Download Chapter 6 Time-Varying Field and Maxwell`s Equations 6

Chapter 6
Time-Varying Field and Maxwell’s Equations
6-1 Overview
6-2 Faraday’s Law of Electromagnetic Induction
6-3 Maxwell’s Equations
6-4 Potential Functions
6-5 Time-Harmonic Fields
6-1 Overview
¾ Fundamental governing equations for electrostatic and magnetostatic models:
JG
∇ ⋅ D = ρv
JG
∇× E = 0
For linear and isotropic media:
JG JG
D=εE
JG
∇⋅B = 0
JJG JG
∇× H = J
JG
JJG
B = μH
¾ Static charges are the source of an electric field; Moving charges produce a
current, which gives rise to a magnetic field. However, these fields are static
fields, which do not give rise to waves.
¾ We wish to have waves, which may propagate and carry energy and information.
¾ How to generate wave? and how wave propagates?
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6-2 Faraday’s Law of Electromagnetic Induction
¾
Fundamental postulate for electromagnetic induction is:
Integral form:
Differential form:
JG
JG
∂B
∇×E = −
∂t
•
•
JG
JG G
∂ B JG
vC∫ E ⋅ dl = − ∫S ∂t ⋅ d S
Point-function relationship.
E in a region of time-varying B is
non-conservative, and ≠ -∇V
V =
JG
G
v∫ E ⋅ d l
(V)
Electromotive force - emf
C
Transformer emf:
How do we generate emf ?
emf:
tr
Vemf
m
Motional emf / flux-cutting emf: Vemf
tr
Vemf
Section 6-2.1: A stationary circuit in a time-varying magnetic field:
Section 6-2.3: A moving conductor in a static magnetic field:
Section 6-2.4: A moving circuit in a time-varying magnetic field:
m
Vemf
tr
m
Vemf
+ Vemf
(Pls ignore section 6-2.2: transformers)
6-2.1 A stationary circuit in a time-varying magnetic field:
Re-write equation:
JG
JG G
∂ B JG
d JG JG
vC∫ E ⋅ dl = −∫S ∂t ⋅ d S = − dt ∫S B ⋅ d S
JG JG
Φ = ∫ B ⋅ d S (Web)
Magnetic flux crossing surface S
S
¾ Faraday’s Law of EM Induction
tr
=−
Vemf
dΦ
dt
(V)
Electromotive force ( Emf ) induced in a stationary closed circuit is equal to the
negative rate of increase of magnetic flux Φ linking the circuit
Transformer emf: The emf induced in a stationary loop caused by a time-varying
magnetic field is called a transformer emf.
¾ Lenz’s Law: The negative sign is an assertion that induced emf will cause a
current to flow in the closed loop in such a direction as to oppose the change in
the linking Φ.
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Faradays’ Law of Induction
¾ Two experiments:
An ammeter registers a current in the wire loop
when moving magnet with respect to the loop
An ammeter registers a current just
when switch is closed or opened
Induction
Inducted current
Induced emf
Φ is varying with time : dΦ ≠ 0
dt
¾ Faradays’ Law of Induction: The magnitude of the emf induced in a conducting
loop is equal to the rate of change of the magnetic flux through that loop
E=−
dΦB
dt
or
E= −N
dΦB
(coil of N turns)
dt
¾ An emf will be induced if any one (or more ) of the following vary with time:
•
B, A, θ
Φ B = BA cos θ
Lenz’s Law
¾ Lenz’s Law is for determining the direction of an induced current in a loop.
¾ Lenz’s Law : An induced current has a direction such that the magnetic field due to the
current opposes the change in the magnetic flux ( ΔΦ ) that induces the current.
-
Opposition to Pole Movement
An Induced Field is created which attempts to negate the applied field
-
Opposition to Flux Change
The Induced current is such as to OPPOSE the CHANGE in applied field.
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6-2.1 A stationary circuit in a time-varying magnetic field:
Example 6-1: p231
A circular loop of N turns of conducting wire lies in the xy-plane with its center
JG G
at the origin of a magnetic field specified by B = a z Bo cos(π r / 2b) sin(ωt ) , where b
is the radius of the loop and ω is the angular frequency. Find the emf induced in the
loop.
