# Lecture 8: RC Circuits

## Physics II — Electromagnetism, Optics & Modern Physics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Describe the charging and discharging behavior of an RC circuit qualitatively and quantitatively
2. Define the time constant and explain its physical significance
3. Derive and apply the exponential equations for charge, current, and voltage in RC circuits
4. Calculate energy stored and dissipated during charging and discharging
5. Identify applications of RC circuits including filters and timing circuits

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## Lecture Content

### I. Introduction to RC Circuits

An **RC circuit** contains a resistor (R) and a capacitor (C) connected to a voltage source or to each other. Unlike purely resistive circuits, RC circuits exhibit **time-dependent** (transient) behavior: the voltages and currents change with time as the capacitor stores or releases energy. Two fundamental processes characterize RC circuits. During **charging**, the capacitor voltage increases from zero toward the source EMF. During **discharging**, the capacitor voltage decreases from an initial value toward zero. The key parameter governing these processes is the **time constant** tau = RC, which has SI units of (Omega)(F) = seconds. The time constant determines how quickly the circuit responds to changes.

### II. Charging an RC Circuit

Consider a circuit with a battery (EMF epsilon), a resistor R, and an initially uncharged capacitor C in series. Applying Kirchhoff's loop rule gives epsilon - IR - Q/C = 0, which leads to the first-order differential equation epsilon = R(dQ/dt) + Q/C.

The solutions to this equation describe exponential approach to equilibrium. The charge on the capacitor is Q(t) = C epsilon (1 - e^(-t/RC)) = Q_max (1 - e^(-t/tau)). The current is I(t) = (epsilon/R) e^(-t/RC) = I_0 e^(-t/tau). The voltage across the capacitor is V_C(t) = epsilon (1 - e^(-t/tau)), and the voltage across the resistor is V_R(t) = epsilon e^(-t/tau).

At t = 0, the capacitor is uncharged (Q = 0, V_C = 0), the current is at its maximum (I = epsilon/R), and the full voltage appears across the resistor (V_R = epsilon). As t approaches infinity, the capacitor becomes fully charged (Q = C epsilon, V_C = epsilon), the current drops to zero, and V_R = 0. At the characteristic time t = tau = RC, the charge has reached 63.2% of its maximum value and the current has decayed to 36.8% of its initial value. After 5 tau, the circuit is approximately 99.3% complete, effectively fully charged.

<image>Two graphs stacked vertically for an RC charging circuit. Top graph: Charge Q (or capacitor voltage V_C) versus time t. The curve starts at zero and rises exponentially toward Q_max = C*epsilon (shown as a dashed horizontal asymptote). The time constant tau = RC is marked on the t-axis, with a horizontal dashed line at 0.632 Q_max. The equation Q(t) = Q_max(1 - e^(-t/tau)) is displayed. Bottom graph: Current I versus time t. The curve starts at I_0 = epsilon/R and decays exponentially toward zero. At t = tau, the current is 0.368 I_0. The equation I(t) = I_0 e^(-t/tau) is displayed. Both graphs have consistent tau markings at 1tau, 2tau, 3tau, 4tau, 5tau along the time axis.</image>

### III. Discharging an RC Circuit

When a charged capacitor (initial charge Q_0, initial voltage V_0 = Q_0/C) discharges through a resistor R, the loop rule gives -Q/C - IR = 0, leading to R(dQ/dt) = -Q/C. The solutions are pure exponential decays. The charge is Q(t) = Q_0 e^(-t/RC) = Q_0 e^(-t/tau). The current is I(t) = -(Q_0/RC) e^(-t/tau) = -I_0 e^(-t/tau), where the negative sign indicates that the current flows in the opposite direction to the charging current. The voltage across the capacitor is V_C(t) = V_0 e^(-t/tau).

All quantities decay exponentially with the same time constant tau = RC. At t = tau, the charge and voltage have fallen to 36.8% of their initial values. The current is largest at the start and decreases as the capacitor discharges. The half-life of the decay is t_1/2 = tau ln(2) = 0.693 RC, which is the time for the charge to fall to half its initial value.

### IV. Energy in RC Circuits

**During charging**, the battery supplies a total energy of E_battery = Q_max epsilon = C epsilon^2. Of this, exactly half is stored in the capacitor (U_C = (1/2) C epsilon^2) and exactly half is dissipated as heat in the resistor (E_R = (1/2) C epsilon^2). This remarkable result holds regardless of the value of R: a larger resistance means slower charging, but the same total heat is produced.

**During discharging**, all the energy initially stored in the capacitor is dissipated as heat in the resistor: E_R = U_C,initial = (1/2) C V_0^2. The instantaneous power dissipated is P(t) = I^2 R = (V_0^2/R) e^(-2t/tau). Integrating this from t = 0 to infinity confirms that the total energy dissipated equals (1/2) C V_0^2.

### V. Time Constant and Circuit Behavior

The time constant tau = RC controls the speed of the transient response. A large RC means slow charging and discharging, because a large capacitor stores more charge and a large resistor limits the current. A small RC means fast charging and discharging.

A practical rule of thumb provides a quick assessment of circuit progress: after 1 tau, the process is approximately 63% complete; after 2 tau, 86%; after 3 tau, 95%; and after 5 tau, 99.3%, at which point the circuit is essentially at steady state.

In circuit analysis, it is often useful to consider the extreme behavior of capacitors. At t = 0, an uncharged capacitor behaves like a **short circuit** (a wire), because it has no voltage across it and does not impede the current. At t = infinity, a fully charged capacitor behaves like an **open circuit**, because it blocks all DC current once it is fully charged.

<image>A table showing the percentage of completion for charging and discharging at multiples of the time constant tau. Columns: Time (t/tau), e^(-t/tau), Charge during charging (% of Q_max), Charge during discharging (% of Q_0). Rows for t/tau = 0, 1, 2, 3, 4, 5. Values: at t=0: 1.000, 0%, 100%; at t=tau: 0.368, 63.2%, 36.8%; at t=2tau: 0.135, 86.5%, 13.5%; at t=3tau: 0.050, 95.0%, 5.0%; at t=5tau: 0.007, 99.3%, 0.7%. A visual bar graph accompanies the table.</image>

### VI. Applications of RC Circuits

RC circuits are ubiquitous in electronics and biology. In **timing circuits**, the RC time constant sets precise delay times, as used in 555 timer ICs, camera flash circuits, and turn signals.

In **filtering** applications, RC circuits selectively pass or block signals based on frequency. A **low-pass filter** passes slow (low-frequency) signals and blocks high-frequency ones; the output is taken across the capacitor, with a cutoff frequency of f_c = 1/(2 pi RC). A **high-pass filter** passes high-frequency signals and blocks low-frequency (DC) signals; the output is taken across the resistor.

**Signal smoothing** uses capacitors to reduce voltage fluctuations in power supplies. **Debouncing** circuits use the RC time constant to prevent multiple triggering from the mechanical bouncing of switches.

A fascinating biological analogy exists as well: the cell membrane acts as a capacitor, with ion channels serving as resistors. The membrane time constant determines how quickly neurons respond to stimuli, with typical neuronal time constants of roughly 10-20 ms. **Defibrillators** use the RC circuit principle directly: a capacitor is charged to high voltage and then discharged through the patient's chest, with the RC time constant determining the pulse duration of approximately 5-10 ms.