1

ElectrostaticsElectrostatics

ElectrostaticsElectrostatics is the branch of is the branch of electromagnetics dealing with electromagnetics dealing with the effects of electric charges the effects of electric charges at rest.at rest.

The fundamental law of The fundamental law of electrostaticselectrostatics is is Coulomb’s lawCoulomb’s law. .

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Electric ChargeElectric Charge

Electrical phenomena caused by Electrical phenomena caused by friction are part of our everyday friction are part of our everyday lives, and can be understood in lives, and can be understood in terms of terms of electrical chargeelectrical charge..

The effects of The effects of electrical chargeelectrical charge can be can be observed in the attraction/repulsion observed in the attraction/repulsion of various objects when “charged.” of various objects when “charged.”

Charge comes in two varieties called Charge comes in two varieties called “positive” and “negative.”“positive” and “negative.”

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Electric ChargeElectric Charge

Objects carrying a net positive charge Objects carrying a net positive charge attract those carrying a net negative charge attract those carrying a net negative charge and repel those carrying a net positive and repel those carrying a net positive charge.charge.

Objects carrying a net negative charge Objects carrying a net negative charge attract those carrying a net positive charge attract those carrying a net positive charge and repel those carrying a net negative and repel those carrying a net negative charge.charge.

On an atomic scale, electrons are negatively On an atomic scale, electrons are negatively charged and nuclei are positively charged.charged and nuclei are positively charged.

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Electric ChargeElectric Charge

Electric charge is inherently Electric charge is inherently quantized such that the charge on quantized such that the charge on any object is an integer multiple of any object is an integer multiple of the smallest unit of charge which is the smallest unit of charge which is the magnitude of the electron the magnitude of the electron charge charge ee = 1.602 = 1.602 10 10-19-19 CC..

On the macroscopic level, we can On the macroscopic level, we can assume that charge is “continuous.”assume that charge is “continuous.”

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Coulomb’s LawCoulomb’s Law

Coulomb’s lawCoulomb’s law is the “law of action” is the “law of action” between charged bodies.between charged bodies.

Coulomb’s lawCoulomb’s law gives the electric force gives the electric force between two between two point chargespoint charges in an in an otherwise empty universe.otherwise empty universe.

A A point chargepoint charge is a charge that occupies a is a charge that occupies a region of space which is negligibly small region of space which is negligibly small compared to the distance between the compared to the distance between the point charge and any other object. point charge and any other object.

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Coulomb’s LawCoulomb’s Law

2120

2112 4

ˆ12 r

QQaF R

Q1

Q212r

12F

Force due to Q1

acting on Q2

Unit vector indirection of R12

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Coulomb’s LawCoulomb’s Law

The force on The force on QQ11 due to due to QQ22 is equal is equal in magnitude but opposite in in magnitude but opposite in direction to the force on direction to the force on QQ22 due due to to QQ11..

1221 FF

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Electric FieldElectric Field

Consider a point Consider a point charge charge QQ placed at placed at the the originorigin of a of a coordinate system coordinate system in an otherwise in an otherwise empty universe.empty universe.

A test charge A test charge QQtt brought near brought near QQ experiences a experiences a force:force:

204

ˆr

QQaF t

rQt

Q

Qt

r

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Electric FieldElectric Field

The existence of the force on The existence of the force on QQtt can can be attributed to an be attributed to an electric fieldelectric field produced by produced by QQ..

The The electric fieldelectric field produced by produced by QQ at a at a point in space can be defined as the point in space can be defined as the force per unit charge acting on a force per unit charge acting on a test charge test charge QQtt placed at that point.placed at that point.

t

Q

Q Q

FE t

t 0lim

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Electric FieldElectric Field The basic units of electric field The basic units of electric field

are are newtons per coulombnewtons per coulomb.. In practice, we usually use In practice, we usually use volts volts

per meterper meter..

