Problem 29 - Physics

Question

A \(10.0-\Omega\) resistor, a \(12.0-\mu \mathrm{F}\) capacitor, and a \(17.0\) -mH inductor are connected in series with a 155-V generator. (a) At what frequency is the current a maximum? (b) What is the maximum value of the rms current?

Step-by-Step Solution

Answer

(a) 354 Hz; (b) 15.5 A.

Step 1: Understanding Resonant Frequency

In a series RLC circuit, the current is maximum at the resonant frequency. This frequency can be found using the formula \[ f_0 = \frac{1}{2 \pi \sqrt{L C}} \] where \( L \) is the inductance and \( C \) is the capacitance.

Step 2: Calculating the Resonant Frequency

Given the inductance \( L = 17.0 \text{ mH} = 17.0 \times 10^{-3} \text{ H} \) and capacitance \( C = 12.0 \mu \text{F} = 12.0 \times 10^{-6} \text{ F} \), plug these values into the resonant frequency formula.\[ f_0 = \frac{1}{2 \pi \sqrt{(17.0 \times 10^{-3}) (12.0 \times 10^{-6})}} = 354 \, \text{Hz} \]

Step 3: Maximum RMS Current Calculation

At resonance, the impedance of the RLC circuit is equal to the resistance only because the inductive reactance and capacitive reactance cancel each other out. The maximum rms current is given by \[ I_{\text{rms}} = \frac{V_{\text{rms}}}{R} \]where \( V_{\text{rms}} = 155 \, \text{V} \) and \( R = 10.0 \Omega \).\[ I_{\text{rms}} = \frac{155}{10.0} = 15.5 \, \text{A} \]

Step 4: Conclusion Compilation

To summarize, the resonant frequency at which the maximum current flows is \( 354 \, \text{Hz} \), and the maximum rms current value is \( 15.5 \, \text{A} \). This is because at resonance, the reactive components cancel, leaving only the resistive component.

Key Concepts

Resonant Frequency

The concept of resonant frequency is crucial in understanding the behavior of an RLC circuit. When we connect a resistor, inductor, and capacitor in series, the circuit will have a unique frequency known as the resonant frequency. At this frequency, the inductive and capacitive reactances cancel each other out, maximizing the current flow. The resonant frequency \( f_0 \) can be calculated using the formula:\[f_0 = \frac{1}{2 \pi \sqrt{L C}}\]where \( L \) represents the inductance and \( C \) represents the capacitance. In practical scenarios, finding this frequency helps in tuning circuits to a specific frequency range, which is a common requirement in radio and communication devices.

RMS Current

RMS, or Root Mean Square Current, is a measure of the effective value of alternating current. It represents the equivalent DC current that provides the same power to a resistor as the AC current does over one cycle. The RMS value simplifies calculations in AC circuits, especially for power and energy measurements. The formula for calculating RMS current in a series circuit at resonance is:\[I_{\text{rms}} = \frac{V_{\text{rms}}}{R}\]where \( V_{\text{rms}} \) is the RMS voltage across the circuit and \( R \) is the resistance. This formula highlights the maximum current flow at the point where inductive and capacitive effects are neutralized.

Series Circuit

In a series circuit, components are connected end-to-end so that there is a single path for the current to flow. The behavior of a series circuit is dictated by the sum of the voltages across each component. Key characteristics include:

• The same current flows through all components because there is only one path.
• The total resistance is the sum of individual resistances.
• Voltage is divided across the components based on their individual resistance or reactance.

Series circuits are prevalent in various electrical applications, offering easier current control and voltage distribution throughout the circuit.

Inductive Reactance

Inductive reactance is a property of an inductor in an AC circuit, representing its opposition to changes in current. It is influenced by the frequency of the AC source and the inductance of the coil. Inductive reactance \( X_L \) can be calculated using:\[X_L = 2\pi f L\]where \( f \) is the frequency and \( L \) is the inductance. In a series RLC circuit, inductive reactance increases with frequency. This behavior is crucial in filtering and tuning applications, where inductors can pass or block signals of certain frequencies.

Capacitive Reactance

Capacitive reactance is another essential concept in AC circuit analysis. It is the opposition a capacitor offers to changes in voltage. Capacitive reactance \( X_C \) depends on the frequency of the applied voltage and the capacitance value, calculated as:\[X_C = \frac{1}{2\pi f C}\]where \( f \) is the frequency and \( C \) is the capacitance. As frequency increases, the capacitive reactance decreases, making capacitors useful in managing and manipulating AC signals. In a resonant circuit, capacitive reactance complements inductive reactance, allowing for fine-tuning of circuit response to certain frequencies.