Whites, EE 382
Lecture 6
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Lecture 6: Maxwell’s Equations,
Boundary Conditions.
In the last four lectures, we have been investigating the behavior
of dynamic (i.e., time varying) electric and magnetic fields.
In the previous lecture, we discussed “Maxwell’s law” (i.e.,
Ampère’s law with the added displacement current term). For a
capacitor, we found that displacement current completes the
path of the current where conduction ends.
Notice in the definition of capacitor displacement current
dV
Id = C
dt
that a time varying electric field in space is producing a
conduction current, which subsequently produces a time varying
magnetic field. Amazing!
Conversely, in Faraday’s law
dψ m
dt
a magnetic field effect produces an emf (a “source” voltage).
emf = −
This is a beautiful “duality” between these two effects:
© 2012 Keith W. Whites
Whites, EE 382
Lecture 6
•
Maxwell’s law:
•
Faraday’s law:
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∂E ( t )
→ B (t )
∂t
∂B ( t )
→ E (t )
∂t
Since a time varying electric field produces a magnetic force and
vice versa, we now speak of an electro-magnetic field, rather
than electric and magnetic fields separately.
Because of this duality, we will see shortly that electromagnetic
signals can propagate as waves! It is because of this fantastic
circumstance that there exists light, radio communications,
satellite remote sensing, RADAR, fiber optic networks, CAT
scans, etc.
Maxwell’s Equations
The laws of classical electromagnetics can be neatly
summarized into a concise collection called Maxwell’s
equations.
In point form, Maxwell’s equations read:
Whites, EE 382
∂B ( t )
∂t
∂D ( t )
∇ × H (t ) =
+ J (t )
∂t
∇ ⋅ D ( t ) = ρv ( t )
∇ ⋅ B (t ) = 0
∇ × E (t ) = −
Lecture 6
Faraday’s law
Ampère’s law
Gauss’ law, I
Gauss’ law, II
In integral form, Maxwell’s equations read
d
E
t
⋅
dl
=
−
B ( t ) ⋅ ds
(
)
v∫
∫
dt s( c )
c( s )
d
D ( t ) ⋅ ds + ∫ J ( t ) ⋅ ds
v∫
∫
dt s( c )
c( s )
s( c )
v∫ D ( t ) ⋅ ds = ∫ ρv ( t ) dv
s( v )
v( s )
v∫ B ( t ) ⋅ ds = 0
s( v )
H ( t ) ⋅ dl =
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Faraday’s law
Ampère’s law
Gauss’ law, I
Gauss’ law, II
In addition, the continuity equation (conservation of charge)
reads in point form:
∂ρ ( t )
∇ ⋅ J (t ) = − v
∂t
and in integral form
d
J
t
⋅
ds
=
−
(
)
v∫
∫ ρv ( t ) dv
dt
s( v )
v( s )
Whites, EE 382
Lecture 6
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These laws describe all of classical (i.e., non-quantum
mechanical) electromagnetism. Maxwell’s equations are an
amazingly short and concise set of equations. However, these
equations are usually difficult to solve for real-world problems.
Interdependent Equations
As it turns out, not all of these equations are independent for
dynamic fields. For example, if we take the divergence of
Ampère’s law:
⎛ ∂D ⎞
∇ ⋅ (∇ × H ) = 0 = ∇ ⋅ J + ∇ ⋅ ⎜
⎟
⎝ ∂t ⎠
we find that it reduces to
∂ρ
∇⋅J = − v
∂t
which is the continuity equation.
There are other examples of interdependencies among
Maxwell’s equations for dynamic fields.
Constitutive Equations
For dynamic electromagnetism, the constitutive equations are
still applicable:
Whites, EE 382
Lecture 6
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• D (t ) = ε E (t )
• B (t ) = μ H (t )
• J (t ) = σ E (t )
However, for sinusoidal steady state problems, the material
parameters are often a function of frequency. That is
ε = ε (ω )
μ = μ (ω )
σ = σ (ω )
Boundary Conditions
The boundary conditions for dynamic EM fields remain the
same as were derived earlier in EE 381 for static fields:
â21
Tangential components –
• aˆ21 × ⎡⎣ E2 ( t ) − E1 ( t ) ⎤⎦ = 0
• aˆ21 × ⎡⎣ H 2 ( t ) − H1 ( t ) ⎤⎦ = J s ( = K )
Normal components –
• aˆ21 ⋅ ⎡⎣ D2 ( t ) − D1 ( t ) ⎤⎦ = ρ s ( t )
• aˆ21 ⋅ ⎡⎣ B2 ( t ) − B1 ( t ) ⎤⎦ = 0
Whites, EE 382
Lecture 6
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Example N6.1: The electric field E ( x, t ) = aˆ z Eo cos (ωt + β x )
V/m exists in free space. Determine H ( t ) and β consistent with
this electric field and all of Maxwell’s equations.
• From Faraday’s law: ∇ × E = −
aˆ x
∂
∂x
0
Therefore,
∂By
or
So,
aˆ y
∂B
. For this example
∂t
aˆ z
∂
( = 0)
∂y
0
∂
∂E
∂B
( = 0 ) = −aˆ y z = −
∂z
∂x
∂t
Ez
∂Ez ∂
= ⎡ Eo cos (ωt + β x ) ⎤⎦
∂t
∂x ∂x ⎣
∂
= Eo ( −1) sin (ωt + β x ) ⋅ (ωt + β x )
∂x
∂By
= − β Eo sin (ωt + β x )
∂t
=
By =
β
E cos (ωt + β x ) + C
ω o
The constant C cannot be a function of time. It is often taken
as zero for dynamical problems if there are no sources present
for constant magnetic fields. Therefore,
H ( t ) = aˆ y
β
E cos (ωt + β x ) [A/m]
ωμ0 o
Whites, EE 382
Lecture 6
• From Ampere’s law: ∇ × H =
aˆ x
∂
∂x
0
aˆ y
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∂D
+ J 0 . Then
∂t
aˆ z
∂
( = 0)
∂y
Hy
∂H y
∂
∂E
ˆ
ε
0
=
=
=
a
( ) z
0
∂z
∂x
∂t
0
Therefore,
⎡
⎤
⎤
∂ ⎡ β
∂⎢
Eo cos (ωt + β x ) ⎥ = ε 0
Eo cos (ωt + β x ) ⎥
⎢
∂x ⎣ ωμ0
∂t ⎢ ⎥
⎦
= Ez
⎣
⎦
or
∂ ( ωt + β x )
∂ ( ωt + β x )
β
Eo ( −1) sin (ωt + β x )
= ε 0 Eo ( −1) sin (ωt + β x )
ωμ0
∂x
∂t
=β
So that
=ω
β2
= ε 0ω or β = ±ω μ0ε 0 [rad/m]
ωμ0
• ∇ ⋅ D = ∇ ⋅ (ε 0 E ) = ε 0
∂Ez
= 0.
∂z
9
consistent with Maxwell’s
equations.
∂B
• ∇ ⋅ B = y = 0 . 9 consistent with Maxwell’s equations.
∂y