Download Maxwell`s Equations

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