Charging of capacitor and inductor

Edited By Vishal kumar | Updated on Jul 02, 2025 07:49 PM IST

The charging of capacitors and inductors plays a crucial role in understanding the behaviour of electrical circuits, particularly in energy storage and conversion. A capacitor stores energy in an electric field, while an inductor stores energy in a magnetic field. These components are essential in devices like cameras, where a capacitor helps store energy for the flash, or in power supplies, where inductors smooth out voltage variations. In real life, charging a capacitor is similar to filling a water tank—initially fast, but slowing down as it nears full capacity. Similarly, charging an inductor involves building up a magnetic field gradually, much like winding up a spring. These principles are fundamental to various applications, from smartphones to electric vehicles, showcasing their significance in modern technology.

This Story also Contains

Charging of a Capacitor
Discharging of Capacitors
Charging and Discharging of an Inductor
Solved Examples Based on Charging of capacitor and inductor
Summary

Charging of capacitor and inductor

Charging of a Capacitor

When a capacitor with zero charges is connected to a battery of emf V through connecting wires, the total resistance including internal resistance of the battery and of the connecting wires is R then after a time t let the charge on the capacitor be q, current be i and $V_c=\frac{q}{C}$,

charge deposited on the positive plate in time $d t$ is so that

$
\begin{aligned}
& d q=i d t \\
& i=\frac{d q}{d t}
\end{aligned}
$
Using Kirchhoffs loop law, $\quad \frac{q}{C}+R i-V=0$
or,

$
R i=E-\frac{q}{C}
$

or, $\quad R \frac{d q}{d t}=\frac{V C-q}{C}$
or, $\quad \int_0^q \frac{d q}{V C-q}=\int_0^t \frac{1}{C R} d t$
or, $\quad-\ln \frac{V C-q}{V C}=\frac{t}{C R}$
or, $\quad 1-\frac{q}{V C}=e^{-t / C R}$
or, $\quad q=V C\left(1-e^{-t / C R}\right)$

Where RC is the time constant $(\tau)$ of the circuit.
At $t=\tau=R C$
$q=C V\left(1-\frac{1}{e}\right)=0.63 C V$

Discharging of Capacitors

If initially a capacitor has a charge Q and is discharged through an external load. Let after a time t the remaining charge in the capacitor be q then

Using Kirchhoff's loop law,

$
\frac{q}{C}-R i=0
$
Here $i=-\frac{d q}{d t}$ because the charge $q$ decreases as time passes.
Thus, $\quad R \frac{d q}{d t}=-\frac{q}{C}$
or, $\quad \frac{d q}{q}=-\frac{1}{C R} d t$
or, $\quad \int_Q^q \frac{d q}{q}=\int_0^t-\frac{1}{C R} d t$
or, $\quad \ln \frac{q}{Q}=-\frac{t}{C R}$
or, $\quad q=Q e^{-t / C R}$

Note: A steady-state capacitor connected to the DC battery acts as an open circuit. The capacitor does not allow a sudden change in voltage.

Charging and Discharging of an Inductor

When an inductor is connected to a DC source of emf v through a resistance R the inductor charges to maximum current $\left(i_0=\frac{V}{R}\right)$ at steady state. If the inductor current is increased from zero at time $=0$ to i at times t then-current i is given by $i=i_0\left(1-e^{\frac{-t}{T}}\right)$ where $\tau=\frac{L}{R}$

$\tau$ is the time constant of the circuit. Here the current is exponentially increasing.
While the discharging of inductor current decreases exponentially and is given by

$
i=i_0\left(e^{\frac{-t}{\tau}}\right)
$

Note: At steady state, an inductor connected to the DC battery acts as a short circuit. The inductor does not allow a sudden change in current.

