Problem 1 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

What will be the resistance of a copper wire \(0.08\) in. in diameter and \(100 \mathrm{ft}\) long if its resistivity is \(1.7 \mu \Omega \cdot \mathrm{cm}\) ?

Step-by-Step Solution

Answer

Resistance is approximately 0.1607 ohms.

Step 1: Convert Units

First, convert the diameter of the copper wire from inches to centimeters. Given the diameter is \(0.08\) inches, we know \(1\) inch is \(2.54\) cm, so:\[ 0.08 ext{ in} = 0.08 imes 2.54 ext{ cm} = 0.2032 ext{ cm} \]

Step 2: Calculate the Radius

Next, calculate the radius of the wire. Given the diameter is \(0.2032\) cm, the radius \(r\) is half of that:\[ r = \frac{0.2032}{2} \text{ cm} = 0.1016 ext{ cm} \]

Step 3: Compute the Cross-sectional Area

Calculate the cross-sectional area \(A\) of the wire using the formula for the area of a circle, \(A = \pi r^2\):\[ A = \pi \times (0.1016)^2 \text{ cm}^2 \approx 0.0324 \text{ cm}^2 \]

Step 4: Convert Length to Centimeters

Convert the length of the wire from feet to centimeters. Given the length is \(100\) feet and \(1\) foot equals \(30.48\) cm, we have:\[ 100 ext{ ft} = 100 \times 30.48 ext{ cm} = 3048 ext{ cm} \]

Step 5: Apply Ohm's Law for Resistance

Calculate the resistance \(R\) using the formula for resistance in terms of resistivity \(\rho\), length \(L\), and cross-sectional area \(A\):\[ R = \frac{\rho \times L}{A} \]Substitute the known values:\[ R = \frac{1.7 \times 10^{-6} \Omega \cdot \text{cm} \times 3048 \text{ cm}}{0.0324 \text{ cm}^2} \approx 0.1607 \Omega \]

Step 6: Result

The resistance of the copper wire is approximately \(0.1607\) ohms.

Key Concepts

Unit Conversion in Physics

In physics, unit conversion is essential to ensure that all measurements are consistent and comparable. It's like speaking the same language in mathematics. When calculating electrical resistance or other physics problems, always convert your units to the standard system you're working with. For instance, in our problem, we converted inches to centimeters and feet to centimeters.
This is because the resistivity is given in micro-Ohm centimeters.

• To convert inches to centimeters, multiply by 2.54, since 1 inch equals 2.54 cm.
• To convert feet to centimeters, multiply by 30.48, because 1 foot equals 30.48 cm.

This ensures our calculation is consistent and accurate. Always double-check your conversions to avoid errors!

Ohm's Law

Ohm's Law is a fundamental principle in physics that relates voltage, current, and resistance in an electrical circuit. It is expressed as:\[ V = IR \]where:

• \( V \) is the voltage across the conductor in volts (V),
• \( I \) is the current flowing through the conductor in amperes (A),
• \( R \) is the resistance of the conductor in ohms (Ω).

In our exercise, we focus on calculating resistance using resistivity, which is a variation of Ohm's Law. The formula for resistance in terms of resistivity is:\[ R = \frac{\rho \cdot L}{A} \]Understanding this relationship helps us predict how changing certain factors like material or dimensions will affect the resistance of a wire.

Resistivity

Resistivity is a property of a material that describes how strongly it opposes the flow of electric current. It is denoted by \( \rho \) and is specific to each material. For instance, copper has a low resistivity of \( 1.7 \times 10^{-6} \Omega \cdot \text{cm} \), making it a good conductor.

• A low resistivity indicates that a material can conduct electric current easily, like copper.
• A high resistivity means that a material resists the flow of electric current, like rubber.

When calculating the resistance of a wire, the resistivity is a critical factor because it determines how much resistance the material itself adds to the electrical flow. Understanding resistivity helps in selecting the right materials for electrical projects.

Cross-sectional Area Calculation

The cross-sectional area of a wire is crucial when calculating its electrical resistance. This is because resistance is inversely proportional to the cross-sectional area. As the area increases, resistance decreases and vice versa.

• The area is calculated using the formula for the area of a circle, since wires are generally cylindrical: \( A = \pi r^2 \).
• The radius \( r \) is half of the diameter of the wire.

In our example, the diameter was converted from inches to centimeters, and then halved to find the radius. From the radius, we calculated the area in square centimeters. Accurately calculating the cross-sectional area is key to determining the wire's resistance accurately, as it directly influences the outcome of our resistance calculation.