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ELEC 3105 Basic EM and Power Engineering

Electric dipoleForce / torque / work on electric dipole

Z

The Electric Dipole

x

z

+q

-q

d

P(x, z)

1r

r

2r

Consider electric field and potential produced by

2 charges (+q, -q) separated by a distance d.

E l e c t r i c f i e l d ( c h a r g e d i s t r i b u t i o n )

2rr

x

y

zq 1

q 2

P

r

1rr

1r

2r

2

22

2

2

1

12

1

1

rr

rr

rr

kq

rr

rr

rr

kqE

T w o p o i n t c h a r g e s

E l e c t r i c p o t e n t i a l e x a m p l e

A p o i n t c h a r g e Q i s l o c a t e d a t t h e o r i g i n . W h a t i s t h e p o t e n t i a l o n t h e x a x i s .

x

kQV A s s u m e s V i s z e r o

a s r e f e r e n c e p o t e n t i a l

x

x dEVV

yz

xQ

xx

kQE ˆ

2

dxxd ˆ

x

x dxxxx

kQVV ˆˆ

2

The Electric Dipole

x

z

+q

-q

d

P(x, z)

1r

r

2r

The dipole is represented by a vector of magnitude qd and pointing from –q to

+q.

p

p

Note: small letter p Units

{p} dipole moment; Coulomb meter {Cm}

The Electric Dipole

x

z

+q

-q

d

P(x, z)

1r

r

2r

Suppose (x, z) >>> d

p

34

,r

zpzxV

o

The Electric Dipole

x

z

+q

-q

d

P(r, , )

1r

r

2r

Suppose (x, z) >>> d

p

24

cos,,

r

prV

o

Spherical coordinates (r, , )

The Electric Dipole

x

z

+q

-q

d

P(r, , )

1r

r

2r

Now to compute the electric field expression

p

Spherical coordinates (r, , )

E l e c t r i c p o t e n t i a l a n d t h e g r a d i e n t o p e r a t o r

VE

E

VE

3224,

zx

zpzxV

o

P(x, z))Cartesian coordinates (x, z)

The Electric DipoleNow to compute the electric field expression

VE

3224,

zx

zpzxV

o

zz

Vy

y

Vx

x

VE ˆˆˆ

0

y

V

zExEE zx ˆˆ

522

3

4

,

zx

xzp

x

zxVE

ox

1cos3

4

13

4

, 23

225

22

2

oo

z

p

zxzx

zp

z

zxVE

zpx

zx

xzpE

oo

ˆ1cos34

ˆ3

42

522

The Electric Dipole

Spherical coordinates (r, , )

VE

32 44

cos,,

r

rp

r

prV

oo

34

1

r

rpE

o

p

r

rrp

rE

o

23

3

4

1

No dependence

The Electric Dipole

zpx

zx

xzpE

oo

ˆ1cos34

ˆ3

42

522

Force on a dipole in a uniform electric field

-q

d

+q

Here consider dipole as a rigid

charge distribution

F

F

p

No net translation since

E

FF

Oppositedirection

FF ˆˆ

Force on a dipole in a non-uniform electric field

-q

d

+q

Here consider

dipole as a rigid charge distribution F

F

p

E

FF

And / Or FF ˆˆ

net translation since

FFFnet

Force on a dipole in a non-uniform electric field

-q

d

+qF

F

E),( yyxxEqF

FFFnet

x

y

),( yxEqF

y

x

),(),( yxEqyyxxEqFnet

Manipulate expression to get simple useful form

netF

Force on a dipole in a non-uniform electric field

-q

d

+qF

F

E

x

y

y

x

xx EpF

After the manipulations end we get:

yy EpF

zz EpF

EpFnet

netF

We will obtain this expression using a different technique.

p

Torque on a dipole

-q

d/2

+qF

p

d/2

F

Here consider

dipole as a rigid charge distribution

E

The torque components + and - act in the same rotational direction trying to rotate the dipole in the electric field.

