Problem 10 - Applied Physics

Question

Find the length of a copper wire with resistance \(0.0262 \Omega / \mathrm{ft}\) and total resistance \(5.62 \Omega\)

Step-by-Step Solution

Answer

The length of the copper wire is approximately 214.50 feet.

Step 1: Understanding the Problem

We are given the resistance per foot of the copper wire, which is \(0.0262 \Omega/\text{ft}\), and the total resistance, which is \(5.62 \Omega\). The goal is to find the length of the wire in feet.

Step 2: Setting Up the Equation

To find the length, we use the formula for resistance: \( R = r \times L \), where \(R\) is the total resistance, \(r\) is the resistance per unit length, and \(L\) is the length of the wire. We need to solve for \(L\).

Step 3: Rearranging the Formula

Rearrange the formula to solve for \(L\): \( L = \frac{R}{r} \). Now, \(R\) is the total resistance \(5.62 \Omega\) and \(r\) is the resistance per foot \(0.0262 \Omega/\text{ft}\).

Step 4: Calculate the Length

Substitute the known values into the equation: \[ L = \frac{5.62}{0.0262} \text{ ft} \]. Perform the division to find \(L\).

Step 5: Compute Final Result

Calculate the length to find \[ L \approx 214.50 \text{ ft} \].

Key Concepts

Copper Wire

Copper wire is a widely used metal in electrical circuits due to its excellent conductivity. Its ability to conduct electricity efficiently makes it an ideal choice for many wiring applications. Copper has a low resistance, meaning it allows electric current to pass through it easily. This property is crucial in electrical circuits, as it reduces energy loss and ensures effective power transmission.

• Copper is not only conductive but also durable and flexible, allowing it to be used in various shapes and forms.
• Its resistance to heat and corrosion further adds to its desirability in electrical installations.

Understanding copper wire's properties helps in analyzing how resistance affects electrical circuits, particularly when exploring the relationship between wire length and resistance.

Resistance Per Unit Length

Resistance per unit length is a measure of a material's inherent resistance that applies over a specific length of wire. For copper wires, this is expressed as resistance per foot (or meter), indicating how much resistance a single unit of length contributes to the total resistance of the wire.

• The resistance per unit length depends on the material's properties and cross-sectional area.
• In our exercise, the copper wire has a resistance of 0.0262 Ω/ft, which is considered low, in alignment with copper's known conductivity.

This value is crucial for determining the overall resistance of a wire, especially if the length of the wire isn't specified and needs to be calculated.

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering, relating voltage, current, and resistance in circuits. The law is usually expressed as \( V = IR \), where \(V\) is voltage, \(I\) is current, and \(R\) is resistance.
While this specific exercise doesn't directly use Ohm's Law, understanding it helps with grasping the concept of resistance in circuits. Resistance is a key component that affects how current flows, and changes in resistance alter the current through any conductor. In situations where we know either the voltage or current, Ohm's Law can be used to find the missing quantity when resistance is known, further reinforcing its importance in electrical calculations.

Calculating Wire Length

To find the length of a wire given its resistance, we use the formula: \( R = r \times L \), where \( R \) represents the total resistance, \( r \) is resistance per unit length, and \( L \) is the length of the wire. The formula is rearranged to calculate the wire's length: \( L = \frac{R}{r} \).

• This formula allows us to solve for any one variable when the other two are known.
• In our exercise, the given values were a total resistance of 5.62 Ω and a resistance per foot of 0.0262 Ω/ft.

Plugging these values into the formula provides us with the wire length, which is approximately 214.50 ft. This calculation is critical in real-world applications such as constructing electrical systems, ensuring that the right length of wire is used to achieve desired resistance levels.