� Charged conductors in electrostatic equilibrium.

� Ground. Electrostatic influence. Electric shield.

� The parallel-plate capacitor. Capacitance.

� Stored energy in a capacitor.

� Combination of capacitors.

� Electric dipole. Dielectrics.

� Capacitors with dielectric

Unit 2: Electric properties of conductors and dielectrics.

� Conductors: Materials whose electric charge(electrons) can move from any point to other dueto an electric field.

By adding e-⇒ Net charge –

By removing e-⇒ Net charge +

� Dielectrics: The electrons are firmly linked toatoms and net charge can not change.Dielectrics can only be polarized.

Tipler, chapter 22, section 22.5

Charged conductors in electrostatic equilibrium

� Conductors in electrostatic equilibrium: There isn’tnet movement of the charges (F=0).

� As electric forces are due to an electric field:

� Electric field inside a conductor in electrostaticequilibrium is zero at any point of the conductor.

Tipler, chapter 22, section 22.5

Charged conductors in electrostatic equilibrium

⇒== 0EqFrr

0=Er

� Electric charge in a conductor must reside on the conductor’s surface.

� conductor’s inside0=E 0=⋅=⇒ ∫S

dSEφ

0εφ

∑= iQ

0=iQ

Gauss’s surface (S)

E

Gauss’stheorem

Charged conductors in electrostatic equilibrium

Electric charge must reside on conductor’s surface

Charged conductors in electrostatic equilibrium

� Any conductor’s point has equal electric potential:

⇒=⋅−=− ∫B

AAB dVV 0lE

A

B

BA VV =0

Tipler, chapter 23, section 23.5

� Electric field is perpendicular to conductor’s surface.

� If electric field wasn’t perpendicular, the tangentialcomponent Et should move the charges and so theconductor wouldn’t be in equilibrium.

E

nE

tE

Charge moving

E

Charge not moving

Charged conductors in electrostatic equilibrium

tqEF =

� Coulomb’s theorem: at points near conductor’s surface

0εσ /=E

E

Sn̂

Charged conductors in electrostatic equilibrium

It can be demonstrated by applying Gauss’s law

Charged conductors in electrostatic equilibrium

� Summary of properties of charged conductors in electrostatic equilibrium:

� E=0 inside the conductor.

� All the charge must be on the surface as σ. Thereisn’t charge inside the conductor.

� Electric potential is constant in all the conductorV=cte.

� Electric field near the conductor’s surface isperpendicular to the surface, with a value: Es= σ/ε0

� The behaviour of a hollow conductor without charges inside is the same as solid conductor:

0=Er

0=Er

cteV =

0=iσ

Hollow conductor

cteV =

q

The Tip shape effect.St. Elmo’s fire (fuego de San Telmo)

https://www.youtube.com/watch?v=kdNjKdmpkOs

� When we put an electric chargenear a conductor, electrostaticinfluence divides the chargeinside the conductor.

E

0=iE

Electrostatic influence

0=iE

� Total Electrostatic influence between twoconductors occurs when all the field lines startingfrom a conductor end in the other conductor.

� Surfaces with total influence have the same chargebut different sign

+Q-Q

+Q

-Q

Total electrostatic influence

� Electric potential of a spheric

conductor is given by:

� As Earth has a very big radius (R→∞) related to anyobject, electric potential of earth (ground) is zero forany charge Q. Ground can take or give any chargewithout change its electric potential (it’s like the sea level)

0=GV

Ground

R

QV

04πε=

Connecting a device to ground means safety

for people

� Linking a conductor to Ground ( ) means:

� 1. Electric potential is 0 (V=0)

� 2. The conductor can change its charge by taking or giving electrons to Ground.

0=Er

0=V

Linking a conductor to Ground

Without charges inside

E=0

V=0 q

� A hollow conductor linked to ground divideselectrically the inner and outer spaces. It’sknown as an electric shield. Outer charges don’tinfluence inner space………

qEr

0=Er

0=V eσ

0=iσ

Electric shield or Faraday’s cage

� And inner charges don’t influence outer space.

qEr

0=Er

0=V

0=eσ

iσ

Electric shield or Faraday’s cage

The parallel-plate capacitor

� It’s made up by two parallelplate conductors being itssurface much more greater thanthe distance between them(Total electrostatic influence).