Torque on a dipole Review of the concept of torque

F

Pivot

r

Torque:

Fr

sinFr

sinrFMoment arm lengthForceAngle between vectors r and F

Torque on a dipole

-q

d

+q

For simplicity consider the dipole in a uniform electric field

F

F

p

E

sin2 Fd

sinrF

sin2 Fd

sindF

sin)( Eqd

sinpE

Ep

Act in samedirection

dqpEqF

Ep

Also valid for small dipoles in a non-uniform electric field.

Work on a dipole

-q

d

+q

Consider work dW required to rotate dipole through an angle d

F

F

p

Ep

By definition

ddW If we integrate oversome angle range then

dpEdW sin

cospEW

EpW

When you have rotation

Work on a dipole

Ep

EpW

cospEW

E

For = 90 degreesW = 0. Thus = 90 degrees is reference orientation for the dipole. It corresponds to the zero of the systems potential energy as well. U=W

p6

Work on a dipole

Ep

EpW

cospEW

0

E

For = 0 degreesW = -pE. Thus = 0 degrees is the minimum in energy and corresponds to having the dipole moment aligned with the electric field.

p6

Work on a dipole

Ep

EpW

cospEW 180

E

For = 180 degreesW = pE. Thus = 180 degrees is the maximum in energy and corresponds to having the dipole moment anti-aligned with the electric field.

p6

Force on a dipole

-q

d

+qF

F

y

x

After the manipulations end we get:

EpFnet

We will obtain this expression using a different technique.p

EpWWork

Force WF

Recall principle of virtual work and force

Exam question: once upon a time

+Q

-Q

2R2r

(0,0)

Stator dipole

Rotor dipole a) E on +qb) F on +q)c on +q)d on rotor

+q -q

Exam question: Once upon a time

+Q

-Q

2R

e) on dipole

D>>R

Polarization

No external electric field

Positive nucleus

Negative electron cloud

With an external electric field

Charge polarization occurs in the presence of electric field

Atom

p

E

Polarization

Positive nucleus

Negative electron cloud

E

With an external electric field

Each atom acquires a small dipole moment . For low intensity electric fields the polarization is expected to be proportional to the field intensity:

p

Atomic polarizability of the atom

p

Ep o

Atomic Polarizability And Ionization Potential

Polarization

No external electric field With an external electric field

If the density of particles per cubic meter is N, the net polarization is:

pNP

}{P

has units of {C/m2}

Polarization

O2-

H+ H+

Some molecules have built in dipole moments due to ionic bonds.

Negative region

Positive region

Water

𝑝 𝑝

Polarization

O2-

H+ H+

Negative region

Positive region

For low intensity applied fields this polarization of the material is again proportional to the field intensity so a more general

expression for polarization is:

EP oC

With the electric susceptibility of the materialC

𝑝 𝑝

Polarization

For low intensity applied fields this polarization of the material is again proportional to the field intensity so a more general

expression for polarization is:

EP oC

Materials where is proportional to are called DIELECTRIC materials.

P

E

Dielectrics

ormedium Why is the dielectric constant in the medium different than the dielectric constant in vacuum?

The answer is contained in the nature of the material being placed in the electric field.

Dielectrics

Consider the slab of material “dielectric” immersed in an external field .oE

oE

The molecules inside will be polarized due to the presence of the electric field.

d

Endface Area A

Dielectrics

Consider the slab of material “dielectric” immersed in an external field .oE

oE

The molecules inside will be polarized due to the presence of the electric field.

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+

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+

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++

+

++

++

+++

++

++

++

++

++

++

++

++

+

++

++

++

++

++

++

++

++

+++

++

+-dipole

SLAB

Polarization induced surface charge densitysp Area A

Dielectrics

For the single dipole, the dipole moment is:

dqp

+-dipole

p

p

For the dipoles along one line between the endfaces, the dipole moment is:

dqpp

d

o

dqp

dqp

d

Dielectrics

The total dipole moment of the slab is then .

dQp

oE

Q: Total charge on one face of the slab.

