Maxwell’s Equations

By: Dr. Ahmed M. AttiyaBy: Dr. Ahmed M. Attiya

Maxwell’s equations

The behavior of electric and magnetic waves can be fully described by a set of four tiequations.

Faraday’s Law of induction

Ampere’s Law

Gauss’s Law for electricity

Gauss’s Law for magnetism

continuity equation

Boundary ConditionsBoundary Conditions

d d d dBoundary ConditionsBetween two different media

Boundary Conditionson a PEC surface

Static PotentialsStatic Potentials

0=×∇ E 0×∇ E

0=⋅∇ B 0∇ B

Time Varying Electric Field

Time Varying Magnetic Field

Lorentz Condition for Potentials

Time Varying Magnetic FieldTime Varying Magnetic Field

Time Harmonic Fields• Time-harmonic field is one that varies

d ll d ll hperiodically or sinusoidally with time.

• Any periodic field can be expanded in terms of y p psinusoidal field by using Fourier transform.

Si id il d i h• Sinusoids are easily expressed in phasors.

• A phasor z is a complex number that can be p pwritten as:

Phasor MathematicsPhasor Mathematics

Time Harmonic FieldTime Harmonic Field

Time Harmonic Maxwell’s Equations in Phasor form (Differential Form)

Time Harmonic Maxwell’s Equations in Phasor form (Integral form)

Example 1Example 1

Example 2Example 2

Example 3Example 3