Problem 37 - Fundamentals of Physics

Question

An electric motor has an effective resistance of \(32.0 \Omega\) and an inductive reactance of \(45.0 \Omega\) when working under load. The voltage amplitude across the alternating source is \(420 \mathrm{~V}\). Calculate the current amplitude.

Step-by-Step Solution

Answer

The current amplitude is approximately 7.61 A.

Step 1: Identify the Given Values

We are given the resistance \( R = 32.0 \, \Omega \), the inductive reactance \( X_L = 45.0 \, \Omega \), and the voltage amplitude \( V = 420 \, \text{V} \). We are asked to find the current amplitude.

Step 2: Calculate the Impedance

The total impedance \( Z \) of the circuit can be calculated by the formula for impedance in an RL circuit: \( Z = \sqrt{R^2 + X_L^2} \). First, compute \( R^2 = (32.0)^2 = 1024 \) and \( X_L^2 = (45.0)^2 = 2025 \). Then, \( Z = \sqrt{1024 + 2025} = \sqrt{3049} \approx 55.2 \, \Omega \).

Step 3: Use Ohm's Law for AC Circuits

The current amplitude can be found using Ohm's law for AC circuits: \( I = \frac{V}{Z} \). Substitute the given values to get \( I = \frac{420}{55.2} \approx 7.61 \, \text{A} \).

Key Concepts

Impedance in RL Circuits

Impedance in RL circuits is a crucial concept when dealing with alternating current (AC) circuits that include resistors and inductors. Unlike direct current (DC) circuits where only resistance affects the flow of current, AC circuits are subject to both resistance and inductive reactance.

This impedance is represented as the total opposition that a circuit offers to the flow of current. In an RL circuit, the total impedance (denoted as \( Z \)) can be calculated using the formula:

• \( Z = \sqrt{R^2 + X_L^2} \)

This formula stems from the fact that resistance (\(R\)) and inductive reactance (\(X_L\)) are perpendicular components in a vector diagram of the circuit's impedance.

Understanding this concept is essential for calculating current in circuits with both resistive and inductive elements, especially since dependence on frequency makes AC circuit analysis more complex than DC.

Resistance

Resistance is a fundamental property in electrical circuits, defined as the opposition to the flow of direct current (DC). In any electrical device, resistance is inherent and measured in ohms (\( \Omega \)).

Within an electric motor or any other circuit, resistance is the result of collisions between the charge carriers flowing through the conductor and the lattice structure of the conductor. These collisions cause energy losses in the form of heat, contributing to the overall resistance.

In the example exercise, the motor's resistance is given as \(32.0 \ \, \Omega \). This value doesn't change with frequency and remains constant irrespective of the type of current flowing through it.

Inductive Reactance

Inductive reactance is another important aspect of AC circuits, especially those containing inductors. Inductors present a type of resistance to the changing current, which is frequency-dependent, unlike simple resistance in DC circuits.

The inductive reactance (often denoted as \( X_L \)) can be calculated using the formula:

• \( X_L = 2 \pi f L \)

where \( f \) is the frequency of the AC source and \( L \) is the inductance of the coil. The unit for inductive reactance is also ohms (\( \Omega \)), similar to resistance.

In the underlying problem, the inductive reactance is given as \(45.0 \, \Omega\), related to its role in creating lag between voltage and current phases in the circuit, further influencing the total impedance.