Unit 2: Electric properties of conductors and dielectrics. Charged - - PowerPoint PPT Presentation

• 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 due to an electric field.

By adding e- Net charge – By removing e-  Net charge +

• Dielectrics: The electrons are firmly linked to

atoms 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’t

net movement of the charges (F=0).

• As electric forces are due to an electric field:
• Electric field inside a conductor in electrostatic

equilibrium is zero at any point of the conductor.

Tipler, chapter 22, section 22.5

Charged conductors in electrostatic equilibrium

F qE E = =  =

• Electric charge in a conductor must reside on the

conductor’s surface.

• conductor’s inside

= E

= ⋅ = 



S

dS E φ

ε φ  =

i

Q

Gauss’s surface (S)

E

Gauss’s theorem

Charged conductors in electrostatic equilibrium

Electric charge must reside on conductor’s surface

i

Q =

Charged conductors in electrostatic equilibrium

• Any conductor’s point has equal electric potential:

B B A A B A

V V d V V − = − ⋅ =  =

 E

l

A B

Tipler, chapter 23, section 23.5

• Electric field near the conductor’s surface is

perpendicular to conductor’s surface.

• If electric field wasn’t perpendicular, the tangential

component Et should move the charges and so the conductor wouldn’t be in equilibrium.

Charge moving Charge not moving

Charged conductors in electrostatic equilibrium

t

qE F =

t

E

n

E E E

• Coulomb’s theorem: at points near conductor’s

surface

S

Charged conductors in electrostatic equilibrium

It can be demonstrated by applying Gauss’s law

E σ ε = E

u

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. There

isn’t charge inside the conductor.

• Electric potential is constant in all the conductor

V=cte.

• Electric

field near the conductor’s surface is perpendicular to the surface, with a value: Es= σ/ε0

• The behaviour of a hollow conductor without charges

inside is the same as solid conductor:

Hollow conductor q

E =

• E =
• V

cte = V cte =

i

σ =

Sharp tip effect: due to the high electric field near a sharp tip:

• Lightning rod
• An umbrella during a storm
• St. Elmo’s fire (fuego de San Telmo)
• https://www.youtube.com/watch?v=kdNjKdmpkOs

Sharp tip effect

Exercise 2.2

1 2

Q q Q + = Q q V k k R r

1 2

= =

Sharp tif effect:

Q R Q1 R r r q2 V V

q

Q QR Q V E R R r R R r R R

1 1 1 1 2 2

4 ( )4 4 ( ) σ σ π π ε πε = =  = = = + + q Qr Q V E r R r r R r r r

2 2 2 2 2 2

4 ( )4 4 ( ) σ σ π π ε πε = =  = = = + +

The lower the radius the higher the electric field near the conductor. Sharp tips attract the electric charges

E2 > E1

Solution:

QR Q R r

1 =

+ Qr q R r

2 =

+ Q V k R r ( ) = +

• Electric potential of a spheric

conductor is given by:

• As Earth has a very big radius (R→∞) related to any
• bject, electric potential of earth (ground) is zero for

any charge Q. Ground can take or give any charge without change its electric potential (it’s like the sea level)

Ground

R Q V 4πε =

Connecting a device to ground means safety for people

G

V =

• 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.

Linking a conductor to Ground

Without charges inside E=0 V=0

q

E =

• V =

• When we put an electric charge

near a conductor, electrostatic influence divides the charge inside the conductor.

Electrostatic influence

i

E =

i

E = E

• Total

Electrostatic influence between two conductors occurs when all the field lines starting from a conductor end in the other conductor.

• Surfaces with total influence have the same charge

but different sign

+Q

• Q

+Q

• Q

Total electrostatic influence

• A hollow conductor linked to ground divides

electrically the inner and

• uter

spaces. It’s known as an electric shield. Outer charges don’t influence inner space………

q Electric shield or Faraday’s cage

E =

• V =

i

σ =

e

σ

E

• And inner charges don’t influence outer space.

q Electric shield or Faraday’s cage

E =

• V =
• E

σ = 0

e

σ i

The parallel-plate capacitor

• It’s made up by two parallel

plate conductors being its surface much more greater than the distance between them (Total electrostatic influence).

Tipler, chapter 24, section 24.1

• If a parallel-plate capacitor is

charged with a charge Q (+Q

• n a plate and –Q on the
• ther plate) (in vacuum):

and the difference

• f

potential between the plates:

+Q

• Q

The parallel-plate capacitor. Capacitance

• σ

+σ

S Q = σ

d S

E ε σ = E

ε σd d E r d E V V V = ⋅ = ⋅ = − =



− + − +

• The rate Q/V is known as the capacitance (C) of

the capacitor, and it’s depending

• n

the geometry (size, shape and relative position), and not depending of the charge of the capacitor:

The parallel-plate capacitor. Capacitance

d S d S V Q C ε ε σ σ = = =

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

C

Some parallel-plates capacitors

( )

1 2

/ ln 2 r r L C ε π =

Other capacitors. Cilindric capacitor

Combination of capacitors. Capacitors in series

• When many capacitors

are connected in series, all the capacitors have the same charge.

Tipler, chapter 24, section 24.3

i eq 1 2 3 i

1 1 1 1 1 ... C C C C C = + + + =

Combination of capacitors. Capacitors in parallel

Tipler, chapter 24, section 24.3

• When

many capacitors are connected in parallel, all the capacitors have the same difference of potential.

eq 1 2 3 i i

C C C C ... C = + + + =

Tipler, chapter 24, section 24.2

dq C q vdq dU = =

Stored energy in a capacitor

• To charge a capacitor means to carry charge from

a plate to another plate (negative charge from + to -, or

positive charge from – to +). Let us take the situation

where the charge and the potential of capacitor are q and V. To increase a dq charge, must be done a work (dU):

• C

q v =

Tipler, chapter 24, section 24.2

2 2

2 1 2 1 2 1 CV QV C Q U = = =

C Q dq C q vdq dU U

Q Q 2

2 1 = = = =

  

Stored energy in a capacitor

• To charge a discharged capacitor until Q charge,

the work done (stored as energy on the electric field) will be:

• From capacitance definition:

Tipler, chapter 24, sections 24.5 and 24.4

• Dielectrics. Dipolar polarization.
• Dielectrics

have no free electrons. But their polar molecules (dipoles) can be oriented by an electric field (dipolar polarization). They are randomly oriented when no electric field is acting.

F=qE

Polar molecule water

E

Tipler, chaper 24, sections 24.5 and 24.4

• Dielectrics. Ionic polarization.
• It occurs on dielectrics with non polar molecules. When an electric

field acts, molecules become polars, they turn and polarization occurs (ionic polarization).

• 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.

E

• Dielectrics. Behaviour on an electric field.
• Whatever the type of polarization(dipolar or ionic), an
• pposite (Ed) to the original electric field (E0) appears.

The resulting electric field E is lower than the original.

Ed E0 E=Eo-Ed =E0/εr < Eo

εr (or k) is characteristic for

each material, and it’s called relative dielectric permitivity

• r dielectric constant.

εr≡k goes from 1 to ∞

Isolated Capacitor without dielectric Isolated Capacitor with dielectric

Tipler, chapter 24, section 24.4

V

• Q

Q

S Qd d E V ε = = Q E S σ ε ε = = d S V Q C ε = =

V

Q

r r

S Q E E ε ε ε = =

r r

C C d S V Q C > = = = ε ε ε

Capacitor with dielectric.

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

r r

V S Qd Ed V ε ε ε = = =

r

C C C > = ε

Let’s take an isolated capacitor with some charge Q