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Week 9
Maxwell’s Equations
Symbols Used
E
Electric field intensity
ρ
charge density
I
B
Magnetic flux density
ε0
Permittivity
current density
D
Electric displacement
μ0
Permeability
c
speed of light
H
Magnetic field intensity
M
Magnetization
P
Polarization
electric current
J

Demonstrated that electricity,
magnetism, and light are all
manifestations of the same
phenomenon:
the electromagnetic field

Electric charges generate fields

Fields interact with each other

Fields act upon charges

Electric charges move in space
◦ Charges generate electric fields
◦ Moving charges generate magnetic fields
◦ changing electric field acts like a current, generating vortex
of magnetic field
◦ changing magnetic field induces (negative) vortex of
electric field
◦ electric force: same direction as electric field
◦ magnetic force: perpendicular both to magnetic field and to
velocity of charge

Gauss’ Law for Electricity

Gauss’ Law for Magnetism
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Faraday’s Law of Induction

Ampere’s Law

Integral Form

Differential form

B and E must obey the same relationship

Show that E = Eo cos (ωt - kz) ax satisfies the
wave equation



Frequency f (cycles per second or Hz)
Wavelength λ (meter)
Speed of propagation c = f λ
Distance (meters)

Determine the frequency of an EM wave with
a wavelength of
◦
◦
◦
◦
1000 m (longwave)
30 m (shortwave)
1 cm (microwave)
500 nm (green light)
E = Eo cos (ωt - kR) aE
H = Ho cos(ωt - kR) aH
where
A is the amplitude
t is time
ω is the angular frequency 2πf
k is the wave number 2π/λ
aE is the direction of the electric field
aH is the direction of the magnetic field
R is the distance traveled

Euler’s Formula
A e+jφ = Acos(φ) + jAsin(φ)
A cos(φ) = Re {Ae+jφ}
Imaginary
A sin(φ) = Im {Ae+jφ}
Real
A e-jφ = A cos(φ) - jA sin(φ)
unit circle

Show that
A cos(φ) = ½ Ae+jφ + ½ Ae-jφ
jA sin(φ) = ½ Ae+jφ - ½ Ae-jφ

Complex field
E = Eo exp (jωt) exp(jψ) aE

Phasor convention

E = Eo exp(jψ) aE
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The frequency must be the same


The plane wave has a constant value on the
plane normal to the direction of propagation
The spacing between planes is the
wavelength


The magnetic field H is perpendicular to the
electric field E
The vector product E x H is in the direction of
the propagation of the wave.

The wave vector is normal to the wave front
and its length is the wavenumber
|k| = 2π/λ

A plane wave propagates in the direction
k = 2ax + 1ay + 0.5az

Determine the following:
◦ wavelength (m)
◦ frequency (Hz)


A plane wave becomes cylindrical when it
goes through a slit
The wave fronts have the shape of aligned
cylinders

A spherical wave can be visualized as a series
of concentric sphere fronts

Poynting Vector (Watts/m2)
S = ½ E x H*

Poynting Vector (Watts/m2)
S = ½ E x H*

For plane waves
S = |E|2/ 2η

Electromagnetic (Intrinsic) Impedance

A plane wave propagating in the +x direction
is described by
E = 1.00 e –jkz ax
H = 2.65 e –jkz ay

Determine the following:
◦ Direction of propagation
◦ Intrinsic impedance
◦ Power density
V/m
mA/m


Read Chapter Sections 7-1, 7-2, 7-6
Solve Problems 7.1 7.3, 7.25, 7.30, and 7.33