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### EEE 498/598Overview of Electrical Engineering

Lecture 3:

Electrostatics: Electrostatic Potential; Charge Dipole; Visualization of Electric Fields; Potentials; Gauss’s Law and Applications; Conductors and Conduction Current

1

Lecture 3 Objectives

- To continue our study of electrostatics with electrostatic potential; charge dipole; visualization of electric fields and potentials; Gauss’s law and applications; conductors and conduction current.

2

Electrostatic Potential Resulting from Multiple Point Charges

Q2

P(R,q,f)

Q1

O

No longer spherically symmetric!

4

Electrostatic Potential Resulting from Continuous Charge Distributions

line charge

surface charge

volume charge

5

-Q

+Q

d

Charge Dipole- An electric charge dipole consists of a pair of equal and opposite point charges separated by a small distance (i.e., much smaller than the distance at which we observe the resulting field).

6

+Q

-Q

Dipole Moment- Dipole momentp is a measure of the strength
- of the dipole and indicates its direction

p isin the direction from the negative point charge to the positive point charge

7

Electrostatic Potential Due to Charge Dipole in the Far-Field

- assume R>>d

- zeroth order approximation:

not good

enough!

11

Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)

- first order approximation from geometry:

q

d/2

d/2

lines approximately

parallel

12

Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)

- Taylor series approximation:

13

Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)

14

Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)

- In terms of the dipole moment:

15

Visualization of Electric Fields

- An electric field (like any vector field) can be visualized using flux lines (also called streamlinesor 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.

17

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.

18

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.

19

Faraday’s Experiment (Cont’d)

- Two concentric conducting spheres are separated by an insulating material.
- The inner sphere is charged to +Q. Theouter sphere is initially uncharged.
- The outer sphere is groundedmomentarily.
- The charge on the outer sphere is found to be -Q.

21

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.

22

Electric (Displacement) Flux Density

- The density of electric displacement is the electric (displacement) flux density, D.
- In free space the relationship between flux density and electric field is

24

Electric (Displacement) Flux Density (Cont’d)

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

25

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.”

26

S

ds

V

Gauss’s Law (Cont’d)By convention, ds

is taken to be outward

from the volume V.

Since volume charge

density is the most

general, we can always write

Qencl in this way.

27

Applications of Gauss’s Law

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

known

unknown

28

Applications of Gauss’s Law (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.

29

Applications of Gauss’s Law (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.

30

Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d)

(1) Assume from symmetry the form of the field

(2) Construct a family of Gaussian surfaces

spherical symmetry

spheres of radius r where

32

Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d)

(3) Evaluate the total charge within the volume enclosed by each Gaussian surface

33

Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d)

(4) For each Gaussian surface, evaluate the integral

surface area

of Gaussian

surface.

magnitude of D

on Gaussian

surface.

35

Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d)

(5) Solve for D on each Gaussian surface

36

a

b

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s LawConsider a spherical shell of uniform charge density:

37

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)

(1) Assume from symmetry the form of the field

(2) Construct a family of Gaussian surfaces

spheres of radius r where

38

a

b

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)- Here, we shall need to treat separately 3 sub-families of Gaussian surfaces:

1)

2)

3)

39

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)

Gaussian surfaces

for which

Gaussian surfaces

for which

Gaussian surfaces

for which

40

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)

(3) Evaluate the total charge within the volume enclosed by each Gaussian surface

41

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)

(4) For each Gaussian surface, evaluate the integral

surface area

of Gaussian

surface.

magnitude of D

on Gaussian

surface.

44

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)

(5) Solve for D on each Gaussian surface

45

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)

46

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)

- Notice that for r > b

Total charge contained

in spherical shell

47

0.7

0.6

0.5

0.4

(C/m)

r

D

0.3

0.2

0.1

0

0

1

2

3

4

5

6

7

8

9

10

R

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)48

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

49

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (Cont’d)

(1) Assume from symmetry the form of the field

(2) Construct a family of Gaussian surfaces

cylinders of radius r where

50

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (Cont’d)

(3) Evaluate the total charge within the volume enclosed by each Gaussian surface

cylinder is

infinitely long!

51

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (Cont’d)

(4) For each Gaussian surface, evaluate the integral

surface area

of Gaussian

surface.

magnitude of D

on Gaussian

surface.

52

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (Cont’d)

(5) Solve for D on each Gaussian surface

53

V

S

Recall the Divergence Theorem- Also called Gauss’s theorem or Green’s theorem.
- Holds for any volume and corresponding closed surface.

55

Applying Divergence Theorem to Gauss’s Law

Because the above must hold for any

volume V, we must have

Differential form

of Gauss’s Law

56

Fields in Materials

- Materials contain charged particles that respond to applied electric and magnetic fields.
- Materials are classified according to the nature of their response to the applied fields.

57

Conductors

- A conductor is a material in which electrons in the outermost shell of the electron migrate easily from atom to atom.
- Metallic materials are in general good conductors.

59

-eConduction Current

- In an otherwise empty universe, a constant electric field would cause an electron to move with constant acceleration.

e = 1.602 10-19C

magnitude of electron charge

60

Conduction Current (Cont’d)

- In a conductor, electrons are constantly colliding with each other and with the fixed nuclei, and losing momentum.
- The net macroscopic effect is that the electrons move with a (constant) drift velocity vd which is proportional to the electric field.

Electron mobility

61

Conductor in an Electrostatic Field

- To have an electrostatic field, all charges must have reached their equilibrium positions (i.e., they are stationary).
- Under such static conditions, there must be zero electric field within the conductor. (Otherwise charges would continue to flow.)

62

Conductor in an Electrostatic Field (Cont’d)

- If the electric field in which the conductor is immersed suddenly changes, charge flows temporarily until equilibrium is once again reached with the electric field inside the conductor becoming zero.
- In a metallic conductor, the establishment of equilibrium takes place in about 10-19 s - an extraordinarily short amount of time indeed.

63

Conductor in an Electrostatic Field (Cont’d)

- There are two important consequences to the fact that the electrostatic field inside a metallic conductor is zero:
- The conductor is an equipotential body.
- The charge on a conductor must reside entirely on its surface.
- A corollary of the above is that the electric field just outside the conductor must be normal to its surface.

64

Macroscopic versus Microscopic Fields

- In our study of electromagnetics, we use Maxwell’s equations which are written in terms of macroscopic quantities.
- The lower limit of the classical domain is about 10-8 m = 100 angstroms. For smaller dimensions, quantum mechanics is needed.

66

Induced Charges on Conductors

- The BCs given above imply that if a conductor is placed in an externally applied electric field, then
- the field distribution is distorted so that the electric field lines are normal to the conductor surface
- a surface charge is inducedon the conductor to support the electric field

68

Applied and Induced Electric Fields

- The applied electric field (Eapp) is the field that exists in the absence of the metallic conductor (obstacle).
- The induced electric field (Eind) is the field that arises from the induced surface charges.
- The total field is the sum of the applied and induced electric fields.

69

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