Electrostatic Potential; Charge Dipole;

EC 2253 EC 2253 ELECTROMAGNETIC ELECTROMAGNETIC

FIELDSFIELDSOverview ofOverview of

Electrostatics: Electrostatic Electrostatics: Electrostatic Potential; Charge Dipole; Potential; Charge Dipole;

Visualization of Electric Fields; Visualization of Electric Fields; Potentials; Gauss’s Law and Potentials; Gauss’s Law and

Applications; Applications;

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Electrostatic Potential of Electrostatic Potential of a Point Charge at the a Point Charge at the

OriginOrigin

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Q

P

r

r

Q

r

rdQ

rdar

QaldErV

r

r

rr

r

02

0

20

44

ˆ4

ˆ

spherically symmetric

Electrostatic Potential Electrostatic Potential Resulting from Multiple Resulting from Multiple

Point ChargesPoint Charges

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Q1

P(R,)

r 1R

1rO

Q2

2r

n

k k

k

R

QrV

1 04

2R

No longer spherically symmetric!

Electrostatic Potential Electrostatic Potential Resulting from Continuous Resulting from Continuous

Charge DistributionsCharge Distributions

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V

ev

S

es

L

el

R

vdrqrV

R

sdrqrV

R

ldrqrV

0

0

0

4

1

4

1

4

1

line charge

surface charge

volume charge

Charge DipoleCharge Dipole An An electric charge dipoleelectric charge dipole consists of a pair of equal consists of a pair of equal

and opposite point charges separated by a small and opposite point charges separated by a small distance (i.e., much smaller than the distance at distance (i.e., much smaller than the distance at which we observe the resulting field).which we observe the resulting field).

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d

+Q -Q

Dipole MomentDipole Moment

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• Dipole moment p is a measure of the strength of the dipole and indicates its direction

dQp +Q

-Q

dp is in the direction from the negative point charge to the positive point charge

Electrostatic Potential Electrostatic Potential Due to Charge DipoleDue to Charge Dipole

April 8, 2023 3

observationpoint

d/2

+Q

-Q

z

d/2

P

Qdap zˆ

R

Rr

Electrostatic Potential Electrostatic Potential Due to Charge Dipole Due to Charge Dipole

(Cont’d)(Cont’d)

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R

Q

R

QrVrV

00 44,

cylindrical symmetry

Electrostatic Potential Electrostatic Potential Due to Charge Dipole Due to Charge Dipole


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d/2

d/2

cos)2/(

cos)2/(

22

22

rddrR

rddrR

R

R

r

P

Electrostatic Potential Electrostatic Potential Due to Charge Dipole in Due to Charge Dipole in

the Far-Fieldthe Far-Field

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• assume R>>d

• zeroth order approximation:

RR

RR

0V

not goodenough!

Electrostatic Potential Due Electrostatic Potential Due to Charge Dipole in the Far-to Charge Dipole in the Far-

Field (Cont’d)Field (Cont’d)

April 8, 2023 3

• first order approximation from geometry:

cos2

cos2d

rR

drR

d/2

d/2

lines approximatelyparallel

R

R

r

Electrostatic Potential Due Electrostatic Potential Due to Charge Dipole in the Far-to Charge Dipole in the Far-

Field (Cont’d)Field (Cont’d)

April 8, 2023 3

• Taylor series approximation:

cos2

111

cos2

11

cos2

11

cos2

111

r

d

rR

r

d

r

r

d

r

dr

R

1,11

:Recall

xnxx n

Electrostatic Potential Due to Electrostatic Potential Due to Charge Dipole in the Far-Field Charge Dipole in the Far-Field


April 8, 2023 3

20

0

4

cos

2

cos1

2

cos1

4,

r

Qd

r

d

r

d

r

QrV

Electrostatic Potential Due to Electrostatic Potential Due to Charge Dipole in the Far-Field Charge Dipole in the Far-Field


April 8, 2023 3

• In terms of the dipole moment:

20

ˆ

4

1

r

apV r

Electric Field of Charge Electric Field of Charge Dipole in the Far-FieldDipole in the Far-Field

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sinˆcos2ˆ4

1ˆˆ

30

aar

Qd

V

ra

r

VaVE

r

r

Faraday’s ExperimentFaraday’s Experiment

April 8, 2023 3

charged sphere(+Q)

+

+

+ +

insulator

metal

Faraday’s Experiment Faraday’s Experiment (Cont’d)(Cont’d)

Two concentric conducting spheres are Two concentric conducting spheres are separated by an insulating material.separated by an insulating material.

