1. Electrical Dipole2. Dielectrics in Static Electric Field3. Electric Flux Density and Dielectric Constant 4. Boundary Conditions for Electrostatic Fields

1. A Dipole

A combination of two electric charges with equal magnitude and opposite sign is called a dipole.

+ -+q -q

d

The charge on this dipole is q (not zero, not +q, not –q, not 2q). The distance between the charges is d. Dipoles are “everywhere” in nature.

This is an electric dipole. Later in the course we’ll study magnetic dipoles.

Dipole

i2

i i

qˆE=k r

r

Example: calculate the electric field at point P, which lies on the perpendicular bisector a distance L from a dipole of charge q.

+ -+q -q

d

P

Lto be worked at the blackboard

3o

qdE

4 r

+ -+q -q

d

P

L

3o

qdE

4 r

Caution! The above equation for E applies only to points along the perpendicular bisector of the dipole.It is not a starting equation.

Electric Dipole in anExternal Electric Field

An electric dipole consists of two charges +q and -q, equal in magnitude but opposite in sign, separated by a fixed distance d. q is the “charge on the dipole.”

Earlier, I calculated the electric field along the perpendicular bisector of a dipole (this equation gives the magnitude only).

3o

qdE .

4 r

The electric field depends on the product qd.

Caution! This is not the general expression for the electric field of a dipole!

q and d are parameters that specify the dipole; we define the "dipole moment" of a dipole to be the vector

p qd,

where the direction of p is from negative to positive (NOT away from +).

+q -q

p

caution: this p is not momentum!

To help you remember the direction of p, this is on the equation sheet:

p q d, from to plus

E

A dipole in a uniform electric field experiences no net force, but probably experiences a torque…

+q

-q

pF+

F-

F F F qE qE 0.

There is no net force on the dipole:

E+q

-q

pF+

F-

½ d sin

If we choose the midpoint of the dipole as the origin for calculating the torque, we find

dsin dsinqE qE qdEsin ,

2 2

and in this case the direction is into the plane of the figure. Expressed as a vector,

p E.

½ d sin

Recall that the unit of torque is N·m, which is not a joule!

E+q

-q

pF+

F-

½ d sin½ d sin

The torque’s magnitude is p E sin and the direction is given by the right-hand rule.

What is the maximum torque?

Energy of an Electric Dipole in anExternal Electric Field

If the dipole is free to rotate, the electric field does work* to rotate the dipole.

E+q

-q

pF+

F-

initial finalW pE(cos cos ).

The work depends only on the initial and final coordinates, and not on how you go from initial to final.

*Calculated using , which you learned in Physics 23.zW d

If a force is conservative, you can define a potential energy associated with it.

Because the electric force is conservative, we can define a potential energy for a dipole. The equation for work

initial finalW pE(cos cos )

suggests we should define

dipoleU pE cos .

E+q

-q

pF+

F-

dipoleU pE cos

With the definition on the previous slide, U is zero when =/2.U is maximum when cos=-1, or = (a point of unstable equilibrium).

U is minimum when cos=+1, or =0 (stable equilibrium).

It is “better” to express the dipole potential energy asdipoleU p E.

Recall that the unit of energy is the joule, which is a N·m, but is not the same as the N·m of torque!

dipole moment ( - + )

Potential energy

Torque felt

Ep

EpV

In discussing dielectric materials

P

PN

Definition

dqp

+

- E

Eq

Eq

+-qq d

Polarization = # of dipole moment / unit volume

=

In dielectrics, there are no free charges, but bound charges, i.e. electric dipoles are present.

Dielectrics

LV

E 00

Electric field inside the parallel plate :

Now, insert a slab of dielectric modify the field to a new value E

2. Dielectrics in Static Electric Field

3. Electric Flux Density and Dielectric Constant

3m

C D

0 E

)( CQdsDdvDdvv v s

Gauss’s Law: The total outward flux of the dielectric displacement (or simply the outward flux) over any closed surface is equal to the total free charge enclosed in the surface

