The charging of a capacitor is the process of storing electrical energy in an electrostatic field by transferring charge onto its plates. When an uncharged capacitor ($C$) is connected in series with a resistor ($R$) and a DC voltage source ($V_s$), the charging process is not instant but exponential, governed by the time constant ($\tau = RC$). [1, 2, 3]
1. Initial State ($t = 0$)
- Conditions: The capacitor is completely discharged. Voltage across the capacitor ($V_c$) is 0V, and charge ($q$) is 0.
- Behavior: When the switch is closed, the capacitor acts like a short circuit (a simple wire).
- Current: The maximum current flows, determined only by the resistor: $I_{max} = V_s / R$. [2, 4, 5, 6, 7]
2. Charging Phase ($0 < t < 5\tau$)
- Mechanism: Electrons are pulled from the plate connected to the positive terminal of the battery and pushed onto the plate connected to the negative terminal.
- Voltage Growth: As charge ($q$) accumulates, the voltage across the capacitor ($V_c$) increases exponentially, according to the formula:$V_c(t) = V_s(1 - e^{-t/RC})$
- Current Decay: As $V_c$ approaches $V_s$, the net voltage pushing current ($V_s - V_c$) drops. The current decreases exponentially:$I(t) = (V_s/R)e^{-t/RC}$
- Time Constant (): The product $RC$ (in seconds) defines how fast the capacitor charges. After one time constant ($t = RC$), the capacitor reaches ~63.2% of its maximum charge. [1, 5, 8, 9, 10]
3. Final State (Steady State: $t > 5\tau$)
- Conditions: After approximately 5 time constants ($5\tau$), the capacitor is considered fully charged.
- Voltage: $V_c$ is equal to the supply voltage $V_s$.
- Current: No more charge can accumulate, so current ($I$) becomes zero.
- Behavior: The capacitor acts as an open circuit (breaking the circuit). [1, 11, 12, 13]
4. Summary Table of Charging
| Parameter [14, 15, 16, 17, 18] | Start ($t=0$) | During | Final ($t>5\tau$) |
| Voltage ($V_c$) | 0 | Increases ($1-e^{-t/RC}$) | $V_s$ |
| Current ($I$) | $V_s/R$ (Max) | Decreases ($e^{-t/RC}$) | 0 |
| Charge ($q$) | 0 | Increases | $CV_s$ (Max) |
| Component | Short Circuit | - | Open Circuit |
Key Takeaways
- Energy Storage: The capacitor stores energy as a potential difference between its plates, equal to $E = \frac{1}{2}CV_s^2$.
- Role of R and C: A larger resistance ($R$) or larger capacitance ($C$) slows down the charging process (increases the time constant).
- Energy Loss: During the charging process, half of the total energy supplied by the source is dissipated in the resistor, while only half is stored in the capacitor. [1, 11, 19, 20]
AI responses may include mistakes.
[14] https://study.com/learn/lesson/capacitor-charging-discharging-construction-equations-examples.html
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