Problem 39 - University Physics with Modern Physics

Question

A coil has a resistance of 48.0\(\Omega .\) At a frequency of 80.0 \(\mathrm{Hz}\) the voltage across the coil leads the current in it by \(52.3^{\circ} .\) Determine the inductance of the coil.

Step-by-Step Solution

Answer

The inductance of the coil is approximately 0.1227 H.

Step 1: Understanding the Problem

The problem involves determining the inductance of a coil. Given parameters include the resistance \(R = 48.0 \, \Omega\), frequency \(f = 80.0 \, \text{Hz}\), and the phase angle \(\phi = 52.3^\circ\). The phase angle \(\phi\) between the voltage and the current is due to the coil's inductance.

Step 2: Use Phase Angle to Find Inductive Reactance

The phase angle \(\phi\) in an RL circuit is related to the resistance \(R\) and inductive reactance \(X_L\) by the formula: \[ \tan(\phi) = \frac{X_L}{R} \]Here, solve for \(X_L\):\[ X_L = R \cdot \tan(\phi) \]Calculate \(\tan(52.3^\circ)\), and then determine \(X_L\).

Step 3: Calculate Inductive Reactance

Calculate \(\tan(52.3^\circ)\), which is approximately 1.2846:\[ X_L = 48.0 \, \Omega \times 1.2846 \approx 61.66 \, \Omega \]

Step 4: Relate Inductive Reactance to Inductance

The inductive reactance \(X_L\) is related to the inductance \(L\) and frequency \(f\) by the formula: \[ X_L = 2 \pi f L \]Rearrange this equation to solve for \(L\):\[ L = \frac{X_L}{2 \pi f} \]

Step 5: Compute the Inductance

Substitute the known values into the formula:\[ L = \frac{61.66 \, \Omega}{2 \pi \times 80.0 \, \text{Hz}} \]Calculate to find \(L\).

Step 6: Final Calculation

Perform the calculation:\[ L \approx \frac{61.66}{502.65} \approx 0.1227 \, \text{H} \]Thus, the inductance of the coil is approximately \(0.1227 \, \text{H}\).

Key Concepts

Phase Angle

In an RL circuit, the phase angle is a measure of the time difference between the peak voltage and the peak current. It is usually expressed in degrees. The voltage across the coil typically leads the current; this lead is represented by the phase angle. In this problem, a phase angle of 52.3° indicates that the voltage reaches its peak before the current does by this exact angle.
The phase angle (\(\phi\)) is crucial because it helps in determining the relationship between resistance and reactance in the circuit. It is measured using trigonometric functions related to the components of the circuit.

• Phase angle is determined through the formula: \(\tan(\phi) = \frac{X_L}{R}\), where \(X_L\) is the inductive reactance.
• This phase lead signifies energy storage temporarily by the inductor.

The phase angle's understanding is essential for calculating other electrical quantities in AC circuits, especially when you need to explore the inductive properties of coils.

Inductive Reactance

Inductive Reactance (\(X_L\)) is a measure of the opposition offered by the inductor to the change in current flowing through it. This happens in AC circuits where the current changes direction periodically.
Inductive reactance depends on both the frequency of the AC signal and the inductance of the coil. Reactance increases with higher frequency or greater inductance.

• It is given by the formula: \(X_L = 2 \pi f L\).
• From the phase angle, \(X_L\) can be calculated using \(X_L = R \times \tan(\phi)\).
• Reactance is measured in ohms (\(\Omega\)).

Understanding how inductive reactance impacts the circuit helps in managing the phase differences and determining energy dissipation through resistive components.

RL Circuit

An RL circuit consists of a resistor (R) and an inductor (L) connected in a specific configuration. These two elements are typically in series or parallel, where they handle AC currents.
Such circuits are critical in filtering signals and setting phase shifts in communication systems. When alternating current flows through the circuit, it experiences different resistive and reactive behaviors.

• The inductor resists changes in current and causes a phase shift (voltage leads current).
• The overall impedance (total opposition to AC flow) combines both resistance and inductive reactance.
• RL circuits are integral parts of power supply units, controlling frequency responses and maintaining stability.

In exploring RL circuits, students gain insights into how physical components affect the overall behavior of electron flow. Understanding the balance in RL circuits progresses electronic tuning and signal processing.

Resistance

Resistance (\(R\)) is a fundamental concept in circuits, representing the opposition to the flow of electric current. It is measured in ohms (\(\Omega\)) and is determined by the material, length, and cross-sectional area of the conductor.
In the context of the RL circuit, resistance plays a part in determining the phase angle and is used along with the inductive reactance to calculate other properties of the circuit.

• It affects the amount of current that can pass at any given voltage.
• In an RL circuit, resistance couched with reactance defines the circuit's impedance.
• Managing resistance involves ensuring that components do not overheat while maintaining operational effectiveness.

Understanding resistance's role helps in designing circuits that optimize energy use and comply with practical limits on electrical dissipation.

Frequency

Frequency (\(f\)) is the number of times a periodic function repeats a complete cycle per second, measured in hertz (Hz).
In electrical circuits, frequency determines how often the AC current changes direction per second, affecting elements like inductance and reactance.

• Frequency plays a direct role in calculating inductive reactance through \(X_L = 2\pi f L\).
• Higher frequencies increase reactance, impacting the current flow and phase angles in the RL circuit.
• It is vital in designing circuits meant for specific signal types and applications.

Thorough understanding of frequency and its implications on circuit behavior can enable effective control of electronic systems in various domains such as telecommunications and audio engineering.