Problem 17 - University Physics with Modern Physics

Question

In an \(L-R-C\) series circuit, the rms voltage across the resistor is \(30.0 \mathrm{V},\) across the capacitor it is \(90.0 \mathrm{V},\) and across the inductor it is 50.0 \(\mathrm{V} .\) What is the rms voltage of the source?

Step-by-Step Solution

Answer

The rms voltage of the source is 50.0 V.

Step 1: Understand the Circuit Components

In an L-R-C series circuit, there are three components: a resistor (R), a capacitor (C), and an inductor (L). The voltages across these components are given as follows: resistor (\(V_R = 30.0\,\mathrm{V}\)), capacitor (\(V_C = 90.0 \,\mathrm{V}\)), and inductor (\(V_L = 50.0 \,\mathrm{V}\)). The task is to determine the rms voltage of the source.

Step 2: Calculate the Impedance and Total Voltage

In an L-R-C series circuit, the voltages are not simply additive due to phase differences between components. We must use the concept of impedance. The voltage of the source (\(V_s\)) is determined by the square root of the sum of the squares of the resistive voltage (real) and the reactive voltage (imaginary part which is the difference between inductive and capacitive voltages). The formula is:\[V_s = \sqrt{V_R^2 + (V_L - V_C)^2}\]

Step 3: Plug in the Known Values

Substitute the given voltages into the formula:\[V_s = \sqrt{(30.0)^2 + (50.0 - 90.0)^2}\]Calculate the squares:\[V_s = \sqrt{900 + (-40.0)^2}\]

Step 4: Compute the Result

Calculate the quantity inside the square root:\[V_s = \sqrt{900 + 1600} = \sqrt{2500}\]Finally, take the square root:\[V_s = 50.0 \,\mathrm{V}\]

Step 5: Conclude the Solution

The rms voltage of the source in the L-R-C series circuit considering both real and reactive components is found to be \(50.0 \,\mathrm{V}\).

Key Concepts

Understanding Impedance in RLC Circuits

Impedance is a crucial concept when dealing with RLC circuits, as it accounts for both the resistive and reactive properties of the elements involved. Unlike a simple resistor, components like capacitors and inductors offer resistance that varies with frequency, known as reactance. Impedance, denoted by the symbol \( Z \), combines resistance (\( R \)) and reactance (\( X \)) to form a comprehensive measure of the circuit's opposition to current flow.

In mathematical terms, impedance is represented in complex form as \( Z = R + jX \), where \( j \) is the imaginary unit. This complex nature signifies that impedance affects both the magnitude and phase of the current. Calculating impedance involves vector addition of individual resistances and reactances of each component.

• Resistive Component (R): Resists current regardless of frequency.
• Inductive Reactance (\(X_L\)): Increases with frequency, calculated as \( X_L = 2\pi f L \).
• Capacitive Reactance (\(X_C\)): Decreases with frequency, calculated as \( X_C = \frac{1}{2\pi f C} \).

The impedance determines how the voltages across different components add up, affecting the total voltage in an RLC circuit. For example, in this exercise, impedance helps in calculating the rms voltage of the source using the phase differences between the elements.

Exploring RMS Voltage

Root Mean Square (rms) voltage is a critical concept used to describe the effective value of an AC voltage or current. It represents the equivalent DC value that would deliver the same power to a resistive load. In simpler terms, rms voltage gives a consistent measure of the actual power capability of an AC voltage source.

For AC voltages, which constantly change direction and magnitude, the rms value allows us to handle them similarly to DC values. The rms value is obtained by taking the square root of the average of the squares of the instantaneous values over one cycle. Mathematically, it's expressed for a sinusoidal voltage as \( V_{rms} = \frac{V_{peak}}{\sqrt{2}} \).

In the context of an RLC circuit, knowing the rms voltage of individual components helps us determine the overall rms voltage of the circuit. The provided exercise makes use of rms voltages across the resistor, capacitor, and inductor, to evaluate the rms voltage of the source, demonstrating its practical utility.

Understanding Phasor Diagrams

Phasor diagrams are powerful graphical tools used to visualize the relationship between different voltages and currents in AC circuits, including RLC circuits. They display the leading or lagging angles between the current and voltage of each component, reflecting their phase differences.

In a phasor diagram, quantities are represented as vectors (phasors) in a complex plane, making it easier to understand how voltages and currents combine. These vectors rotate at the frequency of the AC source:

• Vector Length: Represents the magnitude of the voltage or current.
• Angle to the Horizontal: Indicates the phase angle relative to a reference, usually the voltage across the resistor.

The phasor diagram aids in calculating the combined effect of the voltages across R, L, and C by visually representing the sum of these phasors. This technique simplifies the calculation of the total voltage or current in the circuit, as seen in the step-by-step problem that uses phasors to compute the rms voltage of the source.

Role of Reactive Components

Reactive components, namely inductors and capacitors, play a fundamental role in an RLC circuit by affecting both the voltage and current independently of energy dissipation. Their unique behavior is the result of their ability to store energy temporarily and then release it.

• Inductors (L): Store energy in a magnetic field when current passes through them. They resist changes in current and create a positive phase difference between voltage and current.
• Capacitors (C): Store energy in an electric field, resisting changes in voltage. They create a negative phase difference between voltage and current, essentially opposing the inductor's effects.

In an RLC circuit, these reactive components are crucial as their interplay determines the circuit's total reactance and, consequently, its impedance. As seen in the exercise, the difference in pressures from inductive and capacitive reactances enables calculating the source's rms voltage. Reactive components are key in shaping the AC circuit's voltage characteristics and tuning its behavior, especially under varying frequencies.