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The charging process of a capacitor involves making it store energy by connecting it to a power source or an electromotive force (emf) through a circuit. Initially, the capacitor has no charge. Once connected, the capacitor begins to accumulate charge because of the voltage source, giving rise to a current in the circuit.
Saturating the capacitor, or fully charging it, requires the full battery voltage to be applied across it. As the charge builds up on the capacitor plates, the voltage across its terminals also increases to equal the source voltage. The flow of current diminishes over time, following an exponential decay, until it reaches zero. At this point, the capacitor is fully charged, and no more current flows in the circuit.

When charging a capacitor through a resistor, not all supplied energy is stored. Some of the energy dissipates as thermal energy, primarily by the resistor, during the charging process. This phenomenon often leads to inefficiencies in electrical circuits.
In simple terms, while charging, the energy from the emf device is divided equally: half is stored in the capacitor, while the other half dissipates as heat in the resistor. During the charging phase, the resistor converts some of the electrical energy into heat due to the current flowing through it. This conversion is a significant feature of any real-world circuit utilizing resistors.

In circuits, a resistor in series with a capacitor functions as a crucial component to control the charging rate of the capacitor. This is because a resistor limits the amount of current entering the capacitor, essentially dictating how fast the capacitor charges.
The presence of this resistor leads to a characteristic charging curve described by an exponential function. Without such resistance, the capacitor would charge almost instantaneously, which might not be desired in many practical applications. Additionally, this resistor helps prevent potential damage by regulating the electrical current and ensuring safe operation.

An exponential profile characterizes the rate at which a capacitor charges or discharges. As the charge accumulates, the process begins rapidly and then slows down exponentially over time.
Mathematically, this relationship is depicted using the exponential function: the current decreases exponentially, \[ i(t) = \frac{\mathscr{E}}{R} e^{-t/RC} \]where \(i(t)\) is the instantaneous current, \(\mathscr{E}\) is the emf, \(R\) is the resistance, and \(RC\) is the time constant. The time constant \(\tau = RC\) is crucial as it determines the speed of the charging and discharging processes. When \(t = RC\), the capacitor is about 63% charged. As time proceeds to infinity, the capacitor asymptotically approaches its full charge.