Essentials in ElectromagnetismEssentials in Electromagnetism

Charge. Field. Force

Electrokinetics

History

- 600 B.C – the first observation of an electric effect - Thales of Miletus Greek

mathematician and astronomer – elektron

- a piece of amber, having been rubbed, would attract small bits of

straw and feathers

amber = elektron (greek word)

- William Gilbert (1544-1603), physician to Queen Elisabeth I - investigated

substances that can exhibit the same electrification properties as amber

electrics - materials that we now recognize to be insulatorselectrics - materials that we now recognize to be insulators

nonelectrics - materials, which are difficult to electrify by

rubbing; these materials are, in fact, conductors

- Stephen Gray (1693-1736) - the state of electrification can be transferred

from one object to another when the objects are connected by a piece of metal

electrification is a transitory, not an intrinsic, property of matter;

Charles François du Fay (1698-1739): two types of electrification are possible.

- vitreous materials - glass and mica

- resinous materials - amber, sulphur, hard rubber and resins

When rubbed with silk or cotton:

- the vitreous and resinous materials become oppositely electrified in the sense

that bodies with opposite electrification attract each other

- bodies with the same electrification repel each other

Microscopic Charge Carriers

|e|=1.6021892(46)·10-19 C

Electron -Structure-less point particle - the entire charge of an electron is

concentrated at a point

Contradictions: The energy of the electric field created by a point charge is

infinite and hence the inertial mass of the point charge must also be infinite;

the electron rest mass is me=9.11・10-31 kg

Proton.

A proton consists of two point quarks with a charge +(2/3)·|e| and one point

quark with a charge –(1/3)·|e|

�eutron. A neutron consists of two quarks with a charge -|e|/3 and one

quark with a charge +(2/3)·|e|

Coulomb’s Law

2

21

04

1

r

qqF

πε=

r

r

r

qqF

rr

2

21

04

1

πε=

1772 Cavendish - first experimental verification 1722 but not published !!

2

12

12

2

12

1

0

124

1q

r

r

r

qF

rr

πε= 1

21

21

2

21

2

0

214

1q

r

r

r

qF

rr

πε= 02112 =+ FF

rr

scalar vectorial

12120 21210

The charges q1 and q2 create in the space surrounding them a field named

electric field which is characterized by the strength E

The field strength is a local concept - has a definite value at each point in space.

r

r

r

qrE

rrr

2

04

1)(

πε= EqF

rr'=

- from Newton’s 3rd Law

q’ – test charge

ε0 = 8.854187817.. × 10−12 F/m is the vacuum permittivity

Superposition Principle

23133 FFFrrr

+= EqFrr

33 =2313 EEErrr

+=

The generalization: ∑=i

iEErr

Test charge. The measurement of the electric field is reduced to the measurement

of the force acting on a point charge. The point charge used for measuring the of the force acting on a point charge. The point charge used for measuring the

strength of an electric field is called test charge

The electric field strength is a vector which, at each point of space, points

directly away from q if q is positive and directly toward q if q is negative

Fixed charge. Electrostatics studies electric fields generated by fixed charges.

It is assumed that charges are held at various points in space by the force of no

electrostatic origin

Gauss’s theorem for electric field

a) charge q is inside a volume V bounded by a closed surface S

- the flux of the field E through surface S

∫=Φrr rq

rrr 1

∫=Φ SdErr

r

r

r

qrE

rrr

2

04

1)(

πε=

0

2

0

0

04 επε

qdS

R

q

S

==Φ ∫- the flux of the field E through a spherical surface S0

24

0

RdSS

π=∫

0ε=Φ

qGauss electrostatic theorem for a point charge

q is in the center of a

spherical surface S0But Φ0= Φ

b) The flux of E through a closed surface when a point charge is located outside

the volume bounded by it

'Φ−=Φ dd 0'=Φ+Φ dd

SdEdrr

⋅=Φ ''' SdEdrr

⋅=Φ

- for a point charge outside the volume V

the flux of the field E through a closed

surface is 0

0=Φc) system of point charges:

- applying the principle of superposition ∑= iEErr

∑∫∫ ==ΦS

i

S

SdESdErrrr

QqSdEV

i

S 00

11

εε===Φ ∑∫

rr∑=V

iqQ

∫ρ=V

dVQ

0=Φ

- for a continuous charge distribution with a volume charge density ρ:

Differential form of Coulomb’s law. Maxwell’s equation for div E.