The inner sphere is charged to The inner sphere is charged to ++QQ. . The The outer sphere is initially uncharged.outer sphere is initially uncharged.

The outer sphere is The outer sphere is groundedgrounded momentarily.momentarily.

The charge on the outer sphere is The charge on the outer sphere is found to be found to be --QQ..

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Faraday’s Experiment Faraday’s Experiment (Cont’d)(Cont’d)

Faraday concluded there was a Faraday concluded there was a ““displacementdisplacement” from the charge on the inner ” from the charge on the inner sphere through the inner sphere through sphere through the inner sphere through the insulator to the outer sphere.the insulator to the outer sphere.

The The electric displacementelectric displacement (or (or electric fluxelectric flux) is ) is equal in magnitude to the charge that equal in magnitude to the charge that produces it, independent of the insulating produces it, independent of the insulating material and the size of the spheres.material and the size of the spheres.

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Electric Displacement Electric Displacement (Electric Flux)(Electric Flux)

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+Q

-Q

Electric (Displacement) Electric (Displacement) Flux DensityFlux Density

The density of electric displacement is the The density of electric displacement is the electric electric (displacement) flux density(displacement) flux density, , DD..

In free space the relationship between In free space the relationship between flux densityflux density and electric field is and electric field is

April 8, 2023 3

ED 0

Electric (Displacement) Electric (Displacement) Flux Density (Cont’d)Flux Density (Cont’d)

The electric (displacement) flux The electric (displacement) flux density for a point charge centered density for a point charge centered at the origin is at the origin is

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Gauss’s LawGauss’s Law Gauss’s law states that “the net electric Gauss’s law states that “the net electric

flux emanating from a close surface flux emanating from a close surface SS is is equal to the total charge contained within equal to the total charge contained within the volume the volume VV bounded by that surface.” bounded by that surface.”

April 8, 2023 3

encl

S

QsdD

Gauss’s Law (Cont’d)Gauss’s Law (Cont’d)

April 8, 2023 3

V

Sds

By convention, dsis taken to be outward

from the volume V.

V

evencl dvqQ

Since volume chargedensity is the most

general, we can always write Qencl in this way.

Applications of Gauss’s Applications of Gauss’s LawLaw

Gauss’s law is an Gauss’s law is an integral equationintegral equation for the for the unknown electric flux density resulting unknown electric flux density resulting from a given charge distribution.from a given charge distribution.

April 8, 2023 3

encl

S

QsdD known

unknown

Applications of Gauss’s Applications of Gauss’s Law (Cont’d)Law (Cont’d)

In general, solutions to In general, solutions to integral integral equationsequations must be obtained using must be obtained using numerical techniques.numerical techniques.

However, for certain symmetric However, for certain symmetric charge distributions closed form charge distributions closed form solutions to Gauss’s law can be solutions to Gauss’s law can be obtained.obtained.

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Applications of Gauss’s Applications of Gauss’s Law (Cont’d)Law (Cont’d)

Closed form solution to Gauss’s Closed form solution to Gauss’s law relies on our ability to law relies on our ability to construct a suitable family of construct a suitable family of Gaussian surfacesGaussian surfaces..

A A Gaussian surfaceGaussian surface is a surface to is a surface to which the electric flux density is which the electric flux density is normal and over which equal to normal and over which equal to a constant value.a constant value.

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Electric Flux Density of a Electric Flux Density of a Point Charge Using Point Charge Using

Gauss’s LawGauss’s LawConsider a point charge at the origin:Consider a point charge at the origin:

April 8, 2023 3

Q

Electric Flux Density of a Electric Flux Density of a Point Charge Using Gauss’s Point Charge Using Gauss’s

Law (Cont’d)Law (Cont’d)(1) Assume from symmetry the form of (1) Assume from symmetry the form of

the fieldthe field

(2) Construct a family of Gaussian (2) Construct a family of Gaussian surfacessurfaces

April 8, 2023 3

rDaD rrˆ

spheres of radius r where

r0

spherical symmetry


Law (Cont’d)Law (Cont’d)(3) Evaluate the total charge within the volume (3) Evaluate the total charge within the volume

enclosed by each Gaussian surface enclosed by each Gaussian surface

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V

evencl dvqQ


Law (Cont’d)Law (Cont’d)

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Q

R

Gaussian surface

QQencl


Law (Cont’d)Law (Cont’d)(4) For each Gaussian surface, (4) For each Gaussian surface,

evaluate the integralevaluate the integral

April 8, 2023 3

DSsdDS

24 rrDsdD r

S

magnitude of Don Gaussian

surface.

surface areaof Gaussian

surface.