Solved Examples Based on Charging of capacitor and inductor

Example 1: Two identical capacitors A and B, charged to the same potential 5 V are connected in two different circuits as shown below at time $\mathrm{t}=0$. If the charge on capacitors A and B at time $\mathrm{t}=\mathrm{CR}$ is $Q_A$ and $Q_B$ respectively, then (Here e is the base of natural logarithm)

1) $Q_A=V C, Q_B=C V$
2) $Q_A=\frac{C V}{2}, Q_B=\frac{V C}{e}$
3) $Q_A=\frac{V C}{e}, Q_B=\frac{C V}{2}$
4) $Q_A=V C, Q_B=\frac{V C}{e}$

Solution:

Maximum charge on the capacitor $=5 \mathrm{CV}$

(a) is reverse-biased and (b) is forward-biased

(a) (b)

So, $q=q_{\max }\left[e^{-t / R C}\right]$

$
Q_B=\frac{C V}{e}
$

Hence the correct option is (4).

Example 2:

As shown in the figure, a battery of emf $\epsilon$ is connected to an inductor $L$ and resistance $R$ in series. The switch is closed at $t=0$. The total charge that flows from the battery, between $t=0$ and $t=t_c$ ( $\mathrm{t}_{\mathrm{c}}$ is the time constant of the circuit ) is :

1) $\frac{\epsilon L}{R^2}\left(1-\frac{1}{e}\right)$
2) $\frac{\epsilon L}{R^2}$
3) $\frac{\epsilon R}{e L^2}$
4) $\frac{\epsilon L}{e R^2}$

Solution:

As current at any time $t$ is given as

$
\begin{aligned}
& I=I_0\left(1-e^{\frac{-t}{T}}\right)=\frac{\epsilon}{R}\left(1-e^{\frac{-t}{T}}\right) \\
& \text { So } \mathrm{q}=\int_0^{\mathrm{T}_{\mathrm{C}}} \text { idt }
\end{aligned}
$
So integrating this will give a total charge

$
\begin{aligned}
& q=\frac{\varepsilon}{\mathrm{R}}\left[\mathrm{t}-\frac{\mathrm{e}^{-t / \tau}}{\frac{-1}{\tau}}\right]_0^\tau ;=\frac{\varepsilon}{\mathrm{R}}\left[\tau+\tau \mathrm{e}^{-1}-\tau\right] \\
& q=\frac{\varepsilon}{\mathrm{R}} \times \frac{1}{\mathrm{e}} \times \frac{\mathrm{L}}{\mathrm{R}} ; q=\frac{\varepsilon \mathrm{L}}{\mathrm{R}^2 \mathrm{e}}
\end{aligned}
$
Hence the correct option is (4).

Example 3 : The current (i) at time $t=0$ and $t=\infty$ respectively for the given circuit is:

1) $\frac{5 E}{18}, \frac{10 E}{33}$
2) $\frac{10 E}{33}, \frac{5 E}{18}$
3) $\frac{5 E}{18}, \frac{18 E}{33}$
4) $\frac{18 E}{55}, \frac{5 E}{18}$

Solution:

At t = 0, the inductor is open

So the corresponding equivalent circuit is given below

$
\begin{aligned}
& \mathrm{R}_{\mathrm{eq}}=\frac{6 \times 9}{6+9}=\frac{54}{15} \\
& I(\text { at } t=0)=\frac{15 E}{54}=\frac{5 E}{18}
\end{aligned}
$
At $t=\infty_{\text {, }}$ For steady state inductor is replaced by plane wire

So the corresponding equivalent circuit is given below

We can reduce the above circuit to the below circuit.

$\begin{aligned} & \mathrm{R}_{\mathrm{eq}}=\left(\frac{1 \times 4}{1+4}\right)+\left(\frac{5 \times 5}{5+5}\right)=\frac{4}{5}+\frac{5}{2}=\frac{8+25}{10}=\frac{33}{10} \\ & \mathrm{I}=\frac{\mathrm{E}}{\mathrm{R}_{\mathrm{eq}}}=\frac{10 \mathrm{E}}{33}\end{aligned}$

Hence the correct option is (1).

