Problem 26 - University Physics with Modern Physics

Question

In an \(L_{-} R-C\) series circuit the source is operated at its resonant angular frequency. At this frequency, the reactance \(X_{C}\) of the capacitor is 200\(\Omega\) and the voltage amplitude across the capacitor is 600 \(\mathrm{V}\) . The circuit has \(R=300 \Omega .\) What is the voltage amplitude of the source?

Step-by-Step Solution

Answer

The voltage amplitude of the source is 900 V.

Step 1: Understand Resonance in an L-R-C Circuit

In an L-R-C series circuit, resonance occurs when the reactance of the inductor \(X_L\) is equal to the reactance of the capacitor \(X_C\), i.e., \(X_L = X_C\). At resonance, the total impedance of the circuit is purely resistive, equal to the resistance \(R\).

Step 2: Express Capacitor Voltage Amplitude

At resonance, the voltage amplitude across the capacitor is given by \(V_C = I \, X_C\), where \(I\) is the current amplitude in the circuit. From the problem, \(V_C = 600\, \text{V}\) and \(X_C = 200\, \Omega\).

Step 3: Calculate the Current Amplitude

Rearrange the formula from Step 2 to find the current amplitude: \[ I = \frac{V_C}{X_C} = \frac{600\, \text{V}}{200\, \Omega} = 3\, \text{A} \].

Step 4: Determine Source Voltage Amplitude

Since at resonance, the impedance is equal to the resistance \(R = 300 \Omega\), the source voltage amplitude \(V_S\) is given by \(V_S = I \cdot R\). Substitute the current amplitude and resistance to find \(V_S = 3\, \text{A} \times 300\, \Omega = 900\, \text{V}\).

Key Concepts

Resonance

In electrical circuits, resonance is a fascinating phenomenon that occurs when a circuit, like an L-R-C series circuit, is perfectly tuned to a specific frequency called the resonant frequency. At this frequency, the reactances of the inductor and capacitor cancel each other out because they are equal in magnitude but opposite in phase. This leads to an important result: the circuit's impedance is minimized, and the system behaves purely resistively, with the impedance equaling just the resistance of the circuit.

Understanding resonance is crucial because it allows the circuit to operate with maximum efficiency. In our L-R-C circuit example, when resonance is achieved, the circuit permits the highest possible current for a given voltage. This condition is highly desired in applications such as radio signal tuning and electrical filters because it allows for the precise selection and amplification of specific frequencies.

Reactance

Reactance is a measure of how much a component in an AC circuit, like an inductor or capacitor, opposes the change in current through it. It fluctuates with the frequency of the AC signal, which is why it's a crucial concept when analyzing circuits at different frequencies.

There are two types of reactance:

• Inductive Reactance (\(X_L\)): Opposes changes due to the magnetic field in an inductor and increases with frequency.
• Capacitive Reactance (\(X_C\)): Opposes changes due to the electric field in a capacitor and decreases with frequency.

At resonance in an L-R-C circuit, the inductive and capacitive reactances are equal. This means \(X_L = X_C\), effectively canceling each other out and leaving the circuit with only its resistance to influence the current flow. Reactance is pivotal in designing circuits that need to function optimally at specific frequencies, like in the exercise scenario where \(X_C = 200\, \Omega\).

Impedance

In AC circuits, impedance is a broader term that encompasses both resistance and reactance. Represented usually as \(Z\), impedance can be thought of as the total "opposition" a circuit presents to the flow of alternating current. It's expressed as a complex number because it involves both magnitude and phase.

Impedance is calculated using the formula: \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).

For an L-R-C circuit at resonance, this simplifies significantly! Since \(X_L = X_C\), the reactance terms cancel out, and the impedance is simply \(Z = R\).

This is why resonance is so key—it allows the circuit to transmit maximum power without the extra "drag" of reactance. In our original circuit, this means the impedance equals the resistance, which is \(R = 300\, \Omega\). Therefore, the impedance during resonance directly impacts how we calculate the source voltage.

Voltage Amplitude

Voltage amplitude in AC circuits is a measure of the peak voltage—the maximum value the voltage reaches. In calculations, this is often what we're solving for, like in our L-R-C circuit problem.

During resonance, the voltage across specific components in the circuit can be significantly high due to reduced impedance. This is because the current reaches its maximum possible amplitude. Voltage amplitude relationships depend on the function of different components:

• Across the capacitor: Given by \(V_C = I \cdot X_C\).
• Across the whole circuit (source voltage): \(V_S = I \cdot R \).

For our specific example, with \(V_C = 600 \, \text{V} \,\) and the current \(I = 3 \, \text{A},\) the source voltage amplitude—the peak voltage the source needs to provide to the circuit—is \(V_S = 900 \, \text{V}.\) Understanding voltage amplitude is essential for effectively designing and analyzing AC circuits to ensure they meet the necessary performance criteria.