200

000

m

C E)1( EE

EEPED

re

e

Where- ε is the absolute permittivity (F/m) -εr is the relative permittivity or the dielectric constant of the medium -ε0 is the permittivity of free space -χe is the electric susceptibility (dimensionless)

z

y

x

z

y

x

E

E

E

D

D

D

333231

232221

131211

Homogeneous-εr independent of position

Anisotropic εr is different for different of the electric field

Biaxial, Uniaxial and Isotropic Medium

z

y

x

z

y

x

E

E

E

D

D

D

3

2

1

00

00

00

21

-biaxial

-uniaxial

-isotropic

321

Vacuum 1

Glass 5-10

Mica 3-6

Mylar 3.1

Neoprene 6.70

Plexiglas 3.40

Polyethylene 2.25

Polyvinyl chloride 3.18

Teflon 2.1

Germanium 16

Strontiun titanate 310

Titanium dioxide (rutile) 173 perp

86 para

Water 80.4

Glycerin 42.5

Liquid ammonia(-78°C 25

Benzene 2.284

Air(1 atm) 1.00059

Air(100 atm) 1.0548

Material Dielectric Constants

The maximum electric field intensity that a dielectric material can stand without breakdown

Material Dielectric Strength (V/m)

Air 3e6

Bakelite 24e6

Neoprene rubber 12e6

Nylon 14e6

Paper 16e6

Polystyrene 24e6

Pyrex glass 14e6

Quartz 8e6

Silicone oil 15e6

Strontium titanate 8e6

Teflon 60e6

Dielectric Strength

4. Boundary Conditions for Electrostatic Fields

abcda

ldE 0CONDITION I (tangential components)

Tangential Components of the Electric Field

0 a

d

d

c

c

b

b

a

ldEldEldEldE

0 2/

0

0

2/

02/

0

0

2/0

h

h

w

h

h

w

ldEldE

ldEldEldEldE

02/

0

0

2/

0

2/

0

0

2/0

212

211

h

nnN

h

nnN

w

TTT

h

nnN

h

nnN

w

TTT

adlaEadlaEadlaE

adlaEadlaEadlaE

022

22

21

2211

hE

hE

wEh

Eh

EwE

NN

TNNT

Condition I (tangential components)

21

21

11

TT

TT

DD

EE

CONDITION II (normal components) QsdD

Perpendicular Components of the Displacement Vector

bottomsides top

sdDsdDsdDsdD

sDsD

adsaDadsaDsdD

NN

bottom

nnN

s top

nnN

21

11

)(

s

ssencl sdsQ

Condition II (normal components)

221 m

C )(

21 sNN DDa

The normal component of D field is discontinuous across an interface where a surface charge exists, the amount of discontinuity being equal to the surface charge density

Example: Two dielectric media with permittivity ε1 and respectively ε2 , are separated by a charge free boundary. The electric field intensity in medium 1 at point P1 has a magnitude E1 and makes an angle α1. Determine the magnitude and direction of the electric field intensity at point P2 in medium 2.

α1

α2

P1

P2

0)( since

21

21

11

12

sNN

NN

TT

DDa

DD

EE

222111

2211

coscos

sinsin

EE

EE

2

1

2

1

2

2

1

1

tan

tan

tantan

2

112

12112

2

112

1211

222

222

222

cossin

cossin

cossin21

EEE

EE

EEEEE NT

The magnitude of E2 :

Boundary conditions at a Dielectric/Conductor Interface

-inside a good conductor

E=0 ET=0

D=0 Dn=ρs

2 212 1212 24 x y x y

xy xy VV y xy

x y m

2E a a a a

22

-1.1 2.2 r o x y

nCy xy

m D E a a

The potential field in a material with εr= 10.2 is V = 12 xy2 (V). Find E, P and D.

1 9.2e r

12 22

9.2 8.854 10 -9.8 2.00 e o x y

nCx y xy

m P E E = a a

Practical Problems: Electric Potential

For z ≤ 0, r1 = 9.0 and for z > 0, r2 = 4.0. If E1 makes

a 30 angle with a normal to the surface, what angle does E2 make with a normal to the surface?

1 1 1 2 2 2 1 2sin , sin , and T T T TE E E E E E

More Practical Problems: Boundary Conditions

1 1 1 1 2 2 2 2

1 2

cos , cos ,

and since 0N r o N r o

N N s

D E D E

D D

1 2

1 2

,T T

N N

E E

D D Therefore

and after routine math we find 1 2

2 11

tan tanr

r

Using this formula we obtain for this problem 2 = 14°.

also