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Continuous Distributions Continuous Distributions of Chargeof Charge

Charge can occur asCharge can occur as point chargespoint charges (C) (C) volume chargesvolume charges (C/m (C/m33)) surface chargessurface charges (C/m (C/m22)) line chargesline charges (C/m) (C/m)

most general

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Electrostatic PotentialElectrostatic Potential

An electric field is a An electric field is a force fieldforce field.. If a body being acted on by a If a body being acted on by a

force is moved from one point to force is moved from one point to another, then another, then workwork is done.is done.

The concept of The concept of scalar electric scalar electric potentialpotential provides a measure of provides a measure of the work done in moving charged the work done in moving charged bodies in an electrostatic field.bodies in an electrostatic field.

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Electrostatic PotentialElectrostatic Potential The work done in moving a test charge from one The work done in moving a test charge from one

point to another in a region of electric field:point to another in a region of electric field:

b

a

b

a

ba ldEqldFW

ab

q

F

ld

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Electrostatic PotentialElectrostatic Potential

The electrostatic field is The electrostatic field is conservativeconservative:: The value of the line integral The value of the line integral

depends only on the end points depends only on the end points and is independent of the path and is independent of the path taken.taken.

The value of the line integral The value of the line integral around any closed path is zero.around any closed path is zero.0

C

ldEC

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Electrostatic PotentialElectrostatic Potential

The work done per unit charge in The work done per unit charge in moving a test charge from point moving a test charge from point aa to point to point bb is the is the electrostatic potential electrostatic potential differencedifference between the two points: between the two points:

b

a

baab ldE

q

WV

electrostatic potential differenceUnits are volts.

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Electrostatic PotentialElectrostatic Potential Since the electrostatic field is Since the electrostatic field is

conservative we can writeconservative we can write

aVbV

ldEldE

ldEldEldEV

a

P

b

P

b

P

P

a

b

a

ab

00

0

0

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Electrostatic PotentialElectrostatic Potential

Thus the Thus the electrostatic potentialelectrostatic potential VV is a is a scalar field that is defined at scalar field that is defined at every point in space.every point in space.

In particular the value of the In particular the value of the electrostatic potentialelectrostatic potential at any point at any point PP is given byis given by

P

P

ldErV0 reference point

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Electrostatic PotentialElectrostatic Potential

The The reference pointreference point ( (PP00) is where the ) is where the potential is zero (analogous to potential is zero (analogous to groundground in a circuit).in a circuit).

Often the reference is taken to be at Often the reference is taken to be at infinity so that the potential of a infinity so that the potential of a point in space is defined aspoint in space is defined as

P

ldErV

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Electrostatic Potential Electrostatic Potential and Electric Fieldand Electric Field

The work done in moving a point The work done in moving a point charge from point charge from point aa to point to point bb can be written as can be written as

b

a

abba

ldEQ

aVbVQVQW

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Electrostatic Potential Electrostatic Potential and Electric Fieldand Electric Field

Along a short path of length Along a short path of length ll we havewe have

lEV

lEQVQW

or

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Electrostatic Potential Electrostatic Potential and Electric Fieldand Electric Field

Along an incremental path of Along an incremental path of length length dldl we havewe have

Recall from the definition of Recall from the definition of directional derivativedirectional derivative::

ldEdV

ldVdV

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Electrostatic Potential Electrostatic Potential and Electric Fieldand Electric Field

Thus:Thus:

VE

the “del” or “nabla” operator

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Visualization of Electric Visualization of Electric FieldsFields

An electric field (like any vector field) can An electric field (like any vector field) can be visualized using be visualized using flux linesflux lines (also called (also called streamlinesstreamlines or or lines of forcelines of force).).

A A flux lineflux line is drawn such that it is is drawn such that it is everywhere tangent to the electric field.everywhere tangent to the electric field.

A A quiver plotquiver plot is a plot of the field lines is a plot of the field lines constructed by making a grid of points. An constructed by making a grid of points. An arrow whose tail is connected to the point arrow whose tail is connected to the point indicates the direction and magnitude of indicates the direction and magnitude of the field at that point.the field at that point.

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Visualization of Electric Visualization of Electric PotentialsPotentials

The scalar electric potential can be The scalar electric potential can be visualized using visualized using equipotential surfacesequipotential surfaces..

