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 of the steel rail is approximately 0.54 Ω.

Step 1: Understand the Formula

The resistance (R) of a material is calculated using the formula: \[ R = \frac{\rho \cdot L}{A} \]where \( \rho \) is the resistivity of the material, \( L \) is the length of the material, and \( A \) is the cross-sectional area.

Step 2: Convert Units

Before substituting into the formula, ensure that all units are consistent. The length provided is in kilometers, so we need to convert it to meters (1 km = 1000 m):\[ 10.0 \, \text{km} = 10,000 \, \text{m} \]The cross-sectional area is given in \( \text{cm}^2 \), convert it to \( \text{m}^2 \):\[ 56.0 \, \text{cm}^2 = 0.0056 \, \text{m}^2 \]

Step 3: Substitute Values into the Resistance Formula

Now that all measurements are in consistent units, substitute the known values into the resistance formula:\[ R = \frac{3.00 \times 10^{-7} \, \Omega \cdot \text{m} \times 10,000 \, \text{m}}{0.0056 \, \text{m}^2} \]

Step 4: Perform the Calculation

Calculate the resistance using the formula:\[ R = \frac{3.00 \times 10^{-7} \, \Omega \cdot \text{m} \times 10,000 \, \text{m}}{0.0056 \, \text{m}^2} \]\[ R = \frac{3.00 \times 10^{-3} \, \Omega \cdot \text{m}}{0.0056 \, \text{m}^2} \]\[ R = 0.5357 \, \Omega \]

Key Concepts

Resistivity

Resistivity is a fundamental property of materials that describes how strongly they resist electric current. It is represented by the symbol \( \rho \) and measured in ohm-meters (\( \Omega \cdot \text{m} \)).

Each material has a unique resistivity which affects how easily current can flow through it.

• Materials with low resistivity, like copper or silver, allow current to pass through easily and are called conductors.
• Materials with high resistivity, like rubber or glass, do not allow current to pass easily and act as insulators.

In practical applications, understanding resistivity helps in choosing the right material for electrical wiring and components. In our exercise, we are dealing with steel which has a resistivity of \(3.00 \times 10^{-7} \Omega \cdot \text{m} \). This value is essential for calculating the resistance of the steel rail.

Cross-Sectional Area

Cross-sectional area refers to the area of the slice of an object and is crucial in analyzing how easily current can pass through a material. In our exercise, the cross-sectional area is given as \( 56.0 \, \text{cm}^2 \).

The formula for resistance \( R = \frac{\rho \cdot L}{A} \) reveals that the larger the cross-sectional area, the lower the resistance, since more electrons have the space to flow freely.

• To calculate resistance accurately, the cross-sectional area must be in the same units as the other measurements, typically in square meters (\( \text{m}^2 \)).
• Converting \( 56.0 \, \text{cm}^2 \) to \( \text{m}^2 \) is straightforward: \( 56.0 \, \text{cm}^2 = 0.0056 \, \text{m}^2 \).

The reduction in units makes it compatible with the resistivity units, ensuring accurate resistance calculations.

Unit Conversion

Unit conversion is pivotal in ensuring all components of a formula are in harmony. In calculations involving physical dimensions, using consistent units is crucial.

For the exercise, the length of the steel rail is initially given in kilometers. Since resistivity is measured in meters, we need to convert this length.

• 1 kilometer is equal to 1,000 meters. Therefore, \( 10.0 \, \text{km} \) is converted to \( 10,000 \, \text{m} \) for uniformity.
• The cross-sectional area goes from \( \text{cm}^2 \) to \( \text{m}^2 \) to match the international SI units used in the resistivity formula.

Proper unit conversion simplifies the process of plugging values into formulas and obtaining precise results, leading to accurate predictions and analyses in physics and engineering.