## 19.6 CHARGING AND DISCHARGING OF A CAPACITOR This section describes the behavior of a capacitor in a simple RC (resistor-capacitor) circuit during charging and discharging. The processes are not linear but follow an exponential pattern, governed by the circuit's time constant. ### Charging a Capacitor When a capacitor is connected to a power source (such as a battery with e.m.f. $\varepsilon$) through a resistor, it begins to charge. **Process:** Electrons flow from the battery to one plate of the capacitor and away from the other plate. This creates a potential difference across the capacitor. **Current and Voltage:** Initially, the current is at its maximum, and the voltage across the capacitor is zero. As charge accumulates, it repels incoming charges, causing the current to decrease. The voltage across the capacitor increases. **Completion:** Charging stops when the potential difference across the capacitor equals the e.m.f. of the battery ($\varepsilon$). At this point, the current is zero, and the capacitor holds its maximum charge, $Q_0$. **Charging Equation:** The charge $Q$ on the capacitor at any time $t$ is given by the exponential formula: $$Q = Q_0\left(1 - e^{-t/RC}\right)$$ where $Q_0$ is the maximum charge and $e \approx 2.7182$. ### Discharging a Capacitor When a fully charged capacitor is connected across a resistor, it begins to discharge. **Process:** The stored charge flows from the positive plate to the negative plate through the resistor until the potential difference across the capacitor becomes zero. **Current:** The current is highest at the start of the discharge and exponentially decreases to zero. **Discharging Equation:** The charge $Q$ remaining on the capacitor at any time $t$ is given by: $$Q = Q_0 e^{-t/RC}$$ where $Q_0$ is the initial charge at $t = 0$. ### Time Constant ($\tau$) The **time constant** is a crucial parameter in RC circuits that determines the rate of charging and discharging. **Definition:** It is the product of resistance ($R$) and capacitance ($C$). $$\tau = RC$$ **Significance in Charging:** It is the time taken for the capacitor to charge to **63.2%** of its maximum value. - At $t = RC$, $Q = Q_0(1 - e^{-1}) \approx 0.632Q_0$. **Significance in Discharging:** It is the time taken for the charge on the capacitor to fall to **36.7%** of its initial value. - At $t = RC$, $Q = Q_0 e^{-1} \approx 0.367Q_0$. **Effect on Rate:** A smaller time constant (smaller $R$ or $C$) results in a faster charging and discharging rate. ### General Equations The exponential equations for charging and discharging can be generalized to represent not just charge ($Q$), but also current ($I$) or potential difference ($V$). **For Charging:** $$x = x_0\left(1 - e^{-t/RC}\right)$$ **For Discharging:** $$x = x_0 e^{-t/RC}$$ Here, $x$ can be $Q$, $I$, or $V$, and $x_0$ is the maximum value of that quantity. | Process | Formula for Charge ($Q$) | General Formula (for $Q$, $I$, $V$) | Time Constant ($\tau$) | | :--- | :--- | :--- | :--- | | **Charging** | $Q = Q_0(1 - e^{-t/RC})$ | $x = x_0(1 - e^{-t/RC})$ | $\tau = RC$ | | **Discharging** | $Q = Q_0 e^{-t/RC}$ | $x = x_0 e^{-t/RC}$ | $\tau = RC$ | RC circuits are fundamental in electronics, used in timing circuits, filters, and oscillators. For example, they are used in Capacitor Discharge Ignition (CDI) systems in vehicles.