Electromagnetic Fields
Review of Boundary Conditions
Consider an electromagnetic field at the boundary between two
materials with different properties. The tangent and the normal
component of the fields must be examined separately, in order to
understand the effects of the boundary.
Medium 1
ε 1 ; µ1
G
Hn1
Medium 2
ε2; µ2
G
Hn2
G
H1
boundary
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G
H2
G
H t1
G
Ht 2
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Electromagnetic Fields
Tangential Magnetic Field
Medium 1
ε 1 ; µ1
boundary
G
H n3
Medium 2
ε2; µ2
G
H t1
G
H n4
G
Ht2
b
a
y
.
z
x
Ampère’s law for the boundary region in the figure can be written as
∂H y ∂H x
G
∇×H⇒
−
= J z + jω ε E z
∂x
∂y
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Electromagnetic Fields
In terms of finite differences approximation for the derivatives
H n4 − H n3 H t1 − H t 2
−
= J z + jω ε E z
b
a
If one lets the boundary region shrink, with
than b,
a
going to zero faster
H n3 − H n4
H t 2 − H t1 = lim ( J z a + jωε E z a + a
)
b
a→ 0
for materials with finite conductivity
⇒ H t 2 − H t1 = 0
Tangential components are conserved
for perfect conductors
⇒ H t 2 − H t1 = lim ( J z a) = Js
a→ 0
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(surface current)
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Electromagnetic Fields
For a general boundary geometry
G
G
G
nˆ × (H t1 − H t 2 ) = Js
nˆ = unit vector normal to the surface
In the case of a perfect conductor, the electromagnetic fields go
immediately to zero inside the material, because the conductivity is
infinite and attenuates instantly the fields. The surface current is
confined to an infinitesimally thin “skin”, and it accounts for the
discontinuity of the tangential magnetic field, which becomes
immediately zero inside the perfect conductor.
For a real medium, with finite conductivity, the fields can penetrate
over a certain distance, and there is a current distributed on a thin,
but not infinitesimal, skin layer. The tangential field components on
the two sides of the interface are the same. Nonetheless, the
perfect conductor is often a good approximation for a real metal.
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Electromagnetic Fields
Tangential Electric Field
Medium 1
ε 1 ; µ1
boundary
G
E n3
Medium 2
ε2; µ2
G
E t1
G
Et2
b
G
E n4
a
y
.
z
x
Faraday’s law for the same boundary region can be written as
∂ E y ∂E x
G
∇×E⇒
−
= jω µ H z
∂x
∂y
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Electromagnetic Fields
In terms of finite differences approximation for the derivatives
E n4 − E n3 E t1 − E t 2
−
= jωµ H z
b
a
If one lets the boundary region shrink, with
than b,
a
going to zero faster
E n3 − E n4
E t 2 − E t1 = lim ( jωµ H z a + a
)
b
a→ 0
⇒ E t 2 − E t1 = 0
Tangential components are conserved
For a general boundary geometry
G
G
nˆ × (E t1 − E t 2 ) = 0
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Electromagnetic Fields
Normal components
G
Dn1
Medium 1
ε 1 ; µ1
ρs +
boundary
Medium 2
ε2; µ2
+
+
+
+
G
Dn2
G
Bn1
w
+
G
Bn2
Area
y
.
z
x
Consider a small box that encloses a certain area of the interface
with
ρ s = interface charge density
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Electromagnetic Fields
Integrate the divergence of the fields over the volume of the box:
∫∫∫
G G
∇ ⋅ D dr =
Volume
∫∫∫
G
ρ dr
Volume
w
∫∫
⇓ theorem
G
G
G
D ⋅ n̂ ds = Flux of D out of the box
∫∫∫
G G
∇ ⋅ B dr = 0
Divergence
Surface
Volume
⇓ theorem
G
G
G
B ⋅ n̂ ds = Flux of B out of the box
Divergence
w
∫∫
Surface
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Electromagnetic Fields
If the thickness of the box tends to zero and the charge density is
assumed to be uniform over the area, we have the following fluxes
G
D-Flux out of box = Area ⋅ (D1n − D2 n ) =
= Total interface charge = Area ⋅ ρ s
G
B-Flux out of box = Area ⋅ (B1n − B2 n ) = 0
The resulting boundary conditions are
D1n − D2 n = ρ s
B1n − B2 n = 0
The discontinuity in the normal component of the displacement
field D is equal to the density of surface charge.
The normal components of the magnetic induction field B are
continuous across the interface.
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Electromagnetic Fields
For isotropic and uniform values of ε and µ in the two media
G
G
G
G
Dn1 − D n2 = ε1E n1 − ε 2 E n2 = ρ s
G
G
G
G
Bn1 − Bn2 = µ1H n1 − µ 2H n2 = 0
Even when the interface charge is zero, the normal components of
the electric field are discontinuous at the interface, if there is a
change of dielectric constant .
The normal components of the magnetic field have a similar
discontinuity at the interface due to the change in the magnetic
permeability. In many practical situations, the two media may have
the same permeability as vacuum, µ0, and in such cases the normal
component of the magnetic field is conserved across the interface.
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Electromagnetic Fields
SUMMARY
If medium 2 is
perfect conductor
G
H t1
G
Ht2
G
E t1
G
Et2
G
H n1
G
H n2
G
E n1
G
E n2
ε 1 , µ1
ε 2 , µ2
ε 1 , µ1
ε 2 , µ2
ε 1 , µ1
ε 2 , µ2
ε 1 , µ1
ε 2 , µ2
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G
G
H t1 = H t 2
G
G
nˆ × H t1 = J s
G
Ht2 = 0
G
G
E t1 = E t 2
G
E t1 = 0
G
Et2 = 0
G
G
µ1H n1 = µ 2 H n2
G
G
ε 1E n1 = ε 2 E n2 +ρ s
G
H n1 = 0
G
H n2 = 0
G
E n1 = ρ s ε 1
G
E n2 = 0
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Electromagnetic Fields
Examples:
An infinite current sheet generates a plane wave (free space on
both sides)
x
Js
-z
+z
y
H
G
Js ( t ) = − Jso cos(ω t ) iˆx
G
Phasor J s = − Jso iˆx
The E.M. field is transmitted on both sides of the infinitesimally thin
sheet of current.
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Electromagnetic Fields
BOUNDARY CONDITIONS
G
G
G
nˆ × (H t1 − H t 2 ) = J s
G
G
H t1 − H t 2 = Jso iˆx
G
G
E t1 = E t 2
G
G
E t1 = η 0 H t1
G
G
Symmetry ⇒ H t1 = H t 2
Jso
Jso
H1 =
H2 = −
2
2
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Electromagnetic Fields
A semi-infinite perfect conductor medium in contact with free space
has uniform surface current and generates a plane wave
x
Free Space
Perfect
Conductor
Js
-z
+z
y
H
G
J s = − Jso cos(ω t ) iˆx
The E.M. field is zero inside the perfect conductor. The wave is only
transmitted into free space.
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Electromagnetic Fields
BOUNDARY CONDITIONS
G
G
G
nˆ × (H t1 − H t 2 ) = J s
G
G
G
H t1 − H t 2 = H t1 − 0 = Jso iˆx
G
Et2 = 0
G
G
Asymmetry ⇒ H t1 ≠ H t 2
H t1 = Jso
Ht2 = 0
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