Tipler, chapter 24, section 24.1

� If a parallel-plate capacitor ischarged with a charge Q (+Qon a plate and –Q on theother plate) (in vacuum):

and the difference of potentialbetween the plates:

+Q-Q

The parallel-plate capacitor. Capacitance

-σ

+σ

S

Q=σ

d

S

E

0ε

σ=E

0ε

σddErdEVVV =⋅=⋅=−= ∫

−

+

−+ rr

� The rate Q/V is known as the capacitance (C) ofthe capacitor, and it’s depending on thegeometry (size, shape and relative position), andnot depending of the charge of the capacitor:

The parallel-plate capacitor. Capacitance

d

S

d

S

V

QC 0

0

εε

σ

σ===

[C]=M-1L-2T4I2 Unit: Farad (F)

C

Some parallel-plates capacitors

( )12

0

/ln

2

rr

LC

επ=

Other capacitors. Cilindric capacitor

∑=+++=i

CCCCC ieq

11111

321L

Combination of capacitors. Capacitors in series

� When many capacitorsare connected in series,all the capacitors havethe same charge.

Tipler, chapter 24, section 24.3

∑=++=i

ieq CCCC L21

Combination of capacitors. Capacitors in parallel

Tipler, chapter 24, section 24.3

� When many capacitorsare connected in parallel,all the capacitors havethe same difference ofpotential.

Tipler, chapter 24, section 24.2

dqC

qvdqdU ==

Stored energy in a capacitor

� To charge a capacitor means to carry charge froma plate to another plate (negative charge from + to -, or

positive charge from – to +). Let us take the situationwhere the charge and the potential of capacitorare q and V. To increase a dq charge, must bedone a work (dU):

-C

qv =

Tipler, chapter 24, section 24.2

22

2

1

2

1

2

1CVQV

C

QU ===

C

Qdq

C

qvdqdUU

QQ 2

002

1==== ∫∫∫

Stored energy in a capacitor

� To charge a discharged capacitor until Q charge,the work done (stored as energy on the electricfield) will be:

� From capacitance definition:

+ -

Electric dipole

� In order to understand the behaviourof dielectrics, it’s necessary to knowwhat’s a electric dipole.

� It’s the set of two point charges withthe same value but different sign.

-q+q

d

� Its main feature is the vector dipole moment

� Under an electric field the dipole turns, remaining parallel to E:

Tipler, chapter 21, section 21.4

+-

EF q=+

EF q−=−

EE

dqprr

=

Tipler, chapter 24, sections 24.5 and 24.4

0E

Dielectrics. Dipolar polarization.

� Dielectrics with polar molecules. Suchmolecules act like “dipoles”, randomlyoriented when no electric field is acting.

� Dipoles are oriented when a electric field is acting(dipolar polarization).

F=qE

Polar moleculewater

Tipler, chaper 24, sections 24.5 and 24.4

0E

Dielectrics. Ionic polarization.

� It occurs on dielectrics with non polar molecules.When an electric field acts, molecules becomepolars, they turn and polarization occurs (ionicpolarization).

� Dipoles are oriented when a electric field is acting

F=qE

Acting an external electric field, centers of positive and negative charge are displaced, resulting on electric dipoles.

0

Dielectrics. Behaviour on an electric field.

� When the dielectric is polarized (both by dipolar orionic polarization), it creates an electric field (Ed)opposite to the original E0. The electric field resultingE is lower than the original.

Ed

E0

E=Eo-Ed =E0/εr < Eo

εr (or k) is characteristic for

each material, and it’s calledrelative dielectric permitivityor dielectric constant.

εr≡k goes from 1 to ∞

Capacitor without dielectric Capacitor with dielectric

Tipler, chapter 24, section 24.4

V

-QQ 0

00S

QddEV

ε==

0

0S

QE

ε=

d

S

V

QC 0

0

0

ε==

V

Q

r0r

0

S

QEE

εεε==

00rr0 CC

d

S

V

QC >=== ε

εε

Capacitor with dielectric.

The effect to fill a capacitor with a dielectric is the increasing on capacitance. It’smultipied by the relative dielectric constant:

r

0

r0

V

S

QdEdV

εεε===

00r CCC >= ε