An An equipotential surfaceequipotential surface is a surface over is a surface over which which VV is a constant. is a constant.

Because the electric field is the negative of Because the electric field is the negative of the gradient of the electric scalar the gradient of the electric scalar potential, the electric field lines are potential, the electric field lines are everywhere normal to the equipotential everywhere normal to the equipotential surfaces and point in the direction of surfaces and point in the direction of decreasing potential.decreasing potential.

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Faraday’s ExperimentFaraday’s Experiment

charged sphere(+Q)

+

+

+ +

insulator

metal

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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)

+Q

-Q

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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

ED 0

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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.”

encl

S

QsdD

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

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.

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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.

encl

S

QsdD known

unknown

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

V

evencl

S

dvqQsdD

VS

sd

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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

VS

sd

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

V

ev

VS

dvqdvDsdD

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

evqD Differential formof Gauss’s Law

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The Need for Poisson’s The Need for Poisson’s and Laplace’s and Laplace’s

Equations (Cont’d)Equations (Cont’d) Poisson’s equationPoisson’s equation is a differential equation for is a differential equation for

the electrostatic potential the electrostatic potential VV. Poisson’s . Poisson’s equation and the boundary conditions equation and the boundary conditions applicable to the particular geometry form a applicable to the particular geometry form a boundary-value problem that can be solved boundary-value problem that can be solved either analytically for some geometries or either analytically for some geometries or numerically for any geometry.numerically for any geometry.

After the electrostatic potential is evaluated, After the electrostatic potential is evaluated, the electric field is obtained usingthe electric field is obtained using

rVrE

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Derivation of Poisson’s Derivation of Poisson’s EquationEquation

For now, we shall assume the For now, we shall assume the only materials present are free only materials present are free space and conductors on which space and conductors on which the electrostatic potential is the electrostatic potential is specified. However, Poisson’s specified. However, Poisson’s equation can be generalized for equation can be generalized for other materials (dielectric and other materials (dielectric and magnetic as well).magnetic as well).

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Derivation of Poisson’s Derivation of Poisson’s Equation (Cont’d)Equation (Cont’d)

0

0

ev

evev

qVVE

qEqD

V2

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Derivation of Poisson’s Derivation of Poisson’s Equation (Cont’d)Equation (Cont’d)

0

2

evq

V Poisson’sequation

2 is the Laplacian operator. The Laplacian of a scalarfunction is a scalar function equal to the divergence of thegradient of the original scalar function.

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Laplacian Operator in Laplacian Operator in Cartesian, Cylindrical, and Cartesian, Cylindrical, and

Spherical CoordinatesSpherical Coordinates

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Laplace’s EquationLaplace’s Equation Laplace’s equationLaplace’s equation is the homogeneous form is the homogeneous form

of of Poisson’s equationPoisson’s equation.. We use Laplace’s equation to solve We use Laplace’s equation to solve

problems where potentials are specified problems where potentials are specified on conducting bodies, but no charge on conducting bodies, but no charge exists in the free space region.exists in the free space region.

02 V Laplace’sequation

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Uniqueness TheoremUniqueness Theorem

A solution to Poisson’s or A solution to Poisson’s or Laplace’s equation that satisfies Laplace’s equation that satisfies the given boundary conditions is the given boundary conditions is the the uniqueunique (i.e., the one and only (i.e., the one and only correct) solution to the problem.correct) solution to the problem.

46

Fundamental Laws of Fundamental Laws of Electrostatics in Integral Electrostatics in Integral

FormForm

V

ev

S

C

dvqsdD

ldE 0

ED

Conservative field

Gauss’s law

Constitutive relation

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Fundamental Laws of Fundamental Laws of Electrostatics in Electrostatics in Differential FormDifferential Form

evqD

E

0

ED

Conservative field

Gauss’s law

Constitutive relation

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Fundamental Laws of Fundamental Laws of ElectrostaticsElectrostatics

The integral forms of the fundamental The integral forms of the fundamental laws are more general because they apply laws are more general because they apply over regions of space. The differential over regions of space. The differential forms are only valid at a point.forms are only valid at a point.

