Chapter 4:

Electrostatics

Textbook: Electromagnetics for Engineers

F.T. Ulaby

Electrostatics an Important Field of Study

Many electric and electronic devices and systems are designed

based on electrostatics principles:

oscilloscopes, copying machines, ink-jet electrostatic

printers

LCD’s (liquid crystal displays), capacitance based

keyboards

solid-state control equipment

medical applications: x-ray machines, diagnostic sensors

(electrocardiograms, electroencephalograms, etc.)

4-1 Maxwell’s Equations

4-2 Charge and Current Distributions

4-3 Coulomb’s Law

4-4 Gauss’s Law

4-5 Electric Scalar Potential

4-6 Electrical Properties of Materials

4-7 Conductors

4-8 Dielectrics

4-9 Electric Boundary Conditions

4-10 Capacitance

4-11 Electrostatic Potential Energy

4-12 Image Method

4-1 Maxwell’s Equations

The following four fundamental relations are known as

Maxwell’s equations, and they constitute the base of the first

unified modern electromagnetism theory:

The electric and magnetic field quantities may be functions of time, and they are related by:

, and .

(cont) Maxwell’s Equations in the Static Case

When all charges are permanently fixed in space, or they move at a

steady rate so that and J are constant in time, then all time

derivatives are zero and Maxwell’s equations reduces to

Electrostatics:

Magnetostatics:

Notice that, in the Static case, E is a conservative vector field.

B is always a solenoidal vector field.

4-2 Charge and Current Distributions

At atomic scale the charge distribution is discrete.

At macroscopic scale treat the net charge in an elemental volume

as if it were uniformly distribute with a volume density (measured in

C/m3):

The total charge (in Coulombs) in a volume v is:

In the conductors, the charge may be distributed across the

surface with a surface density (in C/m2):

or along a line with a line charge density (in C/m):

Charge distributions (see Example 4-1, Ulaby)

Current Density

The charges are moving with a mean velocity u a distance

The charge amount, in Figure (a) is:

and for case (b) is:

The corresponding current is

where, J is defined as the current density (A/m2).

The total current flowing through an arbitrary surface is:

(cont) Current Density

The actual movement of electrically charged matter (eg., a charged

cloud driven by wind) generates a so called convection current

In conductors the conduction current is made up by the

movement of electrons (in the outermost electronic shell) from atom

to atom pushed by an applied voltage

Note: the conduction current obeys the Ohm’s law, whereas

convection current does not (it is generated by a different physical

mechanism)

4-3 Coulomb’s Law

Based on the results of experiments on the electrical force

between charged bodies, the Coulomb’s law states that:

(a) an isolated charge q induces an electric field intensity E at

every point in space, and at any specified point P, E is given

by

,

and

(b) the force F acting on a test

charge placed at that point,

and in the presence of the electric

field E, is given by

(cont) Coulomb’s Law

One of the goals is to develop expressions relating the electrical field

quantities, E and D, to any specified charge distribution

with ,

where the electrical permittivity of free space is

and , is called the relative permittivity, or dielectric

constant of the material.

The materials can be classified as linear materials, when is

independent of the magnitude of E, and isotropic materials when

is independent of the direction of E.

Electric Field due to Multiple Point Charges

The expression for the field due to a single charge

can be extended for the case of multiple point charges.

Generalizing,

Electric Field due to a Charge Distribution

This is the case of the field caused by a

continuous charge distribution

Applying the principle of superposition,

the total electric field intensity E is

obtained by integration

Similarly, for the surface distribution,

and for the line charge distribution,

Ring of charge with line density (Example 4-4, Ulaby)

Circular disk of charge with surface charge density (Example 4-5, Ulaby)

4-4 Gauss’s Law

The first Maxwell’s equation is the differential form of Gauss’s law:

This equation can be converted into the integral form by performing

the following two steps:

The divergence theorem states that

Finally, the Gauss’s law in integral form is:

(cont) Gauss’s Law

The electric field D due to point charge q

Applying Gauss’s law:

The electric field induced by an isolated point charge in a medium

with permittivity :

Note: this result is identical with the equation obtained from Coulomb’s

law. Gauss’s law is easier to apply than Coulomb’s law but its

application is limited to symmetrical charge distributions.

Electric field of an infinite line of charge (see Example 4-6, Ulaby)