∫∫ ==ΦVS

dVSdE ρε 0

1rr

0ερ

=≡∂∂

+∂

∂+

∂∂

Edivz

E

y

E

x

E zyxr

- the differential form of Coulomb’s law

- this equation is also true for an arbitrary motion of charges.

-the field lines start where and terminate where 0>Edivr

0<Edivr

∫= dVQ ρ

From

-the field lines start where and terminate where

- the field lines originate at positive charges and terminate at negative ones.

Lines of force:

0>Edivr

0<Edivr

Potential Nature of Electrostatic Field

Work in an electric field

ldEqldFdWrrrr

⋅==

ldEq

ldF

q

dW rrrr

==

- work performed per unit chargeldEW rr

∫=)2(

(J/C)

[J]

- work performed per unit chargeldEq

W

L

∫=),1(

Potential nature of a Coulomb field

A force field is called a potential field if the work done upon a displacement in this

field depends only on the initial and the final points of the path and does not

depends on the trajectory.

∫ =L

ldE 0rr

∑= iEErr

∫ =L

i ldE 0rr

thenand if

∂∂

+∂∂

+∂∂

−=−∇=−=z

ky

jx

igradEϕϕϕ

ϕϕrrrr

Comment. The potential, φ, of a given electric field is defined only to within an

- we can define a scalar potential φ (or V) by:

Potential Nature of Electrostatic Field (cont.)

Comment. The potential, φ, of a given electric field is defined only to within an

additive constant

r

q

r

drqr

r 0

2

0 4

1

4)(

πεπεϕ =−= ∫

∞

- for a point charge q:

)( 212121 ϕϕϕϕ +−=−−=+= gradgradgradEEErrr

∑πε=ϕ

i i

i

r

qzyx

04

1),,(

∫ρ

πε=ϕ

r

dV

4

1∫σ

πε=ϕ

r

dS

4

1

Field potential of continuously distributed charges

or:

( ) ( ) ( )∫−+−+−

=222

0 '''

''')',','(

4

1),,(

zzyyxx

dzdydxzyxzyx

ρπε

ϕ

∫πε=ϕ

r04∫πε

=ϕr04

Ernshaw’s theorem There exists no configuration of fixed charges, which would

be stable in the absence of forces other than the forces of Coulomb’s interaction

between the charges of the system.

or:

σ - surface charge density

Application

Find the field strength due to a very long charged filament with linear charge density γ.

Applying the Gauss theorem:

0εQ

SdES

=∫rr

hQ γ=

The flux of Er

through the bases is equal to 0

hrrESdESdEcylinderSS

⋅⋅== ∫∫ π2rrrr

The flux of Er

through the bases is equal to 0

rE

γπε 02

1=

Electrostatic Field in the Presence of Conductors

In electrostatics, we consider the case when charges are fixed, i.e.:

0=Er

Absence of volume charge inside a conductor

0=ρ � there are no volume charges inside a conductor

Both positive and negative charges exist inside the conductor, but they

compensate each other, and the interior of the conductor is neutral on the whole.

0=jr

compensate each other, and the interior of the conductor is neutral on the whole.

