A Review of Basic Electromagnetic Theories

Important Laws in Electromagnetics• Coulomb’s Law (1785)• Gauss’s Law (1839)• Ampere’s Law (1827)• Ohm’s Law (1827)• Kirchhoff’s Law (1845)• Biot-Savart Law (1820)• Faradays’ Law (1831)

Maxwell Equations(1873)• The governing equations of macroscopic electromagnetic

phenomena.• Predict the existence of electromagnetic waves.• Predict light to be electromagnetic waves.• Verification of electromagnetic waves: Hertz (1887-1891)• Radio communication: Marconi (1901)

: Electric field intensity: Electric flux intensity: Magnetic field intensity: Magnetic flux intensity

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V

an

: Electric current density: Magnetic current density

: Volume electric charge density: Volume magnetic charge densi

tyConstituent relationship

, where

: Permittivity: Permeability

Continuity relationship

Divergence Operator

Physical meaning

Divergence Theorem

Curl Operator

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S

an

C

Physical meaning

Stoke’s theorem:

Integral Forms of Maxwell Equations

Power Relationship

From vector identity

or for simply medium

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or in integral form

That is,

where: supplied energy: flow out energy: dissipated energy

: stored electric and magnetic energy: total stored electric and magnetic energy in a closed

surface .

! Boundary Conditions

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! Time-Harmonic Fields

Time-harmonic:

: a real function in both space and time.: a real function in space.

: a complex function in space. A phaser.

Thus, all derivative of time becomes

.

For a partial deferential equation, all derivative of time can be replacewith , and all time dependence of can be removed and becomesa partial deferential equation of space only.

Representing all field quantities as

,then the original Maxwell’s equation becomes

! Power Relationship

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! Poynting vector:

Solution of Maxwell’s Equations

Vector Potential and Scalar Potential

Since , for simple media, . Thus there exist a vector field, such that

is called vector potential. Then

.

Thus, there exist a scalar potential , such that

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or

Take the divergence of the above, we have

Since only the curl of is specified, the divergence of can be

arbitrarily chosen as . Then

Also

By applying vector identity , we have

The solutions are

where .

For time harmonic fields

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and

where is called the wave number and . Theabove equations are called nonhomogeneous Helmholtz’s equations.

Example:Let , then

HW#1 Prove that the wave functions for electric and magnetic fields insource free region are

or

On Wave in General

A wave function can be specified in complex domain as below:

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where: the magnitude, real,: the phase, real.

The corresponding time domain wave function is

Equal phase surface are defined as

Definitions:1. Plane, cylindrical, or spherical waves: equal phase surfaces are

planes, cylinders, or spheres.2. Uniform waves: amplitude is constant over the equal

phase surface.3. Wave normal: surface normals of the equiphase surfaces. is the

direction and is the curve along which the phase changes mostrapidly.

4. Phase constant: the rate at which the phase decreases in somedirection is called the phase constant in that direction (note: phaseconstant is not necessary a constant). Phase constant can bewritten in vector form as . The maximum phase constant istherefore .

5. Phase velocity: the speed the constant phase surface moves at in agiven direction. The instantaneous equiphase surface of a wave is

For ant increment , the change in is

To keep the phase constant for an incremental increase in time,corresponding incremental change in is necessary. We have

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.In cartesian coordinates

The phase velocity along a wave normal is

and is the smallest.

Alternatively, the wave function can be expressed as

where is a complex function. A complex propagation constantcan be defined as

where is the vector phase constant and is the vectorattenuation constant.

6. Wave impedance: the ratios of components of to . Followright-hand cross-product rule of component rotated into . Forexample,

, ,

Example:

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Plane Wave Propagation and Reflection

From wave equation

where , free space wave number or propagation constant.

In Cartesian coordinates, considering the x component,

.

Assume is independent of x and z, then

.

The solutions are

In time domain,

Constant phase

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, intrinsic impedance

In general,

where

Energy relations

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Define velocity of propagation of energy as

Then,

In general, or

PolarizationIn general,

If and , then

1. Elliptically polarized: in general.2. Linearly polarized: .

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3. Circularly polarized:

Ex: (RHC)

No change in energy and power densities with time or space, steadypower flow.

Plane Waves in Lossy MediaIf the material is conductive

,

we have

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Or if the material has dielectric loss with

where is the attenuation constant, the phase constant.

Low-Loss Dielectrics: , or

and

Good Conductor:

and

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Skin depth or depth of penetration:

Meaning: plane wave decay be a factor of . At microwavefrequencies, is very small for a good conductor, thus confined in avery thin layer of the conductor surface.

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Oblique incidentsPerpendicular Polarization (TE)

,

,

,

,

,

.

The above fields must satisfy Boundary conditions

,

which lead to

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.

.

Note that the condition holds.If at (Brewster (polarization) angle), then

Therefore,

,

or . No solution for .

Snell’s Law

Definition of refraction index

Total reflection occurs at critical angle . When

, is real. At angle larger than critical angle, surface waveexists in dielectric. The wave decays inside the dielectric. For , thepropagation constant in media 2 becomes,

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Note that the minus sign is chosen such that the resulting field quantityin media 2 won’t grow to infinity. In view of the phase term of thepropagation wave in media 2,

the wave decay in the media in direction.

Parallel Polarization (TM)

,

,

,

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,

,

.

.

.

Note that the condition holds.

If at Brewster angle , then .

Therefore,

or .

For nonmagnetic media, ,

Applications: Polarization separation, anti-flare glasses

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