Problem 68 - University Physics with Modern Physics

Question

A resistance \(R,\) capacitance \(C,\) and inductance \(L\) are connected in series to a voltage source with amplitude \(V\) and variable angular frequency \(\omega .\) If \(\omega=\omega_{0},\) the resonance angular frequency, find (a) the maximum current in the resistor; (b) the maximum voltage across the capacitor; (c) the maximum voltage across the inductor; (d) the maximum energy stored in the capacitor; (e) the maximum energy stored in the inductor. Give your answers in terms of \(R, C, L,\) and \(V .\)

Step-by-Step Solution

Answer

(a) \(\frac{V}{R}\); (b) \(\frac{V \sqrt{L/C}}{R}\); (c) \(\frac{V \sqrt{L/C}}{R}\); (d) \(\frac{V^2 L}{2 R^2 C}\); (e) \(\frac{V^2 L}{2 R^2}\).

Step 1: Understand the Resonance Condition

Resonance in an RLC circuit occurs when the inductive reactance equals the capacitive reactance, i.e., \( \omega L = \frac{1}{\omega C} \). Solving this gives the resonance angular frequency \( \omega_0 = \frac{1}{\sqrt{LC}} \). At resonance, the impedance is purely resistive, \( Z = R \).

Step 2: Find Maximum Current in Resistor at Resonance

The current at resonance is given by \( I = \frac{V}{Z} \). Since at resonance, \( Z = R \), the maximum current is \( I_{max} = \frac{V}{R} \).

Step 3: Calculate Maximum Voltage Across Capacitor

The maximum voltage across the capacitor \( V_C \) at resonance, when current is maximum, is given by \( V_C = I_{max} \cdot \frac{1}{\omega_0 C} \). Replace \( \omega_0 \) to find \( V_C = \frac{V}{R} \cdot \frac{1}{\omega_0 C} = \frac{V}{R} \cdot \sqrt{\frac{L}{C}} \).

Step 4: Calculate Maximum Voltage Across Inductor

Similarly, the maximum voltage across the inductor \( V_L \) at resonance is \( V_L = I_{max} \cdot \omega_0 L \). Substitute \( \omega_0 \) to compute \( V_L = \frac{V}{R} \cdot \sqrt{\frac{L}{C}} \cdot L \).

Step 5: Find Maximum Energy Stored in Capacitor

The maximum energy stored in the capacitor \( U_C \) is \( U_C = \frac{1}{2} C V_C^2 \). We substitute \( V_C \) to get \( U_C = \frac{1}{2} C \left(\frac{V \sqrt{L/C}}{R}\right)^2 \). Simplify to find \( U_C = \frac{V^2 L}{2 R^2 C} \).

Step 6: Find Maximum Energy Stored in Inductor

The maximum energy stored in the inductor \( U_L \) is \( U_L = \frac{1}{2} L I_{max}^2 \). Substitute \( I_{max} \) to get \( U_L = \frac{1}{2} L \left(\frac{V}{R}\right)^2 \). Simplify to find \( U_L = \frac{V^2 L}{2 R^2} \).

Key Concepts

Resonance Frequency

In an RLC circuit, the resonance frequency is a specific condition where maximum energy exchange occurs between the inductor and the capacitor. When the circuit reaches resonance, the inductive reactance and capacitive reactance become equal, cancelling each other out. This means that the impedance of the circuit is reduced to the resistance only, allowing all the power from the source to be transferred to the resistive component.
To calculate the resonance angular frequency, the formula used is: \[\omega_0 = \frac{1}{\sqrt{LC}}\]
At this frequency, the circuit's reactance is minimized, leading to maximum current flow throughout the circuit. Understanding resonance is crucial, as it is the condition under which the circuit will operate most efficiently.

Current in Resistance

At resonance, since the circuit's impedance is reduced to just the resistance, the maximum current that can flow through the resistor can be calculated quite simply. The formula to determine this is: \[I_{max} = \frac{V}{R}\]
This relationship implies that at resonance, the voltage source can transfer maximum power to the resistor, as there is no reactive opposition (like capacitance or inductance) to hinder the current flow. This important property of RLC circuits is often leveraged in applications like radio tuning circuits, where clear signal reception is desired.

Energy Stored in Capacitor

Capacitors have the ability to store electrical energy, and understanding this is crucial for analyzing RLC circuits. The maximum energy that a capacitor can store at resonance is calculated with:\[U_C = \frac{1}{2} C V_C^2\]
Substituting for the maximum voltage across the capacitor, which can be determined as:\[V_C = \frac{V}{R} \cdot \sqrt{\frac{L}{C}}\]
We arrive at:\[U_C = \frac{V^2 L}{2 R^2 C}\]
This tells us how much energy can be held momentarily by the capacitor when the circuit is at resonance, and the implications are profound in technologies where energy storage and release are key factors, such as in flash photography lighting and power smoothing systems.

Voltage Across Inductor

Inductors, like capacitors, also play a key role in storing energy in an RLC circuit. The maximum voltage drop across an inductor at resonant frequency can be calculated using:\[V_L = I_{max} \cdot \omega_0 L\]
By substituting the maximum current knowing that:\[I_{max} = \frac{V}{R}\]and the resonance frequency:\[\omega_0 = \frac{1}{\sqrt{LC}}\]the formula becomes:\[V_L = \frac{V}{R} \cdot \sqrt{\frac{L}{C}} \cdot L\]
Understanding how inductors interact with alternating currents and store energy helps in many modern applications such as transformers, induction cooking appliances, and wireless transmitting devices.