Find the charge on the capacitor and the current in the...

Find the charge on the capacitor and the current in the given $L R C$ -series circuit. Find the maximum charge on the capacitor.
$$
\begin{aligned}
&L=1 \mathrm{~h}, R=100 \Omega, C=0.0004 \mathrm{f}, E(t)=30 \mathrm{~V}, q(0)=0 \mathrm{C}, \\
&i(0)=2 \mathrm{~A}
\end{aligned}
$$

Key Concepts

LRC Series Circuit Analysis

An LRC series circuit contains an inductor, a resistor, and a capacitor connected in series. Analyzing such circuits involves applying Kirchhoff’s voltage law, which leads to a second-order differential equation that governs the time-dependent behavior of the charge and current in the circuit.

Kirchhoff's Voltage Law

This fundamental law states that the sum of all voltage drops around any closed loop in a circuit must equal the applied voltage. In an LRC circuit, the voltage drops across the inductor, resistor, and capacitor add up to the driving electromotive force, forming the basis for deriving the differential equation governing the system.

Second-Order Linear Differential Equations

The dynamic behavior of an LRC circuit is modeled by a second-order linear differential equation with constant coefficients. This equation typically includes a term with the second derivative (related to the inductor), a term with the first derivative (related to the resistor), and a term with the function itself (related to the capacitor), describing the evolution of the circuit's state over time.

Transient and Steady-State Response

Analyzing an LRC circuit involves studying both the transient response—the behavior of the circuit immediately after a change in conditions such as a sudden application of voltage—and the steady-state response, which represents the long-term behavior once the transient effects have decayed. The maximum charge on the capacitor is often determined during the transient interval.

Initial Value Problem in Circuit Analysis

The initial conditions, such as the initial charge on the capacitor and the initial current in the circuit, are essential to uniquely determine the solution of the second-order differential equation. These conditions influence the circuit’s transient response and are key to computing quantities like the maximum charge on the capacitor.

Forced Oscillations and Resonance

When an external voltage is applied to the circuit, the system experiences forced oscillations. Understanding how the driving voltage interacts with the circuit’s natural response helps predict behaviors such as resonance, where the amplitude of oscillations—including the capacitor’s charge—can reach a maximum under specific conditions.

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