Example 4: Figure shows a circuit that contains four identical resistors with resistance $R=2.0 \Omega$, two identical inductors with inductance $L=2.0 \mathrm{mH}$ and an ideal battery with emf $E=9 \mathrm{~V}$. The current 'i' just after the switch 's' is closed will be :

1) 9 A

2) 2.25 A

3) 3.0 A

4) 3.37 A

Solution:

Just when switch S is closed, the inductor will behave like an infinite resistance. Hence, the circuit will be like

Given: $\mathrm{V}=9 \mathrm{~V}$
From $\mathrm{V}=\mathrm{IR}$
$\mathrm{I}=\mathrm{V} / \mathrm{R}$
Req. $=2+2=4 \Omega$
$\mathrm{i}=9 / 4=2.25 \mathrm{~A}$

Hence the correct option is (2).

Example 5: In the following figure, the charge (in $\mu C$ ) on each condenser in the steady-state will be

1) 12

2) 6

3) 9

4) 3

Solution:

Charging Of Capacitors
$
Q=Q_0\left(1-e^{\frac{-t}{R_c}}\right)
$

wherein
In the R-C circuit (Transient State).

$
i=\frac{10}{(4+1)}=2 \mathrm{amp}
$

.The potential difference across 4 W resistance is $V=2 * 4=8$ volt
In steady-state current flows through 4-ohm resistance only and it is Hence, the potential difference across each capacitor is 4 V

So charge on each capacitor $Q=3 * 4=12 \mu C$

Summary

The charging and discharging of capacitors and inductors follow exponential processes that are essential in various circuits. A capacitor charges by storing energy in an electric field and discharges through a gradual release of this energy, while an inductor stores energy in a magnetic field and behaves similarly in its charging and discharging process. The time constant, denoted as RC for capacitors and L/R for inductors, determines the rate of these processes. These principles are critical in many real-world applications, from power supplies to electronic devices.

Frequently Asked Questions (FAQs)

1. What is an inductor and how does it store energy?

An inductor is an electrical component that stores energy in a magnetic field. It typically consists of a coil of wire. When current flows through the inductor, it creates a magnetic field. The energy is stored in this magnetic field and can be released when the current changes.

2. How does the current through an inductor change when a voltage is applied?

When a constant voltage is applied to an inductor, the current does not change instantaneously. Instead, it increases gradually, following an exponential curve. This is due to the inductor's property of resisting changes in current, known as self-inductance.

3. What is self-inductance and why is it important?

Self-inductance is the property of an inductor that opposes changes in the current flowing through it. It's important because it determines how quickly current can change in a circuit containing inductors. Higher self-inductance means greater opposition to current changes, which can affect circuit behavior, especially in AC circuits or during transient events.

4. How does the energy stored in an inductor relate to its current?

The energy (E) stored in an inductor is given by E = ½LI², where L is the inductance and I is the current through the inductor. This shows that the energy is proportional to the square of the current, meaning doubling the current quadruples the stored energy.

5. What is the time constant of an RL circuit, and how does it compare to an RC circuit?

The time constant (τ) of an RL circuit is L/R, where L is the inductance and R is the resistance. Like in RC circuits, it represents the time taken for the current to reach about 63.2% of its final value. The key difference is that in RL circuits, the current grows exponentially, while in RC circuits, the voltage grows exponentially.

6. What is a capacitor and how does it store charge?

A capacitor is an electrical component that stores energy in an electric field. It consists of two conducting plates separated by an insulating material (dielectric). When a voltage is applied, opposite charges accumulate on the plates, creating an electric field in the dielectric. This field stores energy, which can be released when the capacitor is discharged.

7. How does the charging process of a capacitor differ from that of a battery?

A capacitor stores energy in an electric field, while a battery stores energy through chemical reactions. Capacitors charge quickly but have limited capacity, whereas batteries charge more slowly but can store more energy. Capacitors can be charged and discharged rapidly many times without degradation, while batteries have a limited number of charge-discharge cycles.

8. What happens to the current and voltage during capacitor charging?

When charging a capacitor, the current starts at a maximum and decreases exponentially to zero as the capacitor becomes fully charged. The voltage across the capacitor, however, starts at zero and increases exponentially to its maximum value, which is equal to the applied voltage.

