Problem 80 - Fundamentals of Physics

Question

A series \(R L C\) circuit is driven by an alternating source at a frequency of \(400 \mathrm{~Hz}\) and an emf amplitude of \(90.0 \mathrm{~V}\). The resistance is \(20.0 \Omega,\) the capacitance is \(12.1 \mu \mathrm{F},\) and the inductance is \(24.2 \mathrm{mH}\). What is the rms potential difference across (a) the resistor, (b) the capacitor, and (c) the inductor? (d) What is the average rate at which energy is dissipated?

Step-by-Step Solution

Answer

(a) 63.64 V, (b) 368.96 V, (c) 60.81 V, (d) 202.02 W.

Step 1: Determine the RMS Current

To find the rms current, we first calculate the angular frequency \( \omega \) using \( \omega = 2\pi f \) where \( f \) is the frequency. Then, calculate the impedance \( Z \) of the circuit using \( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where \( X_L = \omega L \) is the inductive reactance and \( X_C = \frac{1}{\omega C} \) is the capacitive reactance. Finally, use Ohm's law for AC circuits: \( I_{rms} = \frac{V_{rms}}{Z} \). The rms voltage is \( V_{rms} = \frac{V_{max}}{\sqrt{2}} \).

Step 2: RMS Potential Difference Across the Resistor

The rms potential difference across the resistor \( V_{R,rms} \) is calculated using Ohm's law: \( V_{R,rms} = I_{rms} \times R \).

Step 3: RMS Potential Difference Across the Capacitor

Calculate \( V_{C,rms} \) using the formula \( V_{C,rms} = I_{rms} \times X_C \), where \( X_C = \frac{1}{\omega C} \). Substitute the previously calculated \( I_{rms} \) and \( X_C \).

Step 4: RMS Potential Difference Across the Inductor

Compute \( V_{L,rms} \) using \( V_{L,rms} = I_{rms} \times X_L \), where \( X_L = \omega L \). Use the previously calculated \( I_{rms} \) and \( X_L \).

Step 5: Average Rate of Energy Dissipation

The average power \( P \) dissipated in the resistor is given by \( P = I_{rms}^2 \times R \). This represents the average rate at which energy is dissipated in the resistor.

Key Concepts

Understanding Impedance in RLC Circuits

Impedance is a crucial concept when dealing with RLC circuits, as it determines how much the circuit resists the flow of alternating current (AC). Unlike simple resistance, impedance combines resistance with reactance, which accounts for the effects of inductors and capacitors.

In an RLC circuit, impedance (denoted as \( Z \)) is calculated using the formula:\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]where:

• \( R \) is the resistance of the circuit,
• \( X_L \) is the inductive reactance, given by \( \omega L \),
• and \( X_C \) is the capacitive reactance, given by \( \frac{1}{\omega C} \).

This formula ensures we comprehend how both inductors and capacitors influence the circuit's behavior beyond mere resistance. Impedance affects how much current flows at any given voltage and frequency. In practical terms, it helps in designing and tuning circuits for specific applications.

Reactance Explained

Reactance is an essential aspect of AC circuits, primarily focusing on how inductors and capacitors behave. It is measured in ohms and reflects how these components oppose changes in current.

• Inductive reactance \( X_L \) increases with frequency, calculated with \( X_L = \omega L \), where \( L \) is inductance.
• Conversely, capacitive reactance \( X_C \) decreases with frequency, \( X_C = \frac{1}{\omega C} \), where \( C \) is capacitance.

These equations highlight how inductors and capacitors react differently as the frequency changes. Higher frequencies cause inductors to block more current, while capacitors allow more through. Understanding these concepts is key when analyzing AC circuit behaviors, especially when tuning circuits to specific signals or frequencies.

Dive into AC Circuits

Alternating current (AC) circuits, such as those involving RLC components, are different from direct current (DC) circuits because the current and voltage periodically reverse direction. These circuits are the backbone of most household electrical systems. AC circuit analysis requires considering not just resistance, but also how capacitive and inductive components affect current flow. Key points include:

• The use of complex numbers to handle phase differences between voltage and current.
• The importance of frequency, as it impacts reactance and overall circuit behavior.

By focusing on impedance and reactance, we can effectively determine how much current an AC circuit will deliver for a given supply voltage. This understanding is valuable for ensuring devices operate efficiently and safely.

Energy Dissipation in AC Circuits

Energy dissipation in RLC circuits primarily involves the resistor, as it converts electrical energy into heat. This is measured by calculating the average power consumed. To determine the power dissipated, we use the formula:\[ P = I_{rms}^2 \times R \]where \( I_{rms} \) is the root mean square current. This formula emphasizes how energy dissipation in an AC circuit is due to resistance, rather than reactance. Key considerations include:

• Understanding that the **resistor** is the sole component where energy is dissipated as heat.
• The role of **power factor**, which is the cosine of the phase angle between current and voltage, influencing how efficiently energy is used.

Appreciating these concepts helps in designing circuits that ensure minimum energy loss and efficient operation. It is especially important in industries where energy efficiency is critical.