Problem 67 - Physics for Scientists and Engineers with Modern Physics

Question

Suppose you want to run some apparatus that is \(65 \mathrm{~m}\) from an electric outlet. Each of the wires connecting your apparatus to the \(120-\mathrm{V}\) source has a resistance per unit length of \(0.0065 \Omega / \mathrm{m} .\) If your apparatus draws \(3.0 \mathrm{~A},\) what will be the voltage drop across the connecting wires and what voltage will be applied to your apparatus?

Step-by-Step Solution

Answer

The voltage drop across the wires is approximately 2.535 V, and the voltage applied to the apparatus is about 117.465 V.

Step 1: Calculate Total Resistance

Start with finding the total resistance of the wires. The wire resistance per unit length is given as \(0.0065 \, \Omega/\mathrm{m}\). The total length of the wire is \(2 \times 65 \, \mathrm{m}\) because the current travels to the apparatus and back. Thus, the total resistance \(R\) is given by \(R = 2 \times 65 \, \mathrm{m} \times 0.0065 \, \Omega/\mathrm{m}\). Calculate this value.

Step 2: Apply Ohm's Law to Find Voltage Drop

Use Ohm's Law, \(V = IR\), to find the voltage drop \(V_d\) across the wires. The current \(I\) is given as \(3.0 \, \mathrm{A}\) and we just calculated the total resistance \(R\). Substitute these values into the equation to find \(V_d\).

Step 3: Calculate Voltage Applied to the Apparatus

Subtract the voltage drop from the source voltage to find out the voltage actually applied to the apparatus. The source voltage is given as \(120 \, \mathrm{V}\). Therefore, the voltage applied to the apparatus \(V_a\) is \(120 \, \mathrm{V} - V_d\). Substitute the value of \(V_d\) to find \(V_a\).

Key Concepts

Electric Resistance

Electric resistance is a fundamental concept that measures how difficult it is for electric current to flow through a conductor. If we compare it to water flowing through a pipe, resistance is like narrowing the pipe, making it harder for water to flow. A high electric resistance means less current can pass through at a given voltage. It's represented in ohms (Ω).
When calculating total electric resistance for a wire, you need to consider the resistance per unit length and the total length of the wire. In the exercise example, each wire has a resistance per unit length of 0.0065 Ω/m. To find the total resistance for a set of wires that go to the apparatus and back, you multiply the resistance per unit length by the total distance, which was twice the length of 65 m.

• Resistance Formula: Total resistance (R) = Resistance per unit length × Total length
• Wire Resistance: For the example, R = 0.0065 Ω/m × 130 m

This formula gives you the resistance that the electricity encounters as it flows through the wires.

Voltage Drop

The voltage drop is the reduction in voltage as electric current flows through a resistor. It measures how much electric potential energy is "lost" to the resistance of the wires when supplying power to another device or apparatus. This concept is essential because it determines how much voltage will actually reach the end device.
According to Ohm's Law, the voltage drop across a resistor can be found using the formula \( V = IR \), where \( I \) is the current in amperes and \( R \) is the resistance in ohms.

• Ohm's Law for Voltage Drop: Voltage drop (\( V_d \)) = Current (I) × Resistance (R)
• Calculation in Context: In our problem, the current drawn by the apparatus is 3.0 A, and we've calculated the total wire resistance. So, \( V_d = 3.0 A \times R \)

This calculation tells you how much of the source voltage is "used up" by the resistance in the wires before it reaches the apparatus.

Current Electricity

Current electricity refers to the movement of electric charge through a conductor, such as a wire. It is the type of electricity that powers household appliances when you plug them into an outlet. The flow of electric charge is measured in amperes (A), and it is essential for carrying energy from one place to another through the circuit.
The flow of current can be impacted by several factors, including the resistance of the wires. In our exercise, the apparatus draws a current of 3.0 A. This continuous flow is necessary to keep the apparatus functioning at its optimal performance.

• Current Flow: For the apparatus, a consistent flow of 3.0 A is necessary.
• Ampere: The unit of electric current, denoting the flow of charge.

Understanding current electricity helps us know how power is transferred through circuits, enabling the operation of various electrical devices.