A series electric circuit contains a resistance R, a...

A series electric circuit contains a resistance $R$, a capacitance $C$ and a battery supplying a time-varying electromotive force $V(t) .$ The charge $q$ on the capacitor therefore obeys the equation $$ R \frac{d q}{d t}+\frac{q}{C}=V(t) $$ Assuming that initially there is no charge on the capacitor, and given that $V(t)=V_{0} \sin \omega t$, find the charge on the capacitor as a function of time.

A series electric circuit contains a resistance $R$, a capacitance $C$ and a battery supplying a time-varying electromotive force $V(t) .$ The charge $q$ on the capacitor therefore obeys the equation
$$
R \frac{d q}{d t}+\frac{q}{C}=V(t)
$$
Assuming that initially there is no charge on the capacitor, and given that $V(t)=V_{0} \sin \omega t$, find the charge on the capacitor as a function of time.

Key Concepts

RC Circuit

An RC circuit is an electrical circuit consisting of a resistor (R) and a capacitor (C) connected in series. It is used to study the charging and discharging behavior of the capacitor when subjected to various voltage inputs. Understanding this circuit is foundational for analyzing time-dependent voltage and current responses in circuits, which can model many real-world electrical systems.

First-Order Linear Differential Equations

These are differential equations that can be written in the form of a first derivative plus a function times the unknown function, set equal to another function of time. In the context of an RC circuit, the voltage across elements leads to a linear first-order differential equation describing the time evolution of the charge on the capacitor. Solving such equations is a fundamental technique in both mathematics and engineering.

Integrating Factor Method

The integrating factor method is a systematic technique for solving first-order linear differential equations. By multiplying the entire equation by an appropriate function (the integrating factor), the equation can be rewritten as the derivative of a product, which is then integrated to find the general solution. This method is particularly useful in circuits where time-varying sources are present.

Transient and Steady-State Responses

In circuits with time-varying sources, the overall response is typically composed of a transient part, which depends on initial conditions and decays over time, and a steady-state part, which is the response once the circuit has settled. In the analysis of an RC circuit with a sinusoidal voltage input, the transient solution represents the initial charging behavior, while the steady-state solution represents the sinusoidally oscillating charge on the capacitor.

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