Parallel Plate Capacitor - Formula, Definition, Derivation, FAQs

Introduction

A parallel plate capacitor consists of a plate connected to a positive end of a cell and another plate connected to the negative end or earthed. But let us first understand what is meant by a capacitor. A capacitor consists of 2 conducting surfaces that are separated by a layer of an insulating medium also called a dielectric. This dielectric can be any insulating medium, the most common being parallel plate capacitors with air between the plates. The capacity of a conductor can be defined as the ratio between the charge on the conductor to its potential.

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This Story also Contains

1. Introduction
2. Principle of a Parallel Plate Capacitor
3. Units of Capacitance
4. Derivation of Formula for Capacitance of a Parallel Plate Using Electric Field Between the 2 Parallel Plates
5. Combination of Capacitors
6. Energy Stored in a Parallel Plate Capacitor
7. Force Between Parallel Plates of The Capacitor

• C=Q/V, where C is the capacitance of the parallel plate capacitor, ( equation-1)

Q is the charge on the plates and V is the potential between the plates.

However, the capacitance of a parallel plate capacitor depends on its geometrical shape. We will discuss this later in this article regarding how a parallel plate capacitor of capacitance c depends on the distance between plates and the area of the plates.

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See the figure of a parallel plate capacitor with air in between the plates-

(Fig-1)

It is experimentally found that in presence of an earthed plate (see fig-2), the other plate is capable of withdrawing more charge than when it is not. When a parallel plate capacitor is charged by a battery, there is a momentary flow of electrons from one plate to another. As electrons (negative charges) are withdrawn from the first plate, it becomes positively charged and the electrons get collected on the second plate which becomes negatively charged forming a potential difference between the two plates.

Thus, there is a transient flow of electrons giving rise to a charging current and this charging current is maximum when the 2 plates are uncharged. This charging current keeps decreasing and finally becomes 0 when the potential difference between plates becomes equal and opposite to the applied battery emf.

Principle of a Parallel Plate Capacitor

The principle of the parallel plate capacitor is based on the fact that when an earthed conductor is placed in the neighbourhood of a charged conductor, the capacity of the parallel plate capacitor system increases considerably. Let a plate is connected to cell. After V of cell and plate becomes equal, no more charge can be put into the plate. However, if we put another plate of opposite charge parallel to it, V decreases ( since V is added algebraically, therefore V decreases in the first plate). That’s why more charges start flowing and we say that capacity of the plate increases. Thus the property of a capacitor is to store charges and this property is called capacitance.

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Units of Capacitance

In S.I. unit, capacitance is measured in Farad (F).

1F=1Coulomb of charge/1 volt of potential

(By equation 1) we can define 1 Farad as - the capacity of a conductor is 1F, if a charge of 1 Coulomb is required to establish a p.d. of 1 Volt between the plates.

In C.G.S. unit, capacitance is measured in stat farad.

Dimensional formula of capacitance is [M^-1L^-2T⁴A²].

Derivation of Formula for Capacitance of a Parallel Plate Using Electric Field Between the 2 Parallel Plates

(Fig-3)

Q=σA and E=σ/ϵ₀ (for sheets of charges)

E=Q/(Aϵ₀)

Since, E=V/d (magnitudically) => V/d=Q/(Aϵ₀)

• Q/V=Aϵ₀/d

• C=Aϵ₀/d - (equation 2)

So, here parallel plate capacitance C depends on geometrical shape i.e. the distance between the 2 plates or the plate separation d, parallel plate capacitor of area A and ϵ₀.

Generally, we need 2 identical parallel plate capacitors i.e. the area A of plates should be taken equal because if one is smaller and the other is bigger, only the smaller part of the plate is useful, the extra part becomes useless.

The capacitance of parallel plate capacitor with the dielectric slab in between with dielectric constant K is-

• C=Aϵ₀K/d - (equation 3)

Combination of Capacitors

Series Combination- In a series combination, each capacitor is charged with the same charge while they will be raised through different potentials in accordance with their capacities.

Let V1=potential across C1

V2= potential across C2 and so on.

Q=charge given to each capacitor

Since Q=CV

Therefore Q=C₁V₁, Q=C₂V₂, ….. and so on.

Total potential = V=V₁+V₂+….. = Q/C₁+Q/C₂+……

• Q/C=Q/C₁+Q/C₂+……

• 1/Ceff=1/C₁+1/C₂+….. This is the formula for total capacitance in a series combination. (equation 5)

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2. Parallel Combination - The capacitors are said to be connected in parallel combination between any 2 points if we can proceed from 1 point to another along different paths. Since the capacitances are joined through 2 common points, V is constant in all of them.

Q₁=C₁V, Q₂=C₂V,……

Since charges can be added algebraically, we get-

Qtot=Q₁+Q₂+…

C_eff=C₁V+C₂V+…

C_eff = C₁+C₂+… (equation 6)

Energy Stored in a Parallel Plate Capacitor

Let capacitance of the capacitor be C charged by a cell of potential difference V. Let v be the potential between plates of the capacitor at any instant of time.

U=CV²/2 (equation 7)

Force Between Parallel Plates of The Capacitor

The above equation represents the force between the plates of a parallel plate capacitor charged to a potential difference of V. The negative sign implies the force is an attractive force.

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