Problem 26 - College Physics

Question

In an \(R-L-C\) series circuit, \(R=300 \Omega, L=0.400 \mathrm{H},\) and \(C=6.00 \times 10^{-8} \mathrm{~F}\). When the ac source operates at the resonance frequency of the circuit, the current amplitude is 0.500 A. (a) What is the voltage amplitude of the source? (b) What is the amplitude of the voltage across the resistor, across the inductor, and across the capacitor? (c) What is the average power supplied by the source?

Step-by-Step Solution

Answer

(a) 150 V, (b) 150 V, 4,084 V, 4,084 V, (c) 75 W.

Step 1: Resonance Frequency Calculation

The resonance frequency \( f_0 \) of an RLC circuit is given by the formula \( f_0 = \frac{1}{2\pi\sqrt{LC}} \). Substituting the given values, \( L = 0.400 \text{ H} \) and \( C = 6.00 \times 10^{-8} \text{ F} \), we find \( f_0 = \frac{1}{2\pi\sqrt{0.400 \times 6.00 \times 10^{-8}}} \approx 3,250 \text{ Hz} \).

Step 2: Voltage Amplitude across Resistor

At resonance, the impedance is purely resistive, and the voltage across the resistor can be calculated using Ohm's Law: \( V_R = I R \). Given the current amplitude \( I = 0.500 \text{ A} \) and \( R = 300 \Omega \), \( V_R = 0.500 \times 300 = 150 \text{ V} \).

Step 3: Voltage Amplitude across Inductor

At resonance, the voltage across the inductor \( V_L \) is \( V_L = I X_L \), where the inductive reactance \( X_L = 2\pi f_0 L \). Substitute \( f_0 = 3250 \text{ Hz} \) and \( L = 0.400 \text{ H} \) into the formula to find \( X_L = 2\pi \times 3250 \times 0.400 \approx 8,168 \Omega \). Therefore, \( V_L = 0.500 \times 8,168 = 4,084 \text{ V} \).

Step 4: Voltage Amplitude across Capacitor

At resonance, the voltage across the capacitor \( V_C \) is \( V_C = I X_C \), where the capacitive reactance \( X_C = \frac{1}{2\pi f_0 C} \). Substituting \( f_0 = 3250 \text{ Hz} \) and \( C = 6.00 \times 10^{-8} \text{ F} \), \( X_C \approx 8,168 \Omega \). Hence, \( V_C = 0.500 \times 8,168 = 4,084 \text{ V} \).

Step 5: Voltage Amplitude of the Source

The voltage amplitude of the source at resonance is equal to the voltage across the resistor, since the net reactance of the inductor and capacitor cancel out: \( V_{ ext{source}} = V_R = 150 \text{ V} \).

Step 6: Average Power Supplied by the Source

The average power supplied by the source can be calculated as \( P_{ ext{avg}} = I^2 R \). With \( I = 0.500 \text{ A} \) and \( R = 300 \Omega \), \( P_{ ext{avg}} = 0.500^2 \times 300 = 75 \text{ W} \).

Key Concepts

Resonance Frequency

In an RLC circuit, resonance frequency is a key concept. It refers to the frequency at which the circuit naturally oscillates when not externally driven. At this frequency, the inductive and capacitive reactance are equal and opposite, effectively canceling each other out. This leads to a situation where the impedance of the circuit is at its minimum, dictated only by the resistance. The resonance frequency \( f_0 \) is given by the formula: \[f_0 = \frac{1}{2\pi\sqrt{LC}}\] where \( L \) is the inductance and \( C \) is the capacitance. It is essential to understand that at this specific frequency, energy is stored efficiently and oscillates between the inductor and capacitor. This is why RLC circuits are often used in filter applications, radios, and communications devices.

Inductive Reactance

Inductive reactance \( X_L \) is a measure of an inductor's opposition to the change in current flow in an alternating current (AC) circuit. It is directly proportional to the frequency of the AC signal and the inductance of the coil. The formula to calculate inductive reactance is:\[X_L = 2\pi f L\]where \( f \) is the frequency in hertz and \( L \) is the inductance in henries. As the frequency increases, \( X_L \) also increases, which means that the inductor will oppose the current more strongly at higher frequencies. In resonance, \( X_L \) plays a crucial role by becoming equal and opposite to capacitive reactance, thus canceling each other out to allow only the resistive part of the impedance to dictate the current flow.

Capacitive Reactance

Capacitive reactance \( X_C \) is the opposition that a capacitor offers to the change in voltage across its plates in an AC circuit. Unlike inductive reactance, capacitive reactance is inversely proportional to both the frequency and the capacitance:\[X_C = \frac{1}{2\pi f C}\]where \( f \) is the frequency in hertz and \( C \) is the capacitance in farads. As the frequency increases, \( X_C \) decreases. In the context of resonance, \( X_C \) precisely equals \( X_L \), as this balance allows the current to reach maximum amplitude. Capacitors are essential for controlling the phase and amplitude of AC signals, making them useful in radio tuning, timing circuits, and filter applications.

Ohm's Law

Ohm's Law is a fundamental principle used to relate the voltage, current, and resistance in an electric circuit. It is expressed by the equation:\[ V = I R \]where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. This simple law is pivotal in analyzing circuits as it enables the calculation of one quantity if the other two are known. In an RLC circuit operating at resonance, the impedance is purely resistive, simplifying Ohm’s law usage to find the voltage across the resistor. Understanding Ohm's Law is crucial for both AC and DC circuit analysis, as it applies universally to these circuits when considering their resistive elements.

Average Power in AC Circuits

The average power in AC circuits is the power consumed by the resistive elements of the circuit over time. Calculating average power is essential for understanding how much energy is being used in AC circuits. The formula for average power \( P_{\text{avg}} \) in terms of current and resistance is:\[P_{\text{avg}} = I^2 R\]where \( I \) is the rms (root mean square) current, and \( R \) is the resistance. In an RLC circuit at resonance, this calculation becomes straightforward since the only dissipative element is the resistor. The power factor, which equals 1 at resonance, signifies the circuit's efficiency. This means that all the power provided by the source is effectively used by the resistor, making it a point of maximum power transfer within the circuit.