chapter 3: static electric fields -...

Chapter 3:

Static Electric Fields

3-1. Overview

3-2. Fundamental Postulates of Electrostatics in Free Space

3-3. Coulomb’s Law

3-4. Gauss’ Law and Applications

3.1 Overview

Electrostatics is the study of the effect of electric

charges at rest, where the electric fields do not change

in time.

Static electric fields in free space;

Conductors and dielectrics in static electric fields

Deductive approach :

Divergence and curl of E Gauss’s law

Coulomb’s law

3.2 Fundamental Postulates of Electrostatics

in Free Space

Electric field intensity E is defined as the force per unit charge

that a very small stationary test charge experiences when it is

placed in a region where an electric field exists :

(3-1)

According to Eq. (3-1), the force F on a stationary charge q

in an electric field E:

(3-2)

Two fundamental postulates of electrostatics in free space:

(3-3)

3.2 Fundamental Postulates of Electrostatics

in Free Space

(3-4)

(v : the volume charge density [C/m3])

1.

2.

Taking the volume integral of both side of Eq. (3-3) over an

arbitrary volume V,

3.2 Fundamental Postulates of Electrostatics

in Free Space

(3-5)

(3-6)

According to the Divergence theorem,

where Q = the total charge contained in volume V bounded by surface S.

“Gauss’s law”

From Eq. (3-4), i.e., E = 0, integrating E over an open

surface and using Stokes’s theorem :

3.2 Fundamental Postulates of Electrostatics

in Free Space

(3-7)

The scalar line integral of the static electric field intensity

around any closed path vanishes

The scalar product E dl integrated over any path is the

voltage along that path :

“Kirchhoff’s voltage law” : The algebraic sum of voltage drops

around any closed circuit is zero.

3.2 Fundamental Postulates of Electrostatics

in Free Space

Postulates of Electrostatics in Free Space

Differential Form Integral Form

0

v E0

Q

S dsE

0 E 0C dlE

The electric field intensity E due to a single point charge q

at a spherical surface with radius R centered at the origin in free

space can be obtained by applying Gauss’s law, i.e., Eq.(3-6) :

3-3 Coulomb’s Law

0

qdsE

SR

S RR aadsE

0

24

q

REdsE RS

R

or

(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.77)

Eq. (3-8) tells that the electric field intensity of a positive

point charge is in the outward radial direction and has a

magnitude proportional to the charge and inversely

proportional to the square of the distance from the charge.

3-3 Coulomb’s Law

(3-8)

3-3 Coulomb’s Law

)/(4

)(3

0

mVRR

RRqEP

If the charge q is not at the origin of a coordinate system,

(3-11)

Example 3-1

(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.77)

When a point charge q2 is placed in the field of another point

charge q1, a force F12 is experienced by q2 due to electric

field intensity E12 of q1 at q2 :

3-3 Coulomb’s Law

(3-13)

The force between two point charges is proportional to the

product of the charges and inversely proportional to the

square of the distance of separation.

“Coulomb’s law”

3-3.1 Electric Field Due to a System of

Discrete Charges

The total E field at a point is the vector sum of the fields

caused by all the individual charges :

(3-14)

3-3.2 Electric Field due to a Continuous

Distribution of Charge

The contribution of the charge v dv in a differential volume

element dv to the electric field intensity at the field point P :

(3-15)

(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.82)

3-3.2 Electric Field due to a Continuous

Distribution of Charge

(3-16)

(3-17)

(3-18)

(v : the volume charge density [C/m3])

(s : the surface charge density [C/m2])

(s : the line charge density [C/m])

3-3.2 Electric Field due to a Continuous

Distribution of Charge

Example 3-3 As shown in the solution of example 3-3,

we can solve this problem using Eq.(3-18) in principle.

However, we’d better apply Gauss’s law as shown in

Examples 3-4, 3-5, and 3-6.

3-4 Gauss’s Law and Applications

Gauss’s law :

(3-24)

The total outward flux of the E field over any closed

surface in free space is equal to the total charge enclosed in

the surface divided by

+ Divergence theorem

Examples 3-4, 3-5, and 3-6