RC circuits Knowledge-Based Study

Study notes

1. What is an RC Circuit?

   An RC circuit is an electrical circuit composed of a resistor of resistance $R$ and a capacitor of capacitance $C$ connected in series or parallel, driven by a voltage source or discharging through a closed loop. Unlike purely resistive DC circuits where currents and voltages reach steady states almost instantaneously, circuits containing capacitors exhibit time-dependent behavior known as transient response.

   Key components and their roles include:

   • Capacitor ($C$): Stores electric potential energy in the electric field established between its conductive plates, governed by the relation $q = C V_C$, where $q$ is charge and $V_C$ is the potential difference across the plates.
   • Resistor ($R$): Opposes the flow of electric charge, thereby limiting the rate at which charge can accumulate onto or leave the capacitor plates according to Ohm's law, $V_R = I R$.

   The Charging Process

   Consider a series circuit consisting of an ideal battery with electromotive force $\mathcal{E}$, an uncharged capacitor $C$, a resistor $R$, and a switch. At time $t = 0$, the switch is closed.

   Applying Kirchhoff's Loop Rule at any instant gives:

   $$\mathcal{E} - I(t)R - \frac{q(t)}{C} = 0$$

   Since the electric current is the time derivative of charge, $I(t) = \frac{dq}{dt}$, substituting this yields the differential equation:

   $$\mathcal{E} - R\frac{dq}{dt} - \frac{q}{C} = 0$$

   Solving this first-order linear differential equation with the initial condition $q(0) = 0$ gives the time-dependent charge accumulation:

   $$q(t) = C\mathcal{E}\left(1 - e^{-t / RC}\right) = Q_{\max}\left(1 - e^{-t / RC}\right)$$

   Differentiating $q(t)$ with respect to time yields the instantaneous charging current:

   $$I(t) = \frac{\mathcal{E}}{R}e^{-t / RC} = I_0 e^{-t / RC}$$

   • At $t = 0$: The uncharged capacitor has zero potential difference across it ($V_C = 0$). It acts momentarily like a short circuit, allowing maximum initial current $I_0 = \frac{\mathcal{E}}{R}$.
   • As $t o \infty$: The capacitor reaches full charge $Q_{\max} = C\mathcal{E}$, creating an opposing voltage equal to the battery voltage ($V_C = \mathcal{E}$). The current drops to zero ($I = 0$), and the fully charged capacitor acts as an open circuit in direct current.

   The Discharging Process

   Now consider a fully charged capacitor with initial charge $Q_0 = C V_0$ that is disconnected from the battery and allowed to discharge through a resistor $R$. Closing the discharge switch at $t = 0$ produces a loop containing only the capacitor and the resistor.

   By Kirchhoff's Loop Rule:

   $$\frac{q(t)}{C} - I(t)R = 0$$

   Because the charge on the capacitor is decreasing as current flows out of its positive plate, we define $I(t) = -\frac{dq}{dt}$. Substituting this into the loop rule gives:

   $$\frac{q}{C} + R\frac{dq}{dt} = 0 \implies \frac{dq}{dt} = -\frac{q}{RC}$$

   Solutions for Discharging

   Integrating with the initial condition $q(0) = Q_0$ yields:

   $$q(t) = Q_0 e^{-t / RC}$$

   The potential difference across the capacitor is:

   $$V_C(t) = \frac{q(t)}{C} = V_0 e^{-t / RC}$$

   The magnitude of the discharging current through the resistor follows the same exponential decay:

   $$I(t) = -\frac{dq}{dt} = \frac{Q_0}{RC}e^{-t / RC} = I_0 e^{-t / RC}$$

   • During discharge, charge, voltage, and current all decay exponentially toward zero.
   • The electric potential energy initially stored in the capacitor, $U_C = \frac{1}{2}\frac{Q_0^2}{C}$, is completely dissipated as thermal energy through Joule heating in the resistor.

   The Capacitive Time Constant ($au$)

   The rate at which an RC circuit charges or discharges is determined by the product of resistance and capacitance. This product is defined as the time constant of the circuit, denoted by the Greek letter $au$ (tau):

   $$au = RC$$

   The SI unit of $au$ is the second ($\text{s}$):

   $$[ au] = [R][C] = (\Omega)(\text{F}) = \left(\frac{\text{V}}{\text{A}}\right)\left(\frac{\text{C}}{\text{V}}\right) = \frac{\text{C}}{\text{A}} = \frac{\text{A}\cdot\text{s}}{\text{A}} = \text{s}$$

   Significance During Charging and Discharging

   • At $t = au$ (Charging): $q( au) = Q_{\max}(1 - e^{-1}) \approx 0.632\,Q_{\max}$. The capacitor reaches approximately 63.2% of its maximum charge.
   • At $t = au$ (Discharging): $q( au) = Q_0 e^{-1} \approx 0.368\,Q_0$. The remaining charge drops to approximately 36.8% of its initial value.
   • At $t \approx 5 au$: Because $e^{-5} \approx 0.0067$, the circuit is at over 99% of its final state. In practical circuit analysis, a capacitor is considered fully charged or completely discharged after an elapsed time of $5 au$.