Supose that for t=0, ρ(0)≠0

t

et⋅

−

= 0)0()(εσ

ρρ ( ) 0→tρ

- it can be shown that:

σε

τ 0=

relaxation time

- for moderate frequencies free charges in a conductor are distributed over its

surface and volume charges are absent

Metallic screen

- the space charge in the conductor is “assimilated” during the time

τ ~ 10-19 s for Cu

A metallic screen for external fields

A charge surrounded by a closed conducting shell

The earthed closed shell shields the external space from the charges located in

the volume surrounded by this shell. An unearthed shell does not provide such a

screening

unearthed screen

Application for low level signals manipulation

- earthed closed screen shields the connection leads from the exterior charges

- provides immunity to exterior electrical perturbations

Electrostatic Field in the Presence of a Dielectric

The dipole moment

dipole moment

−

πε=

−

πε=ϕ

−+

+−

−+ )()(

)()(

0)()(0 4

11

4)(

rr

rrq

rr

qP

l<<r r(-)-r(+)≈lcosθ and r(-)r(+)≈r2

lqprr

=

- because l<<r � r(-)-r(+)≈lcosθ and r(-)r(+)≈r2

The strength of the electric field generated by a dipole decreases in inverse

proportion to the third power of the distance, i.e. more rapidly than the Coulomb

field of a point charge

3

04

1)(

r

rpp

rr⋅

=πε

ϕ ( ) rrpql /cosrr

=⋅ θwhere

−=−=35

0

)(3

4

1

r

p

r

rrpgradE

rrrrr

πεϕ

- the potential generated by the dipole

The aspect of the field lines generated by

an electric dipole

Equipotential lines for a: charge, dipole and a

constat field

Polarisation of dielectrics

An external electric field tends to displace positive charges in the direction of the

field and the negative charges in the opposite direction. Consequently, the

dielectric acquires a dipolar moment. This process is called polarisation

dielectric polarisation

Molecular pattern of polarisation

V

pP

∆∆

=r

r

Molecular pattern of polarisation

Nonpolar atoms and molecules

- atomic or diatomic molecules consisting of identical atoms: He, H2, O2, N2, Q

- symmetric polyatomic molecules: CO2, CH4, Q

In the absence of an external electric field, such a dielectric is not polarised

Molecules and atoms which possess an electric dipole moment in the absence

of an external electric field are called polar and include CO, N2O, H2O, SO2, etc.

In ionic crystals – ionic lattice polarisation

In electrets and ferroelectrics, in most cases, P≠0 when E=0.

Dependence of polarisation on the electric field strength

For other dielectrics P=0 when E=0.

zyxkjiEEEPj kj

kjijkjiji ,,,,...,

00 =++= ∑ ∑ χεχε

- in general case, the dependence of polarisation on the field strength

- the dielectric is called nonlinear

∑= jiji EP χε 0

or

∑=j

jiji EP χε 0

- the dielectric is called linear

If the properties of such a dielectric are different in different directions, the

dielectric is called anisotropic

The set of 9 quantities χij constitutes the dielectric susceptibility tensor

- linear isotropic dielectric

EPrr

0χε= χ is the dielectric susceptibility

Substance χ

Helium, He

Hidrogen, H2

Carbon dioxide, CO2

Water

Alcohol

Transformer oil

Glass (ordinary)

Sodium chloride

Titanium dioxide

Quartz, Barium titanate

65x10-6

254x10-6

922x10-6

80

25-30

2.24

5

5.62

170

103-104

The role of polarisation - a separation of positive and negative charges, leading to

the appearance of charges in the volume and on the surface of the dielectric. These

charges are called polarisation charges or bound charges - they attached to

different places in the dielectric and cannot move freely in its volume or on its

surface.

Bound charges give rise to an electric field in the same way as free charges, and

are in no way different from them in this respect.

The presence of a dielectric is taken into account by considering the electric

field created by the bound charges induced as a result of polarisation.

(ferroelectric)

Electret

Electret (formed of elektr- from "electricity" and -et from "magnet") is a dielectric

material that has a quasi-permanent electric charge or dipole polarisation.

An electret generates internal and external electric fields, and is the electrostatic

equivalent of a permanent magnet.

