How does the charging and discharging of a capacitor in an RC circuit affect the voltage across it?

Charging of a Capacitor:

When a capacitor is connected to a voltage source (such as a battery) through a resistor in an RC circuit, the capacitor starts to charge. Initially, if the capacitor is uncharged, it acts like a short circuit, and the entire voltage of the source appears across it. As time passes, the capacitor charges up, and its voltage gradually increases towards the source voltage. During the charging process, the voltage across the capacitor (Vc) increases from zero volts to the source voltage (Vs) following an exponential curve.

The charging of a capacitor in an RC circuit is governed by the following equation:

Vc(t) = Vs * (1 - e^(-t / RC))

where:

Vc(t) is the voltage across the capacitor at time t.

Vs is the voltage of the source (battery).

e is the mathematical constant approximately equal to 2.71828.

t is time (in seconds) since the charging process started.

RC is the time constant of the circuit, equal to the product of the resistance (R) and the capacitance (C).

The time constant (RC) determines how quickly the capacitor charges. A larger RC value (either a larger resistance or a larger capacitance) results in slower charging, while a smaller RC value leads to faster charging.

Discharging of a Capacitor:

If a charged capacitor is disconnected from the voltage source and then connected to a resistor in an RC circuit (with the other terminal of the resistor connected to a reference point, like ground), it starts to discharge. Initially, the voltage across the capacitor is equal to its fully charged voltage (let's call this V0), and as time passes, the voltage across the capacitor decreases towards zero volts following an exponential curve.

The discharging of a capacitor in an RC circuit is governed by the following equation:

Vc(t) = V0 * e^(-t / RC)

where:

Vc(t) is the voltage across the capacitor at time t during discharging.

V0 is the initial voltage across the capacitor at the beginning of discharging.

e is the mathematical constant approximately equal to 2.71828.

t is time (in seconds) since the discharging process started.

RC is the time constant of the circuit, as described before.

Similar to the charging process, the time constant (RC) determines how quickly the capacitor discharges. A larger RC value (either a larger resistance or a larger capacitance) results in slower discharging, while a smaller RC value leads to faster discharging.

In summary, the charging and discharging of a capacitor in an RC circuit have a profound effect on the voltage across the capacitor. During charging, the voltage across the capacitor rises from zero to the source voltage, while during discharging, it falls from its initial voltage (V0) to zero. The time constant (RC) of the circuit governs the rate at which these changes occur.