Difference Between Capacitor And Inductor

Capacitors and inductors are key components in electrical and electronic circuits, each serving distinct purposes. Capacitors warehouse energy in an electric field between two conductive plates separated by a dielectric, making them ideal for energy storage, filtering, and timing applications, with quick energy release to stabilize voltage. In contrast, inductors warehouse energy in a magnetic field generated by current through a coil of wire, resisting changes in current. They are vital in transformers and filtering applications that require effective current control. Understanding these differences is essential for circuit design in electrical engineering and physics.

1.0Capacitor

• A Capacitor consists of two conducting bodies separated by a non -conducting medium such that it can store large amounts of electric charge in a small space.
• Capacitor is given by the ratio of magnitude of the charge q to the magnitude of the potential difference V between the conductors.
  $C=\frac{Q}{V}$
• Capacitors are named on the basis of the shape of the conductors named as

1. Parallel plate 
2. Spherical
3. Cylindrical

• On the basis of the name of the dielectric Medium Electrolytic, paper, mica, ceramic
• Capacitor is utilized to store electric charge, it stores electrical energy
• It is used in electrical appliances such as radio sets, television sets, electronic instruments etc.

Symbol of Capacitor

2.0Alternating Voltage Applied to Capacitor

• Alternating source connected to capacitor, such a circuit is known as purely capacitive circuit. The capacitor is periodically charged and discharged when alternating voltage is applied to it.
• Alternating source connected to capacitor,such a circuit is known as purely capacitive circuit.The capacitor is periodically charged and discharged when alternating voltage is applied to it.

$V=V_0\mathrm{Sin}\omega t$

• Potential Difference across Capacitor is,

$V_C=\frac{q}{c}\Rightarrow V_C=V$

$\frac{q}{c}=V_0\mathrm{Sin}\omega t$

$q=C{V}_0\mathrm{Sin}\omega t$

$I=\frac{dq}{dt}\Rightarrow \frac{d}{dt}\left(C{V}_0\mathrm{Sin}\omega t\right)=C{V}_0\omega \mathrm{Cos}\omega t=\frac{V_0}{\frac{1}{C\omega }}\mathrm{Cos}\omega t$

$I_0=\frac{V_0}{\frac{1}{c_{\omega }}}\Rightarrow \text{ is the peak value of a.c }$

$I=I_0\mathrm{Cos}\omega t\Rightarrow \mathrm{Sin}\left(\omega t+\frac{\pi }{2}\right)$

This equation shows that current leads the voltage by an angle of $\frac{\pi }{2}$

Phasor Diagram For Capacitive Circuit

Capacitive Reactance $X_c$

• It is the effective opposition offered by a capacitor to the flow of current in the circuit.
• $X_c=\frac{1}{C\omega }$, SI unit of Capacitive Reactance is Ohm
• For DC, v=0, $X_C=\frac{1}{C\omega }=\frac{1}{C\times 2\pi v}=\frac{1}{0}=\infty$

Capacitor offers infinite opposition to the flow of DC. So direct current cannot pass through a capacitor, however small the capacitance of the capacitor may be.

• For AC, $v=\text{ Finite },X_C=\frac{1}{\text{ Finite Value }}=\text{ Smaller Value }$

Capacitors offer small opposition to the flow of AC, So AC can be considered to pass through a capacitor easily.

• Capacitors behave as a conductor for high frequency alternating current.
• Dimensional Formula- $\left[M{L}^2{T}^{-3}{A}^{-2}\right]$

Graph between $X_c$ and $\nu$

3.0Power Supplied To A Capacitor

$P=VI=\left(V_0\mathrm{Sin}\omega t\right)\left[I_0\mathrm{Sin}\left(\omega t+\frac{\pi }{2}\right)\right]$

$P=V_0{I}_0\mathrm{Cos}\omega t\mathrm{Sin}\omega t=\frac{V_0{I}_0}{2}\mathrm{Sin}2\omega t$

$\Rightarrow \text{ Average of }\left(\frac{V_0{I}_0}{2}\mathrm{Sin}2\omega t\right)$ for the full cycle is zero.