From the integral forms of the From the integral forms of the fundamental laws both the differential fundamental laws both the differential equations governing the field within a equations governing the field within a medium and the boundary conditions at medium and the boundary conditions at the interface between two media can be the interface between two media can be derived.derived.

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Boundary ConditionsBoundary Conditions

Within a homogeneous medium, Within a homogeneous medium, there are no abrupt changes in there are no abrupt changes in EE or or DD. However, at the interface . However, at the interface between two different media between two different media (having two different values of (having two different values of , it is obvious that one or both , it is obvious that one or both of these must change abruptly.of these must change abruptly.

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Boundary Conditions Boundary Conditions (Cont’d)(Cont’d)

To derive the boundary To derive the boundary conditions on the normal and conditions on the normal and tangential field conditions, we tangential field conditions, we shall apply the integral form of shall apply the integral form of the two fundamental laws to an the two fundamental laws to an infinitesimally small region that infinitesimally small region that lies partially in one medium and lies partially in one medium and partially in the other.partially in the other.

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Boundary Conditions Boundary Conditions (Cont’d)(Cont’d)

Consider two semi-infinite media separated Consider two semi-infinite media separated by a boundary. A surface charge may exist at by a boundary. A surface charge may exist at the interface.the interface.

Medium 1

Medium 2x xx xs

52

Boundary Conditions Boundary Conditions (Cont’d)(Cont’d)

Locally, the boundary will look planarLocally, the boundary will look planar

12 22 , DE

11, DEx x x x x x s

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Boundary Condition on Boundary Condition on Normal Component of Normal Component of

DD• Consider an infinitesimal cylinder (pillbox) with cross-sectional area s and height h lying half in medium 1 and half in medium 2:

1

222 , DE

11, DEs

h/2

h/2

x x x x x xs

na

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Boundary Condition on Boundary Condition on Normal Component of DNormal Component of D

(Cont’d)(Cont’d) Applying Gauss’s law to the pillbox, we have Applying Gauss’s law to the pillbox, we have

sqRHS

sDsD

sdDsdDsdDLHS

dvqsdD

es

nn

sidebottomtop

V

ev

S

21

0

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Boundary Condition on Boundary Condition on Normal Component of D Normal Component of D

(Cont’d)(Cont’d) The boundary condition isThe boundary condition is

If there is no surface chargeIf there is no surface charge

snn DD 21

nn DD 21 For non-conductingmaterials, s = 0 unlessan impressed source ispresent.

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Boundary Condition on Boundary Condition on Tangential Component Tangential Component

of Eof E• Consider an infinitesimal path abcd with width w and height h lying half in medium 1 and half in medium 2:

1

2

na

22 , DE

11, DE

h/2

h/2

w

a

bc

d

57

Boundary Condition on Boundary Condition on Tangential Component of Tangential Component of

EE (Cont’d) (Cont’d)

naa

bc

d

sa

tapath alongboundary

the toalr tangentiunit vecto ˆˆˆ

contour by the defineddirection in the

path lar toperpendicur unit vecto ˆ

nst

s

aaa

abcda

58

Boundary Condition on Boundary Condition on Tangential Component of Tangential Component of EE

(Cont’d)(Cont’d) Applying conservative law to the path, we have Applying conservative law to the path, we have

wEE

wEh

Eh

EwEh

Eh

E

ldEldEldEldELHS

ldE

tt

tnntnn

a

d

d

c

c

b

b

a

C

21

121221 2222

0

59

The boundary condition isThe boundary condition is

tt EE 21

Boundary Condition on Boundary Condition on Tangential Component of E Tangential Component of E

(Cont’d)(Cont’d)

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Electrostatic Boundary Electrostatic Boundary Conditions - SummaryConditions - Summary

At any point on the boundary,At any point on the boundary, the components of the components of EE11 and and EE22

tangential to the boundary are equaltangential to the boundary are equal the components of the components of DD11 and and DD22 normal normal

to the boundary are discontinuous to the boundary are discontinuous by an amount equal to any surface by an amount equal to any surface charge existing at that pointcharge existing at that point