is \(56.0 \, \mathrm{cm}^{2}\), which we converted to \(5.60 \times 10^{-3} \, \mathrm{m}^{2}\) for use in our resistance calculation.
Cross-sectional area is essentially the slice of the conductor perpendicular to the direction of the current. When it comes to electrical conductors:

• The area acts as a bottleneck for current flow; larger areas mean less bottlenecking, hence less resistance.
• It's critical in wiring systems to manage current flow and heat dissipation.

Understanding how the cross-sectional area interacts with other factors like resistivity and length helps us make better engineering decisions about material and geometry.

Resistance Formula

The resistance formula \( R = \frac{\rho \cdot L}{A} \) is a straightforward yet powerful equation used to calculate the resistance of a conductor. Here's how each element works in determining the resistance:
- \( R \) represents Resistance, measured in ohms (\( \Omega \)).- \( \rho \) is Resistivity, a material property.- \( L \) is the Length of the conductor, measured in meters.- \( A \) refers to Cross-sectional Area, in square meters.
The formula tells us how these parameters interplay:

• Greater length (\(L\)) increases resistance, as electrons have more obstacle-ridden paths to follow.
• Smaller cross-sectional area (\(A\)) correlates with higher resistance, due to limited paths for current flow.
• High resistivity (\(\rho\)) materials inherently resist current more than low resistivity materials.

In our steel rail problem, we applied this formula by plugging in the resistivity, length, and area to find the total resistance, demonstrating how practical and useful this formula is in real-world applications.