There are two types of electrets:

• Real-charge electrets which contain excess charge of one or both polarities, either

- on the dielectric's surfaces (a surface charge)

- within the dielectric's volume (a space charge)- within the dielectric's volume (a space charge)

• Oriented-dipole electrets contain oriented (aligned) dipoles. Ferroelectric materials

are one variant of these.

Electret materials are quite common in nature. Quartz and other forms of silicon

dioxide, for example, are naturally occurring electrets. Today, most electrets are

made from synthetic polymers, e.g. fluoropolymers, polypropylene,

polyethyleneterephthalate, etc.

The quasi-permanent internal or external electric fields created by electrets can be

exploited in various applications.

Bulk electrets can be prepared by cooling a suitable dielectric material within a

strong electric field (kilovolts/cm), after heating it above its melting temperature. The strong electric field (kilovolts/cm), after heating it above its melting temperature. The

field repositions the charge carriers or aligns the dipoles within the material. When

the material cools, solidification freezes them in position. Materials used for electrets

are usually waxes (ceara), polymers or resins.

Electret microphone

Electric displacement vector

00 ερ

ερ bEdiv +=

r

( ) ρε =+ PEdivrr

0PEDrrr

+= 0ε - displacement vector

ρ=Ddivr

ρ - free charges volume density

ρb - bound charges volume density

- it takes into account the polarisation of the medium

EPrr

0χε=

( ) EEDrrr

εχεε =+= 00)( χ+ε=ε 10 - dielectric constant or permittivity

0/1 εεχε =+=r - relative permittivity

Gauss electrostatic theorem in the presence of dielectrics

∫ =S

QSdDrr

Q represents the total charge inside the volume

EP 0χε=

Capacitor

- passive electronic component consisting of a pair of conductors separated by a

dielectric (insulator).

When there is a potential difference (voltage) across the conductors, a static

electric field develops in the dielectric that stores energy and produces a

mechanical force between the conductors. An ideal capacitor is characterized

by a single constant value, capacitance, measured in farads. This is the ratio

of the electric charge on each conductor to the potential difference between

them.

Battery of four Leyden jars in Museum Boerhaave, Battery of four Leyden jars in Museum Boerhaave,

Leiden, the Netherlands

In October 1745, Ewald Georg von Kleist of Pomerania

in Germany found that charge could be stored by

connecting a high voltage electrostatic generator by a

wire to a volume of water in a hand-held glass jar.

Von Kleist's hand and the water acted as conductors and

the jar as a dielectric

A parallel-plate capacitor

Charge separation in a parallel-plate

capacitor causes an internal electric field.

A dielectric (orange) reduces the field and

increases the capacitance.

The conductors hold equal and

opposite charges on their facing

surfaces and the dielectric

develops an electric field.

SIinFV

C

U

QC ,1

1

1

==

dU

dQC =

Sometimes charge build-up affects the capacitor mechanically, causing its

capacitance to vary. In this case, capacitance is defined in terms of incremental

changes

Energy of Electrostatic Fields – energy storage

Energy of interaction between discrete charges

r

QQW 21

04

1

πε= ( )22112

1

0

12

0 2

1

4

1

4

1

2

1QQQ

r

QQ

r

QW ϕ+ϕ=

πε+

πε=

Formula can be easily generalised for the case of small several charged spheres

∑ϕ= iiQW2

1

1- for a continuous distribution of charges: ∫ϕρ= dVW

2

1

Energy density of an electric field

DEwrr

2

1= - The energy density of the electric field

∫ ==⇒⋅⋅=⇒⋅=U

CUUdUCWdUUCdWdqUdW0

2

2

1

d

SC rεε0= dSVVwDEVW ⋅=⋅=⋅= ;

2

1 rr

C - capacitor

- supercapacitors (up to 3000 farad capacitance);

- they have a much higher power density than batteries or fuel cells;

- Applications: vehicles, complementing batteries, alternative energy (replacing

batteries with capacitors in conjunction with novel energy sources)