electromagnetic theory - web.uettaxila.edu.pk · 1 electromagnetic theory chapter no. 3:...

1

Electromagnetic TheoryChapter No. 3:

Electrostatics:

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

and Conduction Current

By

Engr. Mian Shahzad Iqbal

Lecturer, Telecom Department

University of Engineering and Technology

Taxila

Lecture 32

Chapter No 3

To continue our study of visualization of

electric fields and potentials; Gauss’s law

and applications

Lecture 3

Today’s Quotation

You (humans) think that you are insignificant, while there is a great universe contained in you.By Hazrat Ali (may Allah be pleased with him)

3

Lecture 34

Visualization of Electric Fields

An electric field (like any vector field) can be

visualized using flux lines (also called streamlines

or lines of force).

A flux line is drawn such that it is everywhere

tangent to the electric field.

A quiver plot is a plot of the field lines constructed

by making a grid of points. An arrow whose tail is

connected to the point indicates the direction and

magnitude of the field at that point.

Lecture 35

Visualization of Electric

Potentials

The scalar electric potential can be visualized using

equipotential surfaces.

An equipotential surface is a surface over which V

is a constant.

Because the electric field is the negative of the

gradient of the electric scalar potential, the electric

field lines are everywhere normal to the

equipotential surfaces and point in the direction of

decreasing potential.

Lecture 36

Visualization of Electric Fields

Flux lines are suggestive of the flow of some

fluid emanating from positive charges

(source) and terminating at negative charges

(sink).

Although electric field lines do NOT

represent fluid flow, it is useful to think of

them as describing the flux of something that,

like fluid flow, is conserved.

Lecture 37

Faraday’s Experiment

charged sphere

(+Q)

+

+

+ +

insulator

metal

Lecture 38

Faraday’s Experiment (Cont’d)

Two concentric conducting spheres are

separated by an insulating material.

The inner sphere is charged to +Q. The

outer sphere is initially uncharged.

The outer sphere is grounded momentarily.

The charge on the outer sphere is found to

be -Q.

Lecture 39

Faraday’s Experiment (Cont’d)

Faraday concluded there was a

“displacement” from the charge on the inner

sphere through the inner sphere through the

insulator to the outer sphere.

The electric displacement (or electric flux)

is equal in magnitude to the charge that

produces it, independent of the insulating

material and the size of the spheres.

Lecture 310

Electric Displacement (Electric

Flux)

+Q

-Q

Lecture 311

Electric (Displacement) Flux

Density

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

In free space the relationship between flux densityand electric field is

ED 0

Lecture 312

Electric (Displacement) Flux

Density (Cont’d)

The electric (displacement) flux density for

a point charge centered at the origin is

Lecture 313

Gauss’s Law

Gauss’s law states that “the net electric

flux emanating from a close surface S is

equal to the total charge contained

within the volume V bounded by that

surface.”

encl

S

QsdD

Lecture 314

Gauss’s Law (Cont’d)

V

Sds

By convention, ds

is taken to be outward

from the volume V.

V

evencl dvqQ

Since volume charge

density is the most

general, we can always write

Qencl in this way.

Lecture 315

Applications of Gauss’s Law

Gauss’s law is an integral equation for the

unknown electric flux density resulting

from a given charge distribution.

encl

S

QsdD known

unknown

Lecture 316


(Cont’d)

In general, solutions to integral equations

must be obtained using numerical

techniques.

However, for certain symmetric charge

distributions closed form solutions to

Gauss’s law can be obtained.

Lecture 317


(Cont’d)

Closed form solution to Gauss’s law relies

on our ability to construct a suitable family

of Gaussian surfaces.

A Gaussian surface is a surface to which

the electric flux density is normal and over

which equal to a constant value.

Lecture 318

Electric Flux Density of a Point

Charge Using Gauss’s Law

Consider a point charge at the origin:

Q

Lecture 319


Charge Using Gauss’s Law (Cont’d)

(1) Assume from symmetry the form of the field

(2) Construct a family of Gaussian surfaces

rDaD rrˆ

spheres of radius r where

r0

spherical

symmetry

Lecture 320



(3) Evaluate the total charge within the volume

enclosed by each Gaussian surface

V

evencl dvqQ

Lecture 321



Q

R

Gaussian surface

QQencl

Lecture 322



(4) For each Gaussian surface, evaluate the integral

DSsdDS

24 rrDsdD r

S

magnitude of D

on Gaussian

surface.

surface area

of Gaussian

surface.

Lecture 323



(5) Solve for D on each Gaussian surface

S

QD encl

24ˆ

r

QaD r

2

00 4ˆ

r

Qa

DE r

Lecture 324

Electric Flux Density of a Spherical

Shell of Charge Using Gauss’s Law

Consider a spherical shell of uniform charge

density:

otherwise,0

,0 braqqev

a

b

Lecture 325

Electric Flux Density of a Spherical Shell

of Charge Using Gauss’s Law (Cont’d)



RDaD rrˆ

spheres of radius r where

r0

Lecture 326



Here, we shall need to treat separately 3 sub-

families of Gaussian surfaces:

ar 01)

bra 2)

br 3)

a

b

Lecture 327

Electric Flux Density of a Spherical Shell of


Gaussian surfaces

for which

ar 0

Gaussian surfaces

for which

bra

Gaussian surfaces

for which

br

Lecture 328





V

evencl dvqQ

Lecture 329



0enclQ For

For

ar 0

bra

33

0

3

0

3

00

3

4

3

4

3

4

arq

aqrqdvqQ

r

a

encl

Lecture 330



For

33

0

3

0

3

0

3

4

3

4

3

4

abq

aqbqdvqQ

b

a

evencl

br

Lecture 331




DSsdDS

24 rrDsdD r

S

magnitude of D

on Gaussian

surface.

surface area

of Gaussian

surface.

Lecture 332




S

QD encl

Lecture 333



brr

abqa

r

abq

a

brar

ar

qa

r

arq

a

ar

D

rr

rr

,3

ˆ4

3

4

ˆ

,3

ˆ4

3

4

ˆ

0,0

2

33

0

2

33

0

2

3

0

2

33

0

Lecture 334



Notice that for r > b

24ˆ

r

QaD tot

r

Total charge contained

in spherical shell

Lecture 335



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 3

0

b

a

q

Lecture 336

Electric Flux Density of an Infinite

Line Charge Using Gauss’s Law

Consider a infinite line charge carrying charge per

unit length of qel:

z

elq

Lecture 337

Electric Flux Density of an Infinite Line




DaD ˆ

cylinders of radius where

0

Lecture 338





L

elencl dlqQ

lqQ elencl cylinder is

infinitely long!

Lecture 339




DSsdDS

lDsdDS

2

magnitude of D

on Gaussian

surface.

surface area

of Gaussian

surface.

Lecture 340




S

QD encl

2ˆ elqaD

Lecture 341

Gauss’s Law in Integral Form

V

evencl

S

dvqQsdD

VS

sd

Lecture 342

Recall the Divergence Theorem

Also called Gauss’s

theorem or Green’s

theorem.

Holds for any volume

and corresponding

closed surface.

dvDsdDVS

VS

sd

Lecture 343

Applying Divergence Theorem to

Gauss’s Law

V

ev

VS

dvqdvDsdD

Because the above must hold for any

volume V, we must have

evqD Differential form

of Gauss’s Law

electromagnetic theory - web.uettaxila.edu.pk · 1 electromagnetic theory chapter no. 3:...

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