9. Why does the current eventually become zero in a fully charged capacitor?

The current becomes zero in a fully charged capacitor because the electric field between the plates reaches its maximum strength, equal to the applied voltage. At this point, there is no potential difference to drive further current flow, and the capacitor acts as an open circuit.

10. How does changing the resistance in an RC circuit affect the charging time?

Increasing the resistance in an RC circuit increases the charging time, while decreasing the resistance reduces the charging time. This is because the time constant (τ = RC) is directly proportional to the resistance. A larger resistance limits the current flow, slowing down the charging process.

11. What is the time constant of an RC circuit, and why is it important?

The time constant (τ) of an RC circuit is the product of resistance (R) and capacitance (C), measured in seconds. It represents the time taken for the capacitor to charge to about 63.2% of its full charge or discharge to about 36.8% of its initial voltage. The time constant is important because it determines the charging and discharging rates of the capacitor.

12. How does the energy stored in a capacitor relate to its charge and voltage?

The energy (E) stored in a capacitor is given by E = ½CV², where C is the capacitance and V is the voltage across the capacitor. This shows that the energy is proportional to the square of the voltage, meaning doubling the voltage quadruples the stored energy.

13. How do capacitors and inductors behave differently in AC and DC circuits?

In DC circuits, capacitors block current flow once fully charged, while inductors act like short circuits after the initial charging period. In AC circuits, capacitors allow high-frequency signals to pass more easily (low impedance at high frequencies), while inductors allow low-frequency signals to pass more easily (low impedance at low frequencies).

14. How do capacitors and inductors affect power factor in AC circuits?

Capacitors and inductors introduce reactive power in AC circuits, which affects the power factor. Capacitors provide leading reactive power (current leads voltage), while inductors provide lagging reactive power (current lags voltage). By balancing capacitive and inductive loads, the power factor can be corrected to improve overall system efficiency.

15. What is meant by the "capacitive reactance" of a capacitor?

Capacitive reactance is the opposition a capacitor offers to alternating current (AC). It is inversely proportional to both frequency and capacitance. In AC circuits, capacitors act as frequency-dependent resistors, with their reactance decreasing as frequency increases.

16. What is mutual inductance and how does it differ from self-inductance?

Mutual inductance is the phenomenon where a changing current in one coil induces a voltage in a nearby coil. It differs from self-inductance in that it involves two separate coils, while self-inductance involves a single coil. Mutual inductance is the basis for transformers and some types of coupling in electronic circuits.

17. How do capacitors and inductors behave in resonant circuits?

In resonant circuits, capacitors and inductors work together to create a frequency-selective response. At the resonant frequency, the reactances of the capacitor and inductor cancel out, resulting in maximum current flow (series resonance) or minimum current flow (parallel resonance). This property is used in filters, oscillators, and tuning circuits.

18. What is meant by the "self-resonant frequency" of a capacitor or inductor?

The self-resonant frequency is the frequency at which a capacitor or inductor resonates with its own parasitic inductance or capacitance, respectively. Above this frequency, capacitors begin to act like inductors, and inductors begin to act like capacitors. This limits the useful frequency range of the component.

19. Why does an inductor resist sudden changes in current?

An inductor resists sudden changes in current due to Lenz's law. When the current through an inductor changes, it induces a voltage that opposes this change. This induced voltage creates a magnetic field that opposes the change in the original magnetic field, effectively resisting rapid current changes.

20. How does the inductance of a coil depend on its physical characteristics?

The inductance of a coil depends on several factors: the number of turns in the coil (more turns increase inductance), the cross-sectional area of the coil (larger area increases inductance), the length of the coil (shorter length increases inductance), and the presence of a magnetic core (a ferromagnetic core significantly increases inductance).

21. What is the resonant frequency of an LC circuit?

The resonant frequency of an LC circuit is the frequency at which the inductive and capacitive reactances are equal in magnitude but opposite in phase. It's given by f = 1 / (2π√(LC)), where L is the inductance and C is the capacitance. At this frequency, the circuit can oscillate with minimal external energy input.