6-2.3 A moving conductor in a static magnetic field:
JG
JG
G JG
F = qE + qu × B
¾ Total electromagnetic force:
Electric force
Motional electric field /
Induced electric field:
Lorentz’s force equation
Magnetic force (see p170.)
JG
JG
F m G JG
Em =
= u×B
q
Motional emf / flux-cutting emf:
G 2 G JG G
2 JG
m
Vemf
= V21 = ∫ E ⋅ dl = ∫ (u × B ) ⋅ dl
1
1
G JG G
If moving conductor is a
m
Vemf = v∫ (u × B ) ⋅ dl
part of a closed circuit C
C
--
Negative charge
++
Positive charge
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6-2.3 A moving conductor in a static magnetic field:
Example 6-2 (p236):
A metal bar slides over a pair of conducting rails in a uniform magnetic field
B = az B0 with a constant velocity u, as shown in Fig. 6-4.
1) Determine the open-circuit voltage Vo that appear across terminals 1 and 2
6-2.3 A moving conductor in a static magnetic field:
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6-2.4 A moving circuit in a time-varying magnetic field:
Vemf = V
tr
emf
tr
emf
V
+V
m
emf
m
Vemf
JG
∂ B JG
= −∫
⋅dS
∂t
S
G JG G
= v∫ (u × B) ⋅ dl
C
¾ General form of Faraday’s law:
Vemf = V
tr
emf
+V
m
emf
JG
G JG G
∂ B JG
= −∫
⋅ d S + v∫ (u × B ) ⋅ dl
∂t
S
C
(V)
¾ Another form of Faraday’s law:
'
=−
Vemf
d JG JG
dΦ
B⋅dS = −
∫
dt S
dt
(V)
6-2.4 A moving circuit in a time-varying magnetic field:
Example 6-5
p241
Angular speed: ω
Linear speed: v = r ω
α =ωt
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6-2.4 A moving circuit in a time-varying magnetic field:
Useful Applications
A-C generator
A-C motor
6-3 Maxwell’s Equations
JG
JG
∂B
∇× E = −
∂t
JJG JG
∇× H = J
JG
∇ ⋅ D = ρv
JG
∇⋅B = 0
So far, we have following equations:
Obvious, time-varying B gives rise to E.
Question is:
What about time-varying E, do you think it could induce B ? How ?
Recall the equation of continuity:
JG
∂ρ
∇⋅J = − v
∂t
JG
JJG JG ∂ D
∇× H = J +
∂t
JG
JJG
JG ∂ρ
JG ∂ D
∇ ⋅ (∇ × H ) = 0 = ∇ ⋅ J + v = ∇ ⋅ ( J +
)
∂t
∂t
¾ Introduce displacement current density Jd:
(Time rate of change of D)
Jd =
JG
∂D
∂t
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6-3 Maxwell’s Equations
JG
JJG JG ∂ D
∇× H = J +
∂t
J : Conduction current density
Jd : Displacement current density
¾ Displacement current density Jd:
¾ Displacement current Id:
I d = ∫ J d ⋅ dS = ∫
JG
∂D
⋅ dS
∂t
¾ Displacement current is a result of time-varying electric field
¾ Without the displacement current, electromagnetic waves do not exist
¾ Displacement current flows through free space, or time-varying electric
fields produce magnetic fields
6-3.1 Integral Form of Maxwell’s Equations
Time-varying B gives rise to E.
Time-varying E (D) gives rise to B (H).
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6-3.1 Integral Form of Maxwell’s Equations
Example 6-6 (p246):
An a-c source of amplitude Vo and angular frequency ω, Vc = V0 sin(ωt ) is connected
across a parallel-plate capacitor C, as shown in Fig. 6-7.
(a) Verify that the displacement current in the capacitor is the same as the conduction
current in the wires
(b) Determine the magnetic field intensity at a distance r from the wire
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