1. ELECTROMAGNETICS AND APPLICATIONS
6.013
ELECTROMAGNETICS AND APPLICATIONS
Luca Daniel
Lecture 4 Poynting Vector in Complex Notation. EM Fields and Interfaces. 1

2. Review of Fundamental Electromagnetic Laws
Today’s Outline
Review of Fundamental Electromagnetic Laws
Electromagnetic Waves in Media and Interfaces
The EM waves in homogenous Media
Electromagnetic Power and Energy
The Poynting Theorem
Wave Intensity
Poynting Theorem in Complex Notation
EM Fields at Interfaces between Different Media
Fields at boundaries: normal components
Fields at boundaries: tangential components
Fields inside perfect conductors
Fields at boundaries of perfect conductors
EM Waves Incident “Normally” to a Different Medium
EM Waves Incident at General Angle to a Different Medium
Today 2

3. Power Flow in Uniform Plane Wavesz
The time average is called “intensity” [W/m2] of the wave

4. Poynting Vector in Complex Notation
Defining a meaningful and relating it to is not obvious.
It is easier to relate it to the intensity (time average):
Note:
(by definition)
Thus the Intensity can put computed directly from the field phasors
and

5. Review of Fundamental Electromagnetic Laws
Today’s Outline
Review of Fundamental Electromagnetic Laws
Electromagnetic Waves in Media and Interfaces
The EM waves in homogenous Media
Electromagnetic Power and Energy
The Poynting Theorem
Wave Intensity
Poynting Theorem in Complex Notation
EM Fields at Interfaces between Different Media
Fields at boundaries: normal components
Fields at boundaries: tangential components
Fields inside perfect conductors
Fields at boundaries of perfect conductors
EM Waves Incident “Normally” to a Different Medium
EM Waves Incident at General Angle to a Different Medium 5

6. Fields at Boundaries: Normal Components
Gauss’s Law: A h
(assuming Lim h 0)
surface charge density rs
surface S
Therefore:
Similarly
i.e. normal B is always continuous!
L3-4

7. Fields at Boundaries: Tangential Components
Faraday’s Law:
(Lim h0) L
E1//
Therefore:
E1// = E2// h A
E2//
Tangential E is always continuous
Ampere’s Law:
Js produces a jump in tangential H:
Alternatively:
Note that is only possible on
perfect conductors
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8. Conducting Media
Constitutive relation for conducting medium (Ohm’s Law):
where σ is the conductivity [Am/V]
In a regular conductor charges are free to move.
If E is applied, J will generate charges on the surface that start cancelling the applied E (charge relaxation).
Electric Fields in perfect conductors :
which would instantaneously generate surface charge that immediately canceling all E.
Therefore inside
perfect conductors:
Eout // = Ein // = 0 => E fields can only be  to a
perfect metal surface
r = 0 rs
r can only be on the surface since any charge inside would produce E and J that would instantaneously distributed it to the surface J rs J rs J rs q J J rs J
L3-8 rs