22. What is the difference between series and parallel connections of capacitors?

In a series connection of capacitors, the total capacitance decreases and is less than the smallest individual capacitance. The voltage is divided among the capacitors. In a parallel connection, the total capacitance is the sum of individual capacitances, and all capacitors have the same voltage across them.

23. How do series and parallel connections of inductors differ?

In a series connection of inductors, the total inductance is the sum of individual inductances. The current is the same through all inductors. In a parallel connection, the reciprocal of the total inductance equals the sum of reciprocals of individual inductances, and the voltage is the same across all inductors.

24. What is meant by the "quality factor" (Q factor) of an inductor or capacitor?

The quality factor (Q factor) is a dimensionless parameter that describes how under-damped an oscillator or resonator is. For inductors and capacitors, it represents the ratio of energy stored to energy dissipated per cycle. A higher Q factor indicates lower energy loss relative to the stored energy of the component.

25. How do real capacitors and inductors differ from ideal ones?

Real capacitors have some series resistance (ESR) and inductance, while real inductors have some parallel capacitance and series resistance. These non-ideal characteristics can affect their behavior, especially at high frequencies. Real components also have limitations in terms of maximum voltage, current, and power dissipation.

26. What is the skin effect in inductors and how does it affect their performance?

The skin effect is the tendency of alternating current to flow near the surface of a conductor. In inductors, this effect increases the effective resistance at high frequencies, as the current is confined to a smaller cross-sectional area. This can reduce the Q factor and efficiency of the inductor at high frequencies.

27. How do temperature changes affect capacitors and inductors?

Temperature changes can affect both capacitors and inductors. In capacitors, temperature can change the capacitance value and increase leakage current. For inductors, temperature changes can affect the resistance of the wire and the magnetic properties of the core, potentially altering the inductance and Q factor.

28. What is dielectric breakdown in capacitors and how can it be prevented?

Dielectric breakdown occurs when the electric field in a capacitor exceeds the dielectric strength of the insulating material, causing it to become conductive. This can lead to capacitor failure. Prevention methods include using capacitors rated for higher voltages than required, avoiding voltage spikes, and using protective circuits.

29. How do air-core and iron-core inductors differ in their properties and applications?

Air-core inductors have lower inductance but can operate at higher frequencies with less core loss. They're often used in high-frequency applications. Iron-core inductors have higher inductance due to the magnetic core but are limited to lower frequencies due to core losses. They're commonly used in power supplies and low-frequency applications.

30. What is hysteresis loss in inductors and how does it affect their efficiency?

Hysteresis loss occurs in inductors with ferromagnetic cores due to the energy required to repeatedly magnetize and demagnetize the core material. This loss appears as heat and reduces the efficiency of the inductor. It's more significant at higher frequencies and in materials with wider hysteresis loops.

31. What is the role of capacitors and inductors in power factor correction?

Capacitors and inductors are used for power factor correction in AC power systems. Capacitors are commonly used to correct lagging power factor (typically caused by inductive loads like motors), while inductors can correct leading power factor. By balancing reactive power, they help improve system efficiency and reduce power losses.

32. How do parasitic capacitance and inductance affect circuit performance?

Parasitic capacitance and inductance are unwanted capacitive and inductive effects in electronic components and circuit layouts. They can cause resonances, oscillations, and signal distortions, especially at high frequencies. Minimizing these effects through proper circuit design and component selection is crucial for high-frequency and high-speed circuit performance.

33. What is the difference between electrolytic and ceramic capacitors in terms of their charging characteristics?

Electrolytic capacitors typically have higher capacitance but slower charging times due to their construction. They're polarized and suitable for low-frequency applications. Ceramic capacitors have lower capacitance but can charge and discharge very quickly. They're non-polarized and better suited for high-frequency applications.

34. How does the presence of a capacitor or inductor affect the phase relationship between voltage and current in an AC circuit?

