You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. They are connected to form an L-R-C series circuit. (b) What is the current amplitude?

Verified step by step guidance

1. Calculate the angular frequency (\( \omega \)) using the formula for the resonant frequency of an LRC circuit: \( \omega = \frac{1}{\sqrt{LC}} \), where L is the inductance and C is the capacitance.
2. Calculate the impedance (Z) of the circuit at resonance using the formula: \( Z = R \), since the inductive reactance (\( X_L \)) and capacitive reactance (\( X_C \)) cancel each other out at resonance. \( X_L = \omega L \) and \( X_C = \frac{1}{\omega C} \).
3. Verify that \( X_L = X_C \) at resonance to ensure that the impedance is purely resistive and equals R.
4. Calculate the maximum voltage (V) across the circuit. This is typically given by the source voltage in the problem, but if not specified, assume a standard value or use a variable V to represent it.
5. Determine the current amplitude (I) using Ohm's Law: \( I = \frac{V}{Z} \), where V is the voltage and Z is the impedance at resonance.

Key Concepts

Impedance in RLC Circuits

Impedance is the total opposition that a circuit offers to the flow of alternating current (AC) and is represented as a complex number. In an RLC series circuit, the impedance combines the resistance (R), inductive reactance (XL), and capacitive reactance (XC). The formula for impedance is Z = √(R² + (XL - XC)²), where XL = ωL and XC = 1/(ωC), with ω being the angular frequency of the AC source.

Current Amplitude in AC Circuits

The current amplitude in an AC circuit is the maximum value of the current flowing through the circuit, often denoted as I₀. It can be calculated using Ohm's law for AC circuits, I₀ = V₀ / Z, where V₀ is the maximum voltage and Z is the impedance. Understanding how to calculate the current amplitude is essential for analyzing the behavior of RLC circuits under AC conditions.

Resonance in RLC Circuits

Resonance occurs in an RLC circuit when the inductive reactance equals the capacitive reactance (XL = XC), resulting in maximum current amplitude at a specific frequency known as the resonant frequency. At resonance, the impedance is minimized to just the resistance (Z = R), leading to a significant increase in current amplitude. This concept is crucial for understanding how RLC circuits can be tuned to specific frequencies for optimal performance.