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+

++

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+

+

++

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++

+

++

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++

+

++

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++

+

++

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+++

++

++

++

++

++

++

++

++

+

++

++

++

++

++

++

++

++

+++

++

Q

AQ sp

Dielectrics

The polarization or dipole moment per unit volume of the slab is then.

ddA

Q

Ad

dQ

v

pP sp

ˆˆ

oE

+++++++++

++++++++

+

+

+++++++

+++++++

++++++

+++++

++++

+++++

++++++++

+++++++

+++++++++++++++++++++

dP spˆ

d

End face Area A

Electric flux Density

𝐷=𝜀𝐸FLUX CAPACITOR

MEETS THE FUSOR

Dielectrics

Electric field is shown normal to the surface. oE

oE

The electric flux density vectors Do = Dd . We are treating only normal components here.

d

Endface Area A

o

oEo

oD

oD

d

dDdE

DielectricsThese two charge sheets will produce an electric field directed from the positive sheet towards the negative sheet.

Original slab in external electric field

Polarization bound charge sheets of each endface.

In order to determine the magnitude of the electric field, a parallel plate capacitor analysis can be applied to this charge configuration.

Dielectrics

Through Gaussian analysis:

Polarization bound charge sheets of each end face.

iE

o

sp

iE

Induced electric field due to the polarization effect in the dielectric.

Note: The external field Eo and the induced field Ei are in opposite directions.

If electrons where free in the dielectric, then the magnitudes of Eo and Ei would end up the same, their directions would be opposite and the net electric field inside the medium would be zero. Such a material is called a metal due to the free electrons.

PD spi

P

iD

Dielectrics

oE

d

Endface Area A

o

oEo

oD

oD

d

dDdE

Further manipulations of equations required to obtain desired result. Recall that desired result is: Why mediums have a different dielectric constant from that of vacuum?

iE

iD

P

sp

Dielectrics

oE

dEndface Area A

o

d

dE

Vector diagram

iE

spsp

oE

o

iod EEE

inside dielectric

oE

Dielectrics

oE

dEndface Area A

o

d

dE

Vector diagram

iE

spsp

oE

o

odoio EEEP

Then

Since:

oE

o

sp

iE

PD spi

and

Dielectrics

oE

dEndface Area A

o

d

dE

Vector diagram

iE

spsp

oE

o

PEE dooo

Then

And:

oE

with ddoo EE

PED dod

Dielectrics

oE

dEndface Area A

o

d

dE

Vector diagram

iE

spsp

oE

o

The dielectric constant can now be obtain ed using :

oE

ddd ED

dod E

P

With the electric susceptibility of the material.C

Dielectrics

dod E

P EP oC

1od

1r

Why is the dielectric constant in the medium

different than the dielectric constant in

vacuum?Answer: Polarization

and orientation of internal dipoles.

ELEC 3105 Basic EM and Power Engineering

Next slides: forget me not

Dielectric Materials

Polarization Field

P = electric flux density induced by E

Electric Breakdown

Electric Breakdown

Electric flux Density

𝐷=𝜀𝐸

S

E dAE

o

enclosedE

q

From other definitions of flux we can obtain other useful expressions for electrostatics

VS

E dVEdAE

V o

V

o

enclosedE dV

q

Divergence

theorem

dVdVEV V o

V

Divergence theorem

dVdVEV S o

V

o

VE

Integrands must be the same for all dV

Point functionGauss’s law in differential form

VE

Medium dependence

Divergence theorem

dVdVEV S

Vo

VD

Integrands must be the same for all dV

Point functionGauss’s law in differential form

No dependence on the dielectric constant

Boundary conditions Normal Component of D

snn DD 21

ELECTROSTATICS

Gaussian Surface

Air Dielectric

Gaussian surface on metal interface encloses a real net charge s.

Gaussian surface on dielectric interface encloses a bound surface charge sp , but also encloses the other half of the dipole as well. As a result Gaussian surface encloses no net surface charge.

snD 1

021 nn DD

21 nn DD

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