In a purely capacitive circuit, the current leads the voltage by 90 degrees. In a purely inductive circuit, the current lags the voltage by 90 degrees. In real circuits with resistance, the phase difference is between 0 and 90 degrees, depending on the relative magnitudes of resistance and reactance.

35. How do capacitors and inductors affect the bandwidth of a circuit?

Capacitors and inductors can be used to control the bandwidth of a circuit. In low-pass filters, they limit high-frequency signals. In high-pass filters, they block low-frequency signals. The specific arrangement and values of capacitors and inductors determine the cutoff frequency and the sharpness of the frequency response.

36. What is the significance of the LC product in oscillator circuits?

The LC product (inductance × capacitance) determines the resonant frequency of an LC oscillator circuit. A larger LC product results in a lower resonant frequency. This relationship is crucial in designing oscillators for specific frequencies and in understanding the behavior of resonant circuits.

37. How do capacitors and inductors behave in transient response analysis?

In transient response analysis, capacitors resist sudden changes in voltage, while inductors resist sudden changes in current. This leads to exponential charging and discharging curves in RC and RL circuits. Understanding these behaviors is crucial for analyzing circuit responses to step inputs, pulses, and other time-varying signals.

38. What is the concept of "charge redistribution" in switched-capacitor circuits?

Charge redistribution refers to the transfer of charge between capacitors when they are connected or disconnected by switches. This principle is used in switched-capacitor circuits for analog-to-digital conversion, filtering, and other signal processing applications. It allows for the implementation of resistor-like behavior using only capacitors and switches.

39. How do capacitors and inductors contribute to electromagnetic interference (EMI) suppression?

Capacitors and inductors are key components in EMI suppression. Capacitors can shunt high-frequency noise to ground, while inductors can block high-frequency currents. Combinations of capacitors and inductors form low-pass filters that can effectively reduce EMI in power supplies and signal lines.

40. What is the difference between lumped and distributed capacitance and inductance?

Lumped capacitance and inductance refer to discrete components with well-defined values. Distributed capacitance and inductance occur naturally along conductors and in circuit layouts. At high frequencies, distributed effects become more significant and can affect circuit behavior, requiring transmission line analysis rather than simple lumped element models.

41. How do supercapacitors differ from regular capacitors in terms of charging and energy storage?

Supercapacitors, also known as ultracapacitors, have much higher capacitance and energy density than regular capacitors. They can charge and discharge more quickly than batteries but more slowly than regular capacitors. Supercapacitors store energy through both electrostatic means (like regular capacitors) and electrochemical processes, allowing for greater energy storage.

42. What is the role of inductors in switch-mode power supplies?

Inductors play a crucial role in switch-mode power supplies by storing energy in their magnetic field during the switching cycle. This energy storage allows for efficient voltage conversion and current smoothing. The inductor, along with capacitors, helps to filter out high-frequency switching noise and maintain a stable output voltage.

43. How does the concept of "virtual ground" relate to capacitors in op-amp circuits?

In op-amp circuits, a "virtual ground" is a point that remains at or near ground potential due to negative feedback, even though it's not directly connected to ground. Capacitors connected to this virtual ground point can be used to create integrator or differentiator circuits, as the voltage across the capacitor changes while the virtual ground point remains stable.

44. What is the significance of the "voltage coefficient" in capacitors?

The voltage coefficient describes how a capacitor's capacitance changes with applied voltage. Some capacitor types, particularly ceramic capacitors, can show significant changes in capacitance as the applied voltage varies. This effect can be important in applications requiring precise capacitance values or in circuits where the voltage across the capacitor varies significantly.

45. How do air gaps in inductor cores affect their performance?

Air gaps in inductor cores can increase the energy storage capacity of the inductor and prevent core saturation at high currents. However, they also decrease the overall inductance. Air gaps are often used in switch-mode power supply inductors to optimize performance, allowing for higher current handling without saturation while maintaining sufficient inductance.

46. What is the concept of "Miller capacitance" and how does it affect amplifier circuits?

Miller capacitance is an effect where the input capacitance of an amplifier appears larger due to voltage gain. It's caused by