Time varying Fields and Maxwell's Equations
Introduction
Fundamental relations for electrostatic model:
∇× E = 0
∇⋅D = 0
Constitutive relation;
D = εE
Fundamental relations for magneto static model:
∇⋅B = 0
∇× H = J
Constitutive relation;
B = µH
Electrostatic field produced by stationary charges while magneto
static field produced by direct currents.
E and D are independent from B and H . In time varying fields
(electromagnetic waves) we will see that existence of one field (i.e
E or
H ) is associated with the existence of the other, which
means in time varying E and D are dependent on B and H .
Studying the relation between the electric and magnetic fields in
the time varying case is introduced next.
Faraday's Law
(1) A Stationary Circuit in a Time Varying Magnetic Field
It is well known that current
carrying conductor produces a
I
magnetic field, the inverse
of this phenomena is also true.
B
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A magnetic field can produce a current in a closed circuit but this
magnetic flux linkage by the circuit must be changing with time.
I
I
B
B
N
N
Moving
up
Moving
down
S
S
Increasing B
Decreasing B
Referring to the above figure the magnetic field due to the induced
current opposes the magnet movement, thus moving the magnet
up and down produces AC current.
If the loop is open circuited an emf appears at its terminals, i.e
dψ m
emf = V = −
dt
B
(1)
emf
ψ m = total flux linkage
E
If this loop has N turns
V = −N
B
dψ m
dt
emf
The emf induced in the loop is equal to the emf producing field
E , thus
V