Find the charge on the capacitor in an L R C -series circuit...

Find the charge on the capacitor in an $L R C$ -series circuit when $L=\frac{1}{2} \mathrm{~h}, R=10 \Omega, C=0.01 \mathrm{f}, E(t)=150 \mathrm{~V}, q(0)=1 \mathrm{C}$, and
$i(0)=0 \mathrm{~A} .$ What is the charge on the capacitor after a long time?

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Key Concepts

LRC Circuit Analysis

In an LRC series circuit, the behavior of the circuit is modeled by a second-order differential equation that describes how current and voltage change over time. This analysis involves applying Kirchhoff's voltage law, where the sum of voltage drops across the inductor, resistor, and capacitor equals the source voltage. The interplay between these components dictates both the transient and steady-state responses of the circuit.

Transient vs. Steady-State Behavior

After the circuit is energized, it undergoes a transient period during which the current and voltage across the components change dynamically. However, as time approaches infinity, the transient effects decay, and the circuit reaches a steady state where the response becomes constant. In the steady state of a DC circuit, the behavior of reactive components like capacitors and inductors simplifies considerably.

Capacitor Charging in DC Circuits

In a DC circuit, after a long period, a capacitor charges up to the voltage of the DC source because the transient currents have ceased. Since a capacitor stores electrical energy in the form of charge, the final charge on the capacitor can be determined using the relation Q = C × V, where C is the capacitance and V is the voltage across the capacitor.

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