1. Initial Charging: When the circuit is first connected, the capacitor will begin to charge. The resistor limits the flow of current into the capacitor, causing the voltage across the capacitor to gradually increase. The rate at which the capacitor charges depends on the values of the resistor and capacitor.
2. Exponential Voltage Rise: The voltage across the capacitor will rise exponentially towards the voltage of the DC source. The time constant of the circuit, which is determined by the resistance and capacitance, governs the rate of this voltage rise. The voltage across the capacitor can be calculated using the following equation:
```
Vc(t) = V_Source * (1 - e^(-t/RC))
```
where:
- Vc(t) is the voltage across the capacitor at time t
- V_Source is the voltage of the DC source
- R is the resistance
- C is the capacitance
- t is the time elapsed since the circuit was connected
3. Current Flow: As the capacitor charges, current flows through the resistor. The initial current is high and gradually decreases as the capacitor voltage approaches the source voltage. The current can be calculated using Ohm's law:
```
I = (V_Source - Vc(t)) / R
```
where:
- I is the current flowing through the resistor
- V_Source is the voltage of the DC source
- Vc(t) is the voltage across the capacitor at time t
- R is the resistance
4. Steady State: Eventually, the capacitor will reach its maximum voltage, which is equal to the voltage of the DC source. At this point, the current flow through the resistor will become zero, and the circuit will reach a steady state. The capacitor acts as an open circuit in this state, blocking the flow of DC current.
5. Discharging: If the DC source is disconnected or the circuit is opened, the capacitor will begin to discharge through the resistor. The voltage across the capacitor will decrease exponentially, and the current will flow in the opposite direction. The time constant of the circuit will again determine the rate of discharge.
By understanding the behavior of a resistor-capacitor circuit in a DC circuit, engineers can design and analyze electronic circuits involving capacitors and resistors, such as RC filters, timing circuits, and power supply circuits.