Overview
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In a conducting medium, static electric and magnetic field may
both exist and form an electromagnetostatic field.
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A static E in a conducting medium
H.
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E can be completely determined from the static Q or V
distributions.
steady I
steady
H is a consequence; it does not enter into the calculation of E.
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Static electric and magnetic fields do not give rise to waves that
propagate and carry energy and information.
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A changing magnetic field induces an electric field, and vice
versa.
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Maxwell's Equations
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Faraday’s Law of Electromagnetic
Induction
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In 1831, Michael Faraday discovered experimentally that a
current was induced in a conducting loop when the magnetic
flux linking the loop changed.
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The quantitative relationship between the induced emf and the
rate of change of flux linkage, based on experimental
observation, is known as Faraday’s law.
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Faraday’s law of electromagnetic induction
dΦ
V =−
(V),
dt
where V =
∫
C
E ⋅ d = emf induced in a circuit with contour C.
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Maxwell's Equations
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Faraday’s Law of Electromagnetic
Induction
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A stationary circuit in a time-varying magnetic field
∫
C
E⋅d = −
d
B ⋅ ds ⇒
∫
S
dt
∫
C
∂B
⋅ ds
S ∂t
E ⋅ d = −∫
Assume a stationary circuit
∂B
∂B
⋅ ds ⇒ ∇ × E = −
.
⇒ ∫ ∇ × E ⋅ ds = − ∫
S
S ∂t
∂t
Stokes’s theorem
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Any arbitrary S
Lenz’s law
The induced emf will cause a current to flow in the closed loop
in such a direction as to oppose the change in the linking
magnetic flux.
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Maxwell's Equations
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Faraday’s Law of Electromagnetic
Induction
ƒ Transformer emf:
The emf induced in a stationary loop caused by a time-varying
magnetic field.
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Example 6-1
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Transformers
z
A transformers is an alternating-current (a-c) device that
transforms voltages, currents, and impedances.
∵ V1 = − N1
∴
dΦ
dt
and V2 = − N 2
dΦ
,
dt
V1 N1
=
V2 N 2
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Maxwell's Equations
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5
Faraday’s Law of Electromagnetic
Induction
In an ideal lossless transformer, no power is lost in the core,
P1 = P2 .
∵ P1 = I1V1 and P1 = I1V ,
I1 N 2
=
.
I 2 N1
∴
For a load impedance Z L , the effective load seen by the source is
2
( Z1 )eff
z
V ( N N )V ⎛ N ⎞
= 1 = 1 2 2 = ⎜ 1 ⎟ ZL .
I1 ( N 2 N1 ) I 2 ⎝ N 2 ⎠
For real transformers we have the following real-life conditions:
c the existence of leakage flux, 2 noninfinite inductances, 3
nonzero winding resistance, 4 the presence of hysteresis, 5
eddy current losses, and 6 nonlinear nature of the
ferromagnetic core.
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Maxwell's Equations
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Faraday’s Law of Electromagnetic
Induction
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The induced emf will produce local currents in the conducting
core normal to the magnetic flux. These currents are called
eddy currents.
Eddy currents produce ohmic power loss and cause local
heating.
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In transformers this eddy-current power loss is undesirable and
can be reduced by using core materials that have high µ and
low σ. Ferrites are such materials.
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For low-frequency, high-power applications, an economical way
for reducing eddy-current power loss is to use laminated cores.
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Maxwell's Equations
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Faraday’s Law of Electromagnetic
Induction
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The total eddy-current power loss decreases as the number of
lamination increases.
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Maxwell's Equations
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Faraday’s Law of Electromagnetic
Induction
ƒ
z
A moving conductor in a static magnetic field
Em is a motional electric field, which causes the freely moveable
electrons in the conductor to drift toward one end of the
conductor and leave the other end positively charged.
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Maxwell's Equations
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Faraday’s Law of Electromagnetic
Induction
z
This separation of the positive and negative charges creates a
Coulombian force of attraction.
The charge-separation process continues until the electric and
magnetic forces balance each other and a state of equilibrium is
reached.
Flux - cutting emf or motional emf
V=
ƒ
ƒ
∫C (u × B) ⋅ d
(V)
Example 6-2
Example 6-3
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Maxwell's Equations
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Faraday’s Law of Electromagnetic
Induction
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A moving conductor in a time-varying magnetic field
V=
∫
=−
ƒ
ƒ
C
∂B
⋅ ds +
S ∂t
E ⋅ d = −∫
d
B ⋅ ds
∫
S
dt
∫
C
(u × B) ⋅ d
(V).
Example 6-4
Example 6-5
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Maxwell's Equations
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Maxwell’s Equations
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Something wrong with the Ampere’s law?
∇ × H = J → ∇ ⋅ (∇ × H ) = ∇ ⋅ J = 0
But the continuity of equation: ∇ ⋅ J = −
↑
ƒ
∂ρv
≠0
∂t
in a time-varying situation.
Need some modification in ∇ × H = J!
∂ρ v
∂
∂D
= ∇ ⋅ J + (∇ ⋅ D) = ∇ ⋅ (J +
)
∂t
∂t
∂t
∂D
⇒ ∇×H = J +
∂t
∇ ⋅ (∇ × H ) = ∇ ⋅ J +
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Maxwell's Equations
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Maxwell’s Equations
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Differential form of Maxwell’s equations
∂B
∇×E = −
,
∂t
∂D
∇×H = J +
,
∂t
∇ ⋅ D = ρv ,
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Maxwell’s equations are not all independent.
The two divergence equations can be
derived from the two curl equations by
making use of the equations of continuity.
∇ ⋅ B = 0.
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These four equations, together with the equation of continuity
and Lorentz’s force equation form the foundation of
electromagnetic theory.
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These equations can be used to explain and predict all
macroscopic electromagnetic phenomena.
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Maxwell's Equations
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Maxwell’s Equations
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Integral form of Maxwell’s equations
∂B
⋅ ds,
∫C
S ∂t
∂B ⎞
⎛
H
J
⋅
d
=
+
∫C
∫S ⎜⎝ ∂t ⎟⎠ ⋅ ds,
E ⋅ d = −∫
∫
∫
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S
S
D ⋅ ds = ∫ ρv dv = Q,
V
B ⋅ ds = 0.
It is more convenient to use the integral form when dealing
with finite objects of specified shapes and boundaries.
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Maxwell's Equations
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Maxwell’s Equations
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Example 6-6
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Maxwell's Equations
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Maxwell’s Equations
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Electromagnetic boundary conditions
Boundary conditions for tangential component of E
E1t = E2t or nˆ × (E1 − E2 ) = 0.
Boundary conditions for tangential component of H
nˆ × (H1 − H 2 ) = J s .
Boundary conditions for normal component of D
D1n − D2 n = ρ s or nˆ ⋅ (D1 − D2 ) = ρ s .
Boundary conditions for normal component of B
B1n = B2 n or nˆ ⋅ (B1 − B 2 ) = 0.
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Maxwell's Equations
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Maxwell’s Equations
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Boundary conditions for interface between two lossless media
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There are usually no free charges and no surface currents at
the interface between two lossless media.
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Maxwell's Equations
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Maxwell’s Equations
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Boundary conditions for interface between a dielectric and a
perfect conductor
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Good conductors are often considered perfect conductors in
regard to boundary conditions.
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Maxwell's Equations
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