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Fields & Effects

 

Capacitors 1

 

action

capacitance

parallel plate capacitor

relative permitivity

dielectrics

 

 

 

Action

 

Capacitors are electrical components used to store charge. Their construction is simply two equal area conducting plates, with an insultor(dielectric) sandwiched in between.

 

 

capacitor action

 

 

When the switch is turned to the left, there is an instantaneous flow of current.

 

By the action of the battery electrons move in a clockwise sense. They are taken from the lower plate and deposited on the upper one.

 

In a very short time all motion ceases. The p.d. across the plates is now the same as that across the battery, but in the opposite direction.

 

With the positive of the battery connected to the positive of the capacitor, no p.d. exists. So no current flows.

 

In this state the capacitor is said to be 'fully charged'.

 

Charges on upper and lower plates are of opposite type and equal in quantity.

 

 

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Capacitance

 

Capacitance is the measure of a capacitor to store charge.

 

The larger the capacitor the more charge can be stored per volt of p.d. across the plates.

 

capacitance equation

 

where,

 

C is the capacitance in Farads (F)
Q is the charge in Coulombs (C)
V is the p.d. between the plates

 

 

The unit of capacitance is called the Farad.

 

 

defintion: a capacitor has a capacitance of 1 Farad when 1 Coulomb of charge is

 

stored with a p.d. of 1 volt across the plates.

 

 

Hence the units of Farads are Coulombs per volt (CV-1)

 

One Farad is too large a unit for ordinary circuits. Instead smaller derivative units are used, eg microfarads (μF) and picofarads (pF).

 

 

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The Parallel Plate Capacitor

 

The capacitance of a parallel plate capacitor can easily be derived from first principles.

 

 

parallel plate capacitor

 

 

Starting with our basic equation for capacitance,

 

capacitance equation                          (i           

 

The charge Q is equal to the charge density σ multiplied by the area A .

 

charge and charge density

 

Substituting for Q in the first equation (i ,

 

capacitance in terms of charge density, area and potential                        (ii               

 

Taking for granted the results from Gauss's Theorem* giving electric field strength E in terms of charge density σ and permittivity ε :


*an advanced theory not dealt with here

 

field strength in terms of charge density and permittivity

 

Rearranging to make σ the subject,

 

charge density in terms of electric field strength and permittivity

 

Now substituting for σ in equation (ii

 

capacitance equation #7                      (iii               

 

For the uniform field inside a capacitor,

 

elctric field strength in terms of potential and plate separation

 

hence,

 

pd in terms of electric field strength and plate displacement

 

Substituting for V into equation (iii

 

capacitance equation #12C

 

capacitance in terms of permittivity area and plate displacement

 

 

NB The expression for electric field strength E from Gauss's law is for an infinite plate area A . This result is therefore an approximation.

 

 

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Relative Permitivity

 

The definition: relative permittivity is the ratio of the capacitance of a capacitor with a dielectric to that of a capacitor without (ie free space).

 

definition of relative permittivity

 

By looking at capacitance we can obtain an expression linking relative permittivity εr , the permittivity of free space εo and the permittivity of the dielectric being used.

 

capacitance and capacitance of a vacuum

 

We find εr by dividing the first equation by the second :

 

relative permittivity expanded

 

 

permittivity in terms of the permittivity of free space and relative  permittivity

 

 

NB Because εr is a ratio of permittivities, it has no units

 

 

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Dielectrics

 

To understand the action of dielectric materials it is important to appreciate what is happpening on the molecular level.

 

When an electric field is applied to a dielectric the positioning of the components of atoms/molecules is changed.

 

 

polarized molecules in a dielectric

 

 

Positive atomic nuclei are moved to their limit a short distance and point towards the negative plate.

 

Electron clouds around atoms become mis-shapen with the bulk of their charge pointing towards the positive plate.

 

In this way atoms/molecules become polarized, with opposite charges tending to be concentrated at either end.


The result is that the surfaces of the dielectric facing the capacitor's plates become charged. A positive plate opposes the negative face of the dielectric, while a negative plate opposes the dielectric's positive face.

 

 

A dielectric between the plates of a capacitor modifies capacitance in two particular circumstances:

 

i) by being isolated      ii) being part of an electric circuit

 

 

An isolated charged capacitor

 

 

isolated capacitor

 

 

The negative charge on the top surface of the dielectric combines with the positive charge on the top plate to lessen the overall potential in the area. (The area is less positive.)

 

The positive charge on the bottom surface of the dielectric combines with the negative charge on the bottom plate to increase the potential there. (The area is less negative.)

 

The result is that there is less potential difference (V) across the capacitor.

 

Since Q = CV , Q is unchanged and V decreases, then C increases.

 

 

A capacitor in-circuit with a battery

 

 

in-circuit capacitor

 

 

The p.d. V across the plates is maintained by the battery.

 

The surface charges on the dielectric cause more electrons to be drawn from the positive plate and be deposited on the negative plate.


So the overall charge Q on the plates is increased. Since V is constant and Q = CV then C increases.

 

 

For different reasons, the effect of introducing a dielectric substance between the plates of a capacitor has the effect of increasing the capacitance.

 

 

Below is a table of common relative permittivities.

 

Note how the relative permittivity for water is much higher than the rest. This is because water(along with many other liquids), has polar molecules .

 

These are polarized. That is they have + and - ends, without any field being applied.

 

Polar molecules in liquids readily align themselves to an electric field.

 

In this way more surface charges are produced and the effect is greater.

 

 

material

 

rel. permittivity
(εr)

vacuum

1.0

 

 

air

1.00058986 (STP)

ebonite

3

glass

5

mica

7

paper

3.85

polythene

2.25

polystyrene

2.4 - 2.7

rubber

7

water

80.1 (room temp.)

 

 

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