$P_{\text{capacitor }}=0$

4.0Inductor

• Inductor is also called a coil having a number of turns and having a property called self induction. It is the belonging of a coil by merit  of which it opposes the growth or decay of the current flowing through it.
• Self Induction is also known as the Inertia of Electricity.

• Inductors play a crucial role in various electrical and electronic applications, including energy storage, transformers, and filters.

5.0Alternating Voltage Applied to Inductor(L)

• An alternating source connector to an Inductor Lsuch a circuit is known as a purely Inductive circuit.
• Alternating Voltage across an inductor is, $V=V_0\mathrm{Sin}\omega t$
• Induced EMF across an inductor is.

$ϵ=-L\frac{dI}{dt}$

$V+\left(-L\frac{dI}{dt}\right)=0$ (As there is no potential drop across the circuit)

$L\frac{dI}{dt}=V\Rightarrow \frac{dI}{dt}=\frac{V}{L}$

$\frac{dI}{dt}=\frac{V_0\mathrm{Sin}\omega t}{L}=\frac{V_0}{L}\mathrm{Sin}\omega t$

$\int dI=\int \frac{V_0}{L}\mathrm{Sin}\omega t$

$I=\frac{V_0}{L}\left(-\frac{\mathrm{Cos}\omega t}{\omega }\right)=\frac{V_0}{L\omega }(-\mathrm{Cos}\omega t)\left(\therefore I_0=\frac{V_0}{L\omega }\right)$

$\left(\therefore (-\mathrm{Cos}\omega t)=\mathrm{Sin}\left(\omega t-\frac{\pi }{2}\right)\right)$

$I=I_0\mathrm{Sin}\left(\omega t-\frac{\pi }{2}\right)$

$\Rightarrow$ This equation shows that current lags behinds the voltage by an angle $\frac{\pi }{2}$

Phasor Diagram For Inductive Circuit

Inductive Reactance $\left(X_L\right)$

• It is the effective opposition tendered by the inductor to the stream of current in the circuit.
• $X_L=L\omega =L\times 2\pi v$
• For DC, Frequency (v=0)
• $X_L=L\omega =L\times 2\pi \nu =0\Rightarrow X_L=0$

Inductors offer no opposition to the flow of DC, Hence DC can flow easily through an inductor.

• For AC, Frequency $(v=\text{ Finite })$
• $X_L=L\omega =L\times 2\pi \nu =\text{ Finite Value }$

Inductor Offers finite opposition to the flow of AC

• SI unit of Inductive Reactance $\left(X_L\right)$ is Ohm
• Dimensional Formula- $\left[M{L}^2{T}^{-3}{A}^{-2}\right]$
• Graph Between $X_L$ and v 

6.0Power Supplied To An Inductor

$P_L=VI=\left(V_0\mathrm{Sin}\omega t\right)\left[I_0\mathrm{Sin}\left(\omega t-\frac{\pi }{2}\right)\right]$

$P_L=VI=-V_0{I}_0\mathrm{Cos}\omega t\mathrm{Sin}\omega t=-\frac{V_0{I}_0}{2}\mathrm{Sin}2\omega t$

Average of Sin2t is zero

$P_L=0$

7.0Difference Between Capacitor And Inductor

8.0Sample Questions on Difference Between Capacitor And Inductor

Q-1. Calculate the reactance of an inductor in a d.c circuit.

Solution: $X_L=L\omega =L\times 2\pi v=0\Rightarrow X_L=0$

Inductors offer no opposition to the flow of DC, Hence DC can flow easily through an inductor.

Q-2. What is the reactance of a capacitor in a d.c circuit.

Solution: For DC,v=0, $X_C=\frac{1}{C\omega }=\frac{1}{C\times 2\pi \nu }=\frac{1}{0}=\infty$,

Capacitor offers infinite opposition to the flow of DC. So direct current cannot pass through a capacitor, however small the capacitance of the capacitor may be.

Q-3. A $100\mathrm{Hz}$ a.c is flowing in a $14\mathrm{mH}$ coil, calculate its reactance.

Solution : $X_L=L\omega =L\times 2\pi v=14\times 10^{-3}\times 2\times \frac{22}{7}\times 100=8.8\Omega$