To analyze the given LCR circuit, we need to break down the components and their relationships. The circuit consists of an inductor with an inductance of 5 mH (millihenries) and a capacitor with a capacitance of 20 µF (microfarads). The voltage source is given as V = 5 * 10^-2 cos(2t/(LC)^0.5). Let's go through the calculations step by step to find the maximum charge on the capacitor, the maximum current in the circuit, and the phase relationships between the voltages.
Understanding the Circuit Parameters
First, we need to calculate the values of inductance (L) and capacitance (C) in standard units:
- L = 5 mH = 5 * 10^-3 H
- C = 20 µF = 20 * 10^-6 F
Calculating the Resonant Frequency
The resonant frequency (ω₀) of an LCR circuit is given by the formula:
ω₀ = 1 / √(LC)
Substituting the values of L and C:
ω₀ = 1 / √((5 * 10^-3) * (20 * 10^-6))
Calculating this gives:
ω₀ = 1 / √(1 * 10^-7) = 1 / (1 * 10^-3) = 1000 rad/s
Maximum Charge on the Capacitor
The maximum charge (Q_max) on the capacitor can be determined using the formula:
Q_max = C * V_max
Here, V_max is the amplitude of the voltage source, which is 5 * 10^-2 V. Thus:
Q_max = (20 * 10^-6) * (5 * 10^-2) = 1 * 10^-6 C = 1 µC
Maximum Current in the Circuit
The maximum current (I_max) can be calculated using the relationship between maximum voltage and impedance in the circuit. The impedance (Z) of the circuit at resonance is given by:
Z = √(L/C)
Calculating Z:
Z = √((5 * 10^-3) / (20 * 10^-6)) = √(250) = 15.81 Ω
Now, using Ohm's law, we can find the maximum current:
I_max = V_max / Z
I_max = (5 * 10^-2) / 15.81 ≈ 0.00316 A = 3.16 mA
Phase Relationships in the Circuit
In an LCR circuit, the phase relationship between the voltages across the inductor (V_L), capacitor (V_C), and the source voltage (V) is crucial. At resonance, the voltages across the inductor and capacitor are equal in magnitude but opposite in phase:
- V_L leads the current by 90 degrees.
- V_C lags the current by 90 degrees.
- The source voltage V is in phase with the current.
This means that at resonance, the inductor and capacitor effectively cancel each other out, and the circuit behaves as a purely resistive circuit with the source voltage and current in phase.
Summary of Results
- Maximum Charge on the Capacitor: 1 µC
- Maximum Current in the Circuit: 3.16 mA
- Phase Relationship: V_L leads the current by 90 degrees, V_C lags the current by 90 degrees, and V is in phase with the current.
This analysis provides a comprehensive understanding of the LCR circuit's behavior under the given conditions. If you have any further questions or need clarification on any part, feel free to ask!
