When a DC (direct current) supply is connected to a capacitor, the capacitor undergoes a transient process known as charging. The behavior of the capacitor during this process is determined by its capacitance, the applied voltage, and the resistance in the circuit. Let’s explore in detail what happens when a DC supply is connected to a capacitor:

**1. Initial Conditions:**

**Uncharged Capacitor:**- Initially, if the capacitor is uncharged, it acts as a short circuit, allowing current to flow easily.

**Voltage Difference:**- At the moment of connection, the voltage across the capacitor is zero.

**2. Charging Process:**

**Capacitor Charging Equations:**- The voltage across the capacitor (��Vc) during the charging process can be described by the equation ��=�0×(1−�−�/(��))Vc=V0×(1−e−t/(RC)), where �0V0 is the applied voltage, �t is time, �R is the resistance in the circuit, and �C is the capacitance of the capacitor.

**Exponential Charging:**- The charging process is exponential in nature. Initially, the capacitor charges rapidly, but the rate of charging decreases over time.

**3. Time Constant (��RC):**

**Time Constant Definition:**- The time constant (��RC) is a key parameter in the charging process, representing the time required for the voltage across the capacitor to reach approximately 63.2% of its final value.

**Significance:**- A smaller time constant results in faster charging, while a larger time constant leads to a slower charging process.

**4. Capacitor Voltage and Current:**

**Voltage Across Capacitor:**- As time progresses, the voltage across the capacitor increases, approaching the applied voltage (�0V0).

**Current Flow:**- Initially, the current flowing through the circuit is at its maximum, and it gradually decreases as the capacitor charges.

**5. Steady State:**

**Fully Charged Capacitor:**- In the steady state, the capacitor becomes fully charged, and the voltage across it equals the applied voltage (�0V0).

**No Current Flow:**- Once fully charged, the capacitor acts as an open circuit, and no current flows through it.

**6. Charging Characteristics:**

**Dependence on Resistance and Capacitance:**- The time constant (��RC) determines the charging characteristics, and it depends on both the resistance and capacitance values in the circuit.

**Higher Resistance:**- A higher resistance leads to a longer charging time.

**Higher Capacitance:**- A higher capacitance also results in a longer charging time.

**7. Energy Storage:**

**Energy Stored in the Electric Field:**- During the charging process, energy is stored in the electric field between the capacitor plates.

**Stored Energy Equation:**- The energy (�W) stored in a capacitor is given by the equation �=12��2W=21CV2, where �C is the capacitance and �V is the voltage across the capacitor.

**8. Applications:**

**Energy Storage and Timing:**- Capacitors are commonly used in electronic circuits for energy storage, timing, filtering, and various other applications.
**Filtering Capacitors:**- In power supply circuits, capacitors smooth out variations in voltage.

**9. Discharging Process:**

**Discharging Equation:**- If the capacitor is initially charged and is then connected to a resistor, it undergoes a discharging process described by ��=�0×�−�/(��)Vc=V0×e−t/(RC).

**Exponential Discharging:**- The voltage across the capacitor decreases exponentially over time during discharging.

**Conclusion:**

Connecting a DC supply to a capacitor initiates the charging process, where the capacitor gradually accumulates charge and stores energy. The charging behavior is characterized by an exponential increase in voltage over time, influenced by the time constant, resistance, and capacitance in the circuit. Understanding these charging dynamics is essential in the design and analysis of electronic circuits involving capacitors.