Law (Cont’d)Law (Cont’d)(5) Solve for (5) Solve for DD on each Gaussian on each Gaussian

surfacesurface

April 8, 2023 3

S

QD encl

24ˆ

r

QaD r

2

00 4ˆ

r

Qa

DE r

Electric Flux Density of a Electric Flux Density of a Spherical Shell of Charge Spherical Shell of Charge

Using Gauss’s LawUsing Gauss’s LawConsider a spherical shell of uniform charge density:Consider a spherical shell of uniform charge density:

April 8, 2023 3

otherwise,0

,0 braqqev

a

b


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d)

(1) Assume from symmetry the form of (1) Assume from symmetry the form of the fieldthe field


April 8, 2023 3

RDaD rrˆ

spheres of radius r where

r0


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d) Here, we shall need to treat Here, we shall need to treat

separately 3 sub-families of Gaussian separately 3 sub-families of Gaussian surfaces:surfaces:

April 8, 2023 3

ar 01)

bra 2)

br 3)

a

b

Electric Flux Density of a Electric Flux Density of a Spherical Shell of Charge Using Spherical Shell of Charge Using


April 8, 2023 3

Gaussian surfacesfor which

ar 0


bra


br


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d)(3) Evaluate the total charge within the volume (3) Evaluate the total charge within the volume


April 8, 2023 3

V

evencl dvqQ


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d) For For

For For

April 8, 2023 3

0enclQar 0

bra

330

30

300

3

4

3

4

3

4

arq

aqrqdvqQr

a

encl


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d) For For

April 8, 2023 3

330

30

30

3

4

3

4

3

4

abq

aqbqdvqQb

a

evencl

br



(4) For each Gaussian surface, (4) For each Gaussian surface, evaluate the integralevaluate the integral

April 8, 2023 3

DSsdDS

24 rrDsdD r

S


surface.


surface.



(5) Solve for (5) Solve for DD on each Gaussian on each Gaussian surfacesurface

April 8, 2023 3

S

QD encl



April 8, 2023 3

brr

abqa

r

abqa

brar

ar

qa

r

arqa

ar

D

rr

rr

,3

ˆ4

34

ˆ

,3

ˆ4

34

ˆ

0,0

2

330

2

330

2

30

2

330


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d) Notice that for Notice that for r > br > b

April 8, 2023 3

24ˆ

r

QaD tot

r

Total charge containedin spherical shell



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0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

R

Dr (

C/m

)

m 2

m 1

C/m 1 30

b

a

q

Electric Flux Density of an Electric Flux Density of an Infinite Line Charge Using Infinite Line Charge Using

Gauss’s LawGauss’s LawConsider a infinite line charge carrying charge perConsider a infinite line charge carrying charge per

unit length of unit length of qqelel::

April 8, 2023 3

z

elq



(1) Assume from symmetry the form of (1) Assume from symmetry the form of the fieldthe field


April 8, 2023 3

DaD ˆ

cylinders of radius where

0


Gauss’s Law (Cont’d)Gauss’s Law (Cont’d)(3) Evaluate the total charge within the volume (3) Evaluate the total charge within the volume


April 8, 2023 3

L

elencl dlqQ

lqQ elencl cylinder is infinitely long!



(4) For each Gaussian surface, (4) For each Gaussian surface, evaluate the integralevaluate the integral

April 8, 2023 3

DSsdDS

lDsdDS

2


surface.


surface.



(5) Solve for (5) Solve for DD on each Gaussian on each Gaussian surfacesurface

April 8, 2023 3

S

QD encl

2ˆ elqaD

Gauss’s Law in Integral Gauss’s Law in Integral FormForm

April 8, 2023 3

V

evencl

S

dvqQsdD

VS

sd

Recall the Divergence Recall the Divergence TheoremTheorem

Also called Also called Gauss’s theoremGauss’s theorem or or Green’s theoremGreen’s theorem..

Holds for Holds for anyany volume and volume and corresponding corresponding closed surface.closed surface.

dvDsdDVS

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VS

sd

Applying Divergence Applying Divergence Theorem to Gauss’s LawTheorem to Gauss’s Law

April 8, 2023 3

V

ev

VS

dvqdvDsdD

Because the above must hold for any volume V, we must have

evqD Differential formof Gauss’s Law

Fields in MaterialsFields in Materials

Materials contain charged Materials contain charged particles that respond to applied particles that respond to applied electric and magnetic fields.electric and magnetic fields.

Materials are classified Materials are classified according to the nature of their according to the nature of their response to the applied fields.response to the applied fields.

April 8, 2023 3

Documents

Electrostatic Potential; Charge Dipole;