Charging and Discharging of a Capacitor in an RC Circuit: Formulas, Graphs, and Applications
The charging and discharging of capacitor is a core phenomenon in JEE Main Physics, describing how capacitors store and release energy in electrical circuits. Everyday uses are everywhere—camera flashes, power backup circuits and even signal processing. Understanding this topic is essential for analyzing any RC circuit, solving fast-decay electronics numericals, and performing practicals in the Physics lab.
When a capacitor is connected to a battery via a resistor, its charge builds up over time, not instantly. Likewise, when disconnected from the supply and closed through a resistor, it slowly releases stored charge. This gradual change is what 'charging' and 'discharging' mean in Physics and circuit theory.
Charging and Discharging of Capacitor: Core Ideas and Definitions
Let’s clarify key terms needed for JEE Main questions on charging and discharging of capacitor and related circuits.
- Capacitor (C): Device that stores electric charge and energy.
- RC circuit: Circuit with resistor (R) and capacitor (C) in series.
- Time constant (τ = RC): Time to charge up to 63% or discharge to 37% of maximum.
- Charging: Capacitor accumulates charge from applied voltage.
- Discharging: Stored charge leaves the capacitor through the resistor.
- Both processes follow exponential laws important in current electricity analysis.
For detailed background, check electrostatic potential and capacitance and current electricity mock test pages on Vedantu.
Charging of a Capacitor in RC Circuits
When a capacitor connects to a cell through a resistor, charging is not instantaneous. Charge builds up according to an exponential curve dictated by the RC time constant.
The key equations for the charging process are:
| Parameter | Charging Equation |
|---|---|
| Charge on capacitor Q(t) | \( Q(t) = Q_{max}\left(1 - e^{-t/RC}\right) \) |
| Current in circuit I(t) | \( I(t) = \frac{V}{R}e^{-t/RC} \) |
| Voltage across capacitor VC(t) | \( V_C(t) = V\left(1 - e^{-t/RC}\right) \) |
Here, V is the supply voltage, Qmax = CV, R is resistance, C is capacitance, and t is time. The exponential increase is crucial for signal delay and filtering circuits in electronics.
- At t = 0, Q = 0, I is maximum.
- As t increases, I drops and Q rises until full charge CV is reached.
- The RC circuit’s time constant decides the charging speed.
Graphically, the charging curve starts steep and flattens off as it nears maximum charge, never overshooting the applied voltage.
Discharging of a Capacitor: Process and Equations
If you disconnect the supply and complete the circuit with just the resistor and charged capacitor, the energy stored gets released. This is the discharging of capacitor process.
The main equations (for initial charge Q₀ = CV) are:
| Parameter | Discharging Equation |
|---|---|
| Charge remaining Q(t) | \( Q(t) = Q_{0}e^{-t/RC} \) |
| Current I(t) | \( I(t) = -\frac{Q_0}{RC}e^{-t/RC} \) |
| Voltage across capacitor VC(t) | \( V_C(t) = V_0 e^{-t/RC} \) |
Charge, voltage, and current all decrease exponentially. At t = RC (one time constant), only about 37% of initial value remains.
- Discharge is complete after several time constants (usually 5RC).
- No external supply is needed—discharge is due to stored energy.
- These principles govern electrostatics experiments and safety procedures.
Comparing Charging and Discharging of Capacitor for JEE
| Aspect | Charging | Discharging |
|---|---|---|
| Direction of current | From battery towards capacitor | From capacitor through resistor |
| Charge behavior | Increases to CV | Decreases to zero |
| Equation | Q = Qmax(1 - e-t/RC) | Q = Q0e-t/RC |
| Graph shape | Exponential rise | Exponential decay |
| Initial value at t = 0 | Q = 0 | Q = Q0 |
If you need distinction in analyzing inductor and capacitor differences or for MCQs, the above table helps avoid common mistakes.
Capacitor Charging and Discharging Practical: JEE Physics Lab
JEE Main and Class 12 students often perform this as a core practical. Here’s a quick procedure relevant for exam writeups:
- Set up an RC series circuit on a breadboard with capacitor, resistor, battery, and switch.
- Close the switch to start charging. Monitor voltage across the capacitor using a voltmeter at various intervals.
- Plot VC vs. time to observe exponential growth.
- Open battery, close secondary switch across capacitor for discharging. Record voltage drop over time.
- Repeat to confirm the exponential decay pattern on a graph.
Refer to experimental skills mock test for more lab scenarios.
Numerical Example: RC Time Constant Application
A 20 μF capacitor is charged in series with a 100 kΩ resistor and 6 V battery. Find the time to charge to 99% of maximum charge.
- RC = (100 × 103)(20 × 10-6) = 2 s.
- 99% charge ⇒ Q/Qmax = 0.99. Use \( Q = Q_{max}(1 - e^{-t/RC}) \).
- 0.99 = 1 − e−t/2 ⇒ e−t/2 = 0.01.
