Parallel Plate Capacitor

A parallel plate capacitor is a simple yet essential electronic component used to store electrical energy in the form of an electric field. It consists of two conductive plates placed parallel to each other and separated by an insulating material called a dielectric. Widely used in circuits, sensors, and power systems, parallel plate capacitors are crucial for energy storage, signal filtering, and voltage regulation in modern electronics. A parallel plate capacitor is a widely used electronic component designed to store electrical energy in the form of an electric field. It consists of two flat, conductive plates placed parallel to each other, separated by a small gap filled with a dielectric material such as air, glass, or plastic. This configuration allows the capacitor to store and release charge efficiently, making it essential in various electronic circuits.

1.0Principle of Capacitor

• A capacitor works on the principle of electrostatic induction, where a charged conductor induces an opposite charge on a nearby uncharged conductor, creating a potential difference and allowing the system to store electrical energy.

• Consider a positively charged metal plate. Place a nearby uncharged metal plate close to it.
• Due to induction, the face of the uncharged plate closer to the charged plate acquires a negative charge, while its opposite face gains a positive charge.
• The negative charge on the nearby plate reduces the potential of the charged plate. Although the positive charge on the farther face tends to increase the potential, the negative charge being closer has a stronger effect, resulting in an overall slight decrease in the charged plate’s potential.
• As the potential of the charged plate decreases, its capacitance increases slightly.
• If the positive face of the nearby plate is connected to the ground, the positive charge is neutralized by electrons flowing from the earth, while the negative charge remains due to induction.
• The remaining negative charge significantly reduces the potential of the charged plate, causing a much larger increase in capacitance.
• The capacitance of an insulated conductor increases considerably when an earthed conductor is placed close to it. Such a system of two conductors—one charged and the other earthed—placed near each other is called a capacitor.

2.0Definition of Parallel Plate Capacitor

• It consists of two large plane parallel conducting plates separated by a small distance.
• A parallel plate capacitor is an electronic device consisting of two flat, parallel conductive plates separated by a small distance, often filled with a dielectric material, used to store electrical energy in the form of an electric field.

3.0Capacitance of Parallel Plate Capacitor

Surface charge Density $\sigma =\frac{Q}{A}$

Electric Field Intensity Between Plates $E_0=\frac{\sigma }{2{ϵ}_0}+\frac{\sigma }{2{ϵ}_0}=\frac{\sigma }{{ϵ}_0}=\frac{Q}{{ϵ}_{0}A}$

Potential difference between the plates $V=E_0\times d=\frac{Qd}{{ϵ}_{0}A}$

Capacitance of parallel plate capacitor $C_0=\frac{Q}{V}=\frac{Q}{\left(\frac{{\rho }_d}{c_d}\right)}=\frac{{ϵ}_{0}A}{d}$

Also Practice: Parallel Plate Capacitor Questions

4.0Factors Affecting Capacitance of Parallel Plate Capacitor

Overlapping Area $\Rightarrow C\propto A$

Distance between plates $\Rightarrow C\propto \frac{1}{d}$

Medium between plates $\Rightarrow C\propto {ϵ}_r$

Note: Capacitance of a parallel plate capacitor does not depend on thickness and nature of metal of plates.

5.0Capacitance of Parallel Plate Capacitor With Dielectric

Electric field intensity in Vacuum $\Rightarrow E_0$

Electric field intensity in Medium $\Rightarrow E_m$

$E_m=\frac{\sigma }{ϵ}=\frac{\sigma }{{ϵ}_0{ϵ}_r}=\frac{Q}{{ϵ}_0{ϵ}_{r}A}\Rightarrow E_m=\frac{E_0}{{ϵ}_r}$

Potential difference between the plates$V=E_m\times d=\frac{Qd}{{ϵ}_0{ϵ}_{r}A}$

Capacitance of a capacitor in presence of medium

$C_m=\frac{Q}{V}=\frac{{ϵ}_0{ϵ}_{r}A(Q)}{Qd}=\frac{{ϵ}_0{ϵ}_{r}A}{d}={ϵ}_r{C}_0$

Also Read: Capacitors in Combination

6.0Fringing of Electric Field

For the plates of finite area the electric field between the two plates will not be uniform and the field lines bend outward at the edges. This is called "fringing of electric field"

7.0Capacitance With Partial Dielectric Filling

Surface charge density, $\sigma =\frac{Q}{A}$

Electric Field in air or vacuum, $E_0=\frac{\sigma }{{ϵ}_0}=\frac{Q}{{ϵ}_{0}A}$

Electric field in dielectric medium, $E_m=\frac{\sigma }{ϵ}=\frac{Q}{{ϵ}_0{ϵ}_{r}A}$

Potential difference between the plates of capacitor

$V=E_0(d-t)+E_{m}t=\frac{Q}{{ϵ}_{0}A}(d-t)+\frac{Q}{{ϵ}_0{ϵ}_{r}A}t$

$V=\frac{Q}{{ϵ}_{0}A}\left[(d-t)+\frac{t}{{ϵ}_r}\right]$

Capacitance, $C=\frac{Q}{A}$

$\Rightarrow C=\frac{Q}{\frac{Q}{{ϵ}_{A}A}\left[(d-t)+\frac{t}{{ϵ}_r}\right]}=\frac{{ϵ}_{0}A}{(d-t)+\frac{t}{{ϵ}_r}}$

In case of multiple slabs,

$C=\frac{{ϵ}_{0}A}{\left(d-\left(t_1+t_2+\dots \dots \right)\right)+\frac{t_1}{{ϵ}_r}+\frac{t_2}{{ϵ}_r}+\dots \dots \dots ..}$

8.0Force Between Plates of PPC

Force between two plates means force on a plate

$\begin{array}{rl}& F=\text{ Electric field due to plate }(1)\times \text{ charge of plate }(2)\\ & F=\left(\frac{\sigma }{2{ϵ}_0}\right)Q\Rightarrow \sigma =\frac{Q}{A}\\ & F=\frac{Q^2}{2A{ϵ}_0}=\frac{{\sigma }^{2}A}{2{ϵ}_0}\\ & F=\frac{(CV{)}^2}{2A{ϵ}_0}\Rightarrow C=\frac{A{ϵ}_0}{d}\Rightarrow A{ϵ}_0=Cd\\ & F=\frac{C^2{V}^2}{2Cd}=\frac{C{V}^2}{2d}\Rightarrow F=\frac{C{V}^2}{2d}\end{array}$

9.0Pressure on Each Plate Of A Capacitor

$\begin{array}{rl}& F=\frac{{\sigma }^{2}A}{2{ϵ}_0}\\ & P=\frac{F}{A}=\frac{{\sigma }^2}{2{ϵ}_0}\end{array}$

This is known as electrostatic pressure. Electrostatic pressure always acts perpendicular to the surface and outwards.

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