107: ECE Electromagnetism
Set 7: Dynamic fields
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego, CA 92093
1
Maxwells equations
Maxwells equations
B
E =
t
D
H =
+J
t
D =
B = 0
n12 (E2 E1 ) = 0
n12 (D2 D1 ) = s
n12 (H 2 H1 ) = J s
n12 (B 2 B1 ) = 0
D = E
B = H
Goals of this chapter
d
E d l = dt B d s
d
H d l = D d s + I
dt
D ds = Q
B ds = 0
D = E
B = H
To close the loop between dynamics and statics
To explain effects introduced by time variations
To extend some results from static to dynamics
2
Faradays law
d
Faradays law: E d l = dt B d s
Magnetic flux: = B ds
d
Electromotive force voltage: V = ddt = dt B ds
V induces currents in a wire
C
S
S
emf
S
emf
loop. The currents produce the
flux that opposes the inducing
flux (Lenzs law)
Vemf allows for transformers,
generators, motors, etc.! Its
existence is a foundation of
Electromagnetics
3
Faradays law (2)
Transformers:
Current mutual flux
Flux voltage
d
d
, V2 = N 2
dt
dt
V
N
1= 1
V2 N 2
V1 = N1
Energy conservation
I1V1 = I 2V2
I
N
1= 2
I 2 N1
Faradays law (2)
Electric generator:
Rotating loop in a magnetic field
EMF
Electric motor:
Opposite to the generator
An AC current is sent to the coil in
a constant magnetic field coil
spins
Faradays law (2)
Eddy currents:
Time varying magnetic field Electric field (Eddy) current
E=
B
J = E
t
Examples:
Eddy current waist separator
Eddy current break
Transcranial magnetic stimulation
Amperes law
d
Amperes law: H d l = I + dt D d s
d
Displacement current: I = J ds = dt D ds
D
Displacement current density: J = ddt
Displacement current is required to establish
C
S
d
S
d
S
d
consistent equations of Electromagnetics. Its
introduction resulted in Maxwells equations
To explain its importance, consider a capacitor
under an ac voltage.
7
Charge-current continuity relation
Start with the Amperes law H = D + J
t
D
Apply divergence H) = t + J J = t
(
=0
(vector identity)
Continuity relation: J v = v
D
= v
Gauss law
t
J v d s = v I = Q
S
t V
t
The amount of charge change equals the current
needed to compensate the change
For statics
The current and charge are independent
S J v d s = 0 Kirchhoffs current law i Ii = 0
8
Complex permittivity (1)
Modified Amperes law
Amperes law H = + jD J
Constitutive relation D = E
Ji
Current J = J c +
conductivity
J c = E
Modified Amperes
impressed
given source
law H = j E + J c + J i
H = j E + E + J i
j
H = j
E + Ji
c
Loss tangent
Complex permittivity c =
j
= j =| c | e j
tan = = ( )
9
Complex permittivity (2)
Maxwells equations in conducting media
Differential equations
E = jH
H = j E + J
c
D = i
B = 0
Boundary conditions
i
n12 (E2 E1 ) = 0
n ( E E ) =
12
c2
2
c11
1
is
n12 ( H 2 H1 ) = J is
n ( H H ) = 0
12
Continuity relation
2
2
1
J i = jis
1
10
Electromagnetic potentials (1)
Fields radiated in free space by a charge
distribution v (r, t ) and current distribution J v (r, t )
(recall that v and J v are related J v = v t )
H=
1
A
E = V
A
t
retarted
potential
retardation
due to causality
v ( t R v p )
1
V (R, t ) =
dv
4
R
V
A( R , t ) =
4
J v ( t R v p )
dv
R
V
v p = c = 3 108 m s
11
Electromagnetic potentials (2)
Frequency domain fields radiated by a charge
distribution v (r ) and current distribution
(recall that v and J v are related J v = jv )
k=
vp
=
2
wavenumber
1
H= A
E = V j A
1
V (R ) =
4
(R ) =
A
4
v e jkR
R dv
V
J v e jkR
R dv
V
12
Radiation from a short dipole (1)
Consider an electric dipole p = ql z
Its time derivative results in
current d
p = Il z j p = Il z
dt
When l = then I is uniform
Physically, the dipole is
represented by two wires
Vector potential
13
Radiation from a short dipole (2)
Expressions for the radiated fields
0
0 =
= 120
0
When 0
characteristic impedance
of free space
the expressions reduce to the
expression due to a static dipole with p = Il ( j)
Magnetic field is present only in the dynamic case
No electric field is in the direction, hence the
field is TM to z
The short dipole is one of the simplest antennas 14
Radiation from a short dipole (3)
Far field approximation: R ?
or kR ? 1
The terms ~ 1 (kR)2 ,1 (kR)3 can be neglected
Far field approximation fields
Characteristics of far field fields of ANY antenna
Relation between E and H is simply through the impedance
of free space similar to the case of transmission lines
The phase behavior is like for TLs with z replaced by R
The field decays but only as 1 R (compare to 1 R 2 decay
for a static charge and 1 R 3 decay for an electric dipole)
15