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0 Bv Dt
x
D
jH
tx
BE
Maxwell’s equations
constitutive relations
• D = E
• B = H
• j = E
jE
EE
HH
tt
HExHEEH xx
dsHEx dvxHE
Poynting’s theorem
22
x
E1 Hdv
2 t t
dv
E H ds
E j
Poynting’s theorem
• em power that leaves
• = - (stored em energy) / t
• - lost em powerE x H
Spherical radiation
• E x H • ds = EH 42
E2 42
constant
• E 1/
Cylindrical radiation
• E x H • ds = EH 2r L E2 2r L constant
• E 1/(r)1/2
r L
aLa
I
L
Vx o
2
2 dsHE
22 E1 Hdv 0
2 t t
oV Ix 2 aL
L 2 a
E H ds
2o2
V Idv L a
L a
E j
oV Ix 2 aL
L 2 a
E H ds
2E1dv
2 t
0 dvjE21 H
dv 02 t
2
V1 d
dA2 t
21 A V2 d t
xE
ωsinωtd
ε
2
rH
t
B
dlH
dst
D
r
d
V = Vocos t
= 2rH
t
V
d
επr2
Boundary conditions
E1 E2
B1 B2
Pillbox S Loop L
components
Integral form of Maxwell’s equations
dsHx
dvD
dvB
dsEx dlE
dsB
t
dlH dsD
j
t
dsD dvv
dsB 0
Normal component of B
Bn1s + Bn2 s = 0
Bn1 Bn2
Normal components of B are continuous
No magnetic monopoles!!
dvB dsB 0
Normal component of D
Dn1s + Dn2 s = s s
Dn1 Dn2
Normal components of D differ by surface charge
density
Electric charge
dvD dsD dvv dss
Tangential component of H
Ht1L + Ht2 L = js L
Ht1 Ht2
Tangential components of H differ by surface current
density
surface current
dsHx dlHdsD
j
t dljs
Tangential component of E
Et1L + Et2 L = 0 Et1 Et2
Tangential components of E are continuous
dsEx dlE
dsB
t0
example
2n4 E 1 5 3
5 4 1 n tE u u
?2E1= 1 2= 4
s 2C3
m
2 2 4 n tE u u
example
2y 1yB B0
4 1
5 4 1 x yB u u
?2B
A3 ms yJ u
5 16 2 x yB u u
1= 1 2= 4
xy
example
2y 1yB B3
4 1
5 4 1 x yB u u
?2B
A3 ms zJ u
5 28 2 x yB u u
1= 1 2= 4
xy
Tangential components of the electric field intensity are continuous
Normal components of the displacement flux density differ by a surface charge density
Normal components of the magnetic flux density are continuous
Tangential components of the magnetic field intensity differ by a surface current density
Tangential component of Emetal
Et1L + Et2 L = 0 Et1 Et2
Tangential component of E is zero.
but Et2 = 0
dsEx dlE
dsB
t0
images• Charge +Q• Equipotential
contours• Electric field E
• Image charge -Q
• Equipotential contours
• Electric field E
images
• Charge +Q
• Image charge -Q
2d
• Image charge –Q
• Image charge +Q
• Image charge +Q
Antenna on top of the ground
• Underneath an antenna is an array of conductors
• This creates a ground plane• This effectively makes the antenna