Time-Varying Fields and
Maxwell’s Equations
Faraday’s Law of
Electromagnetic Induction (I)
Overview
y So far
∇ × E = 0, ∇ ⋅ D = ρv .
D = ε E for linear and isotropic media.
y Fro the magnetostatic model,
∇ ⋅ B = 0, ∇ × H = J.
B = μ H for linear and isotropic media.
y In a conducting medium, static electric and magnetic fields may
both exist and form an electromagnetostatic field.
E → J → H
JHLin, EM Wave; Faraday's Law
2
Overview
y
Static electric and magnetic fields do not give rise to waves that
propagate and carry energy and information.
y
In this chapter we will see that a changing magnetic field induces
an electric field, and vice versa.
y
The two curl equations need to be modified in order to be
consistent with the law of conservation of charge.
y
Maxwell’s equations.
y
Wave equations.
JHLin, EM Wave; Faraday's Law
EM wave
3
Faraday’s Law of Electromagnetic
Induction
y
A major advance in EM theory was made by Michael Faraday
in 1831.
y
y
Faraday’s law:
d
∂B
d
d
E
⋅
l
=
−
B
⋅
s
∇
×
E
=
−
Or
.
v∫ C
dt ∫∫S
∂t
Lenz’s law – the negative sign in above equations is an
assertion that 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.
y
A Stationary Circuit in a Time-Varying Magnetic Field
The emf induced in a stationary loop caused by a timevarying magnetic field is a transformer emf.
JHLin, EM Wave; Faraday's Law
4
Faraday’s Law of Electromagnetic
Induction
y
Example 6-1 (7-1)
JHLin, EM Wave; Faraday's Law
5
Transformers
y
v1 N1
=
v2 N 2
i1 N 2
=
i2 N1
y Impedance
transformation
2
⎛N ⎞
( Z1 )eff = ⎜ 1 ⎟ Z L.
⎝ N2 ⎠
JHLin, EM Wave; Faraday's Law
6
Transformers
y Real transformers have real-life conditions:
1. leakage flux
2. noninfinite inductances
3. nonzero winding resistances
4. hysteresis
5. eddy-current losses
6. nonlinear nature of the ferromagnetic core
y In transformers eddy-current power loss is undesirable.
1. high μ and low σ (Ferrites)
2. laminated cores
JHLin, EM Wave; Faraday's Law
7
A Moving Conductor in a Static Magnetic
Field
y Motional emf (flux-cutting emf)
Fm = q u × B
2
V21 = ∫ (u × B) ⋅ dl
1
V ′ = v∫ (u × B) ⋅ dl
C
y Example 6-2 (7-2)
JHLin, EM Wave; Faraday's Law
8
A Moving Conductor in a Static Magnetic
Field
y
JHLin, EM Wave; Faraday's Law
9
A Moving Conductor in a Static Magnetic
Field
y Example 6-3 (7-3)
JHLin, EM Wave; Faraday's Law
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