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 $L=\frac{1}{4} \mathrm{~h}, R=20 \Omega, C=\frac{1}{300} \mathrm{f}, E(t)=0 \mathrm{~V}, q(0)=4 \mathrm{C}$, and
$i(0)=0 \mathrm{~A}$. Is the charge on the capacitor everequal to zero?

Key Concepts

RLC Series Circuit

An RLC series circuit combines a resistor (R), an inductor (L), and a capacitor (C) in a single loop. Its analysis involves studying how energy is stored and dissipated in each component. The circuit’s natural response, when no external voltage is applied, is determined by the interplay between the inductive energy, the capacitive energy, and the resistive dissipation.

Differential Equation Modeling

The behavior of an RLC circuit is described by a second?order linear differential equation. This equation characterizes how the charge (or current) evolves over time in response to the circuit’s components. The form of the equation encapsulates the effects of inertia (from the inductor), stiffness (from the capacitor), and damping (from the resistor).

Characteristic Equation and Damping

The differential equation of the RLC circuit leads to a characteristic equation whose roots determine the system’s behavior. Depending on the relative magnitude of the damping factor (related to resistance and inductance) and the natural frequency (dictated by the inductor and capacitor), the circuit can be underdamped, critically damped, or overdamped. Each regime implies a different temporal evolution—oscillatory or non?oscillatory—of the charge on the capacitor.

Initial Conditions in Circuit Analysis

Initial conditions, such as the initial charge on the capacitor and the initial current in the circuit, are crucial for finding the particular solution to the differential equation governing the circuit’s behavior. They allow for the determination of the combination of exponential terms (or sinusoidal terms in the underdamped case) that fully describe the circuit’s time-dependent response.

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