- −t/2 = ln(0.01) → t = −2 × ln(0.01)
- t = 2 × 4.605 = 9.21 s
Final answer: The capacitor charges to 99% in 9.21 seconds. Accuracy with exponentials and time constants is critical for JEE scoring.
More on Charging and Discharging of Capacitor: Exam Tips and Real-World Uses
- Always check units—μF and kΩ must be in SI for RC.
- Short-circuiting a charged capacitor directly is dangerous. Always discharge slowly with large resistors.
- Relevant in electric circuits, filters, timer circuits, and high-speed cameras.
- RC time determines speed of digital and analog devices.
- Capacitor never charges instantly; exponential law always applies due to Ohm’s law and circuit resistance.
- Practice advanced questions using mock tests and practice paper.
For deep dives, refer to the RL and RC circuits page and all related Vedantu resources. Their solutions are trusted for JEE conceptual clarity.
- Understand difference between series and parallel circuit behavior for combined resistances.
- Connect findings with current and potential difference in other electric topics.
- Application expands to AC circuits and inductive charging setups.
Be consistent with exponential notation and keep formulas ready for both charging and discharging of capacitor in quick-revision notes—essential for high JEE Main scores.
FAQs on Charging and Discharging of Capacitor Explained
1. What is the charging and discharging of a capacitor?
Charging and discharging of a capacitor refer to the processes where a capacitor stores or releases electrical energy in an RC circuit.
Charging occurs when a capacitor accumulates charge from a source, increasing its voltage, while discharging is when stored energy flows back into the circuit, decreasing the voltage.
Key points:
- Involves energy storage and release
- Described by exponential curves and time constant (τ = RC)
- Essential for understanding RC circuits in both theory and practical physics
2. What is the formula for charging and discharging of a capacitor?
The formulas describe how charge and voltage change over time in an RC circuit for both charging and discharging.
Charging:
- Charge: Q(t) = Qmax(1 - e-t/RC)
- Voltage: V(t) = V0(1 - e-t/RC)
- Charge: Q(t) = Q0e-t/RC
- Voltage: V(t) = V0e-t/RC
- R = resistance, C = capacitance, t = time, Q = charge, V = voltage
3. How do you know if a capacitor is charging or discharging?
A capacitor is charging when its voltage and charge increase towards the supply voltage, and discharging when these quantities decrease.
Observation points:
- During charging, the voltage across the capacitor rises exponentially.
- During discharging, the voltage and charge drop exponentially.
- Direction of current in the circuit helps indicate the process.
- Use a voltmeter to observe the voltage change over time.
4. What's the difference between charging and discharging of a capacitor?
The main difference is the direction of charge movement and how energy is processed in the RC circuit.
- Charging: Capacitor stores energy; voltage and charge increase to a maximum.
- Discharging: Capacitor releases stored energy; voltage and charge decrease towards zero.
- Both processes follow exponential laws but have opposite trends.
- Time constant (τ = RC) defines the speed of both processes.
5. What is an RC time constant and why is it important?
RC time constant (τ = RC) is the product of the circuit's resistance (R) and capacitance (C). It represents the time required for the capacitor to charge or discharge about 63% of the total change.
Importance:
- Determines the rate of charging/discharging.
- Helps predict capacitor behavior in circuits.
- Relevant for experiments and exam numericals.
6. What is the proper way to discharge a capacitor safely?
To discharge a capacitor safely, connect a resistor across the capacitor’s terminals and wait for the charge to drop.
- Use a high-resistance resistor to control discharge speed.
- Never short the terminals directly as it can be dangerous.
- Confirm zero voltage before handling the capacitor.
7. Can a capacitor charge instantaneously?
No, a capacitor cannot charge instantaneously due to the presence of resistance in real circuits.
- The charging follows an exponential curve defined by the time constant τ = RC.
- At t = 0, current is maximum; charge and voltage gradually build up over time.
8. How does the graph of charging and discharging of a capacitor look like?
The graphs for charging and discharging of a capacitor in an RC circuit are exponential curves.
- Charging graph: Starts at zero and asymptotically approaches maximum value (Qmax or V0).
- Discharging graph: Starts from maximum and decays exponentially toward zero.
- Both can be represented as Q(t) or V(t) versus time.
9. What are real-life applications of capacitor charging and discharging?
Capacitor charging and discharging play a key role in various daily and technological applications.
- Camera flashes
- Timing circuits (e.g., timers, blinkers)
- Pulse generation in oscillators
- Signal filtering in electronics
- Backup power systems
10. How is the charging and discharging of a capacitor demonstrated in a class 12 physics experiment?
In a typical CBSE class 12 physics lab, the charging and discharging of a capacitor are demonstrated using an RC circuit.
- Set up a circuit with a resistor, capacitor, DC power supply, switch, and voltmeter.
- Observe the voltage across the capacitor as it charges (switch closed), recording values at intervals.
- Open the switch to allow the capacitor to discharge, again noting voltage changes over time.
- Plot graphs of voltage vs. time for both processes and verify the exponential nature.
