Maxwell’s Equations
January 16, 2013
1. Maxwell’s Equation, integral form
2. Maxwell’s Eqaution, differential form
3. Conservation of electric charge
4. Maxwell’s equations in static case
5. Maxwell’s equations in source-free dynamic case
6. Maxwell’s Eqaution in phasor form
7. The dependency of Maxwell’s equations
1
Maxwell’s equations in integral form
1.1
Faradya’s Law of induction
The induced emf (electromotive force) in a closed circuit is equal to the negative of rate of change
of magnetic flux pass through it.
‹
˛
∂
∂Φ
B
=−
B̄ · dS̄
Ē · d¯l = −
∂t
∂t S
∂S
1.2
Ampère’s circuital law with Maxwell’s Correction
The mmf (magnetomotive force) in a closed circuit is equal to the rate of change of electric flux and
current pass through it.
)
˛
‹ (
∂ D̄
¯
¯
· dS̄
H̄ · dl =
J+
∂t
∂S
S
1.3
Gauss’s Law for electricity
The electric flux pass through any closed surface is proportional to the enclosed electric charge.
‹
D̄ · dS̄ = ΦD = Q = ρv V
S
1
1.4
Gauss’s Law for magnetism
The magnetic flux pass through any closed surface is zero
‹
B̄ · dS̄ = 0
2
Maxwell’s equations in differential form
2.1
2.1.1
Vector Identities
Gauss’s Divergence Theorem
˚
‹
V̄ · dS̄
∇ · V̄ dV =
∂V
V
2.1.2
Stokes’ Curl Theorem
¨
˛
V̄ · d¯l
∇ × V̄ · dS̄ =
S
2.2
∂S
Faraday’s Law in differential form
Apply Stoke’s Theorem
˛
‹
∂
¯
Ē · dl = −
B̄ · dS̄
∂t S
∂S
‹
∂
∇ × Ē · dS̄ = −
∂t
S
−→
‹
B̄ · dS̄
S
Rearrange
‹
(
S
)
‹ (
∂ B̄
∇ × Ē · dS̄ =
−
· dS̄
∂t
S
)
The integral kernel thus equal to each other
∇ × Ē = −
2.3
∂ B̄
∂t
Ampère’s circuital law in differential form
Apply Stoke’s Theorem
)
˛
‹ (
∂
D̄
· dS̄
H̄ · d¯l =
J¯ +
∂t
∂S
S
)
‹ (
∂
D̄
∇ × H̄ · dS̄ =
· dS̄
J¯ +
∂t
S
S
‹
−→
The integral kernel thus equal to each other
∇ × H̄ = J¯ +
2
∂ D̄
∂t
2.4
Gauss’s Law for electricity in differential form
Apply Gauss’s Divergence Theorem
‹
D̄ · dS̄ = Q
˚
˚
−→
∇ · D̄dV = Q =
S
ρv dV
V
The integral kernel thus equal to each other
∇ · D̄ = ρv
2.5
Gauss’s Law for magnetism in differential form
Apply Gauss’s Divergence Theorem
‹
B̄ · dS̄ = 0
˚
˚
−→
∇ · B̄dV = 0 =
S
0dV
V
The integral kernel thus equal to each other
∇ · B̄ = 0
3
Conservation of electric charge
Consider the follow equations

∂ D̄


 ∇ × H̄ = J¯ +
∂t
∇ · D̄ = ρv


(
)

∇ · ∇ × V̄ = 0
Faraday’s Law
Gauss’s Law
Div of curl is zero
Take the div of Faraday’s Law
(
)
∂
D̄
∇ · ∇ × H̄ = ∇ · J¯ +
∂t
(
)
∂
0 = ∇ · J¯ + ∇ · D̄
∂t
∂ρv
0 = ∇ · J¯ +
∂t
i.e. The rate of current transfer out of a volume eqaul to decreasing rate of charge in that volume
∂ρv
∇ · J¯ = −
∂t
3
4
Maxwell’s equations in static case
Static ⇐⇒
∂
=0
∂t
{
∇ × Ē = 0
∇ × H̄ = J
∇ · B̄ = 0
∇ · D̄ = ρv
∇ · J¯ = 0
∇ × Ē = 0
∇ · D̄ = ρv


 ∇ × H̄ = J
∇ · B̄ = 0


∇ · J¯ = 0
E & D field are generated by ρv
H & B fields are generated
( by J )
J & ρv are independent ∇ · J¯ = 0
E , H are decoupled, the electrostatic field and magnetostatic field are independent.
5
Maxwell’s Equations in source-free dynamic case
Source free ⇐⇒ J = ρv = 0
∂ B̄
∂t
∂ D̄
∇ × H̄ =
∂t
∇ · B̄ = 0
∇ · D̄ = 0
E & H are coupled, they are not independent.
This coupling generate the phenomenon of EM wave propagation
∇ × Ē = −
6
Maxwell’s equations in phasor form
6.1
Phasor Review
Apply Euler’s Eqaution to sinusoidal term
[
]
[(
)
]
V (t) = V0 cos (ωt + ϕ) = ℜe V0 ej(ωt+ϕ) = ℜe V0 ejϕ ejωt
The phasor form is thus
V0 ejϕ
i.e.
V (t) = V0 cos (ωt + ϕ)
• Original time-domain form is real number
• Phasor form is complex number
4
←→
Ve = V0 ejϕ
6.2
Phasor Differentiation and Integration
[(
)
]
∂
∂
V (t) = V0 cos (ωt + ϕ) = −ωV0 sin (ωt + ϕ) = ℜe jωV0 ejϕ ejωt
∂t
∂t
Therefore
∂
V (t) ←→ jω Ve
∂t
ˆ
ˆ
V (t)dt =
1
V0 cos (ωt + ϕ) dt = V0 sin (ωt + ϕ) = ℜe
ω
[(
)
]
1
jϕ
jωt
V0 e
e
jω
Therefore
ˆ
V (t)dt ←→
6.3
1 e
V
jω
Maxwell’s Equations in Phasor Form
∇ × Ē = −jω B̄
∇ × H̄ = jω D̄ + J¯
∇ · B̄ = 0
∇ · D̄ = ρv
∇ · J¯ = −jωρv
The equations are now complex and time-independent.
7
The dependency in Maxwell’s Equations
There are 4 equations, but the other equations can be derived form the 2 curl equations with some
vector identities.
∇ × Ē = −jω B̄
∇ × H̄ = jω D̄ + J¯
∇ · B̄ = 0
∇ · D̄ = ρv
∇ · J¯ = −jωρv
7.1
Conservation of charge derived from Ampere’s Law
Already shown previously
5
7.2
Gauss’s Law for electricity derived from Ampere’s law & Conservation
of Charge
Apply div of curl is zero into Ampere’s Law
∇ × H̄ = jω D̄ + J¯
(
)
(
)
∇ · ∇ × H̄ = ∇ · jω D̄ + J¯
−→
0 = jω∇ · D̄ + ∇ · J¯
Apply Conservation of charge ∇ · J¯ = −jωρv
0 = jω∇ · D̄ − jωρv
i.e.
∇ · D̄ = ρv
7.3
Gauss’s Law for magnetism derived from Faraday’s Law
Apply div of curl is zero into Faraday’s Law
∇ × Ē = jω B̄
(
)
(
)
∇ · ∇ × Ē = ∇ · jω B̄
−→
(
)
0 = ∇ · jω B̄ = jω∇ · B̄
i.e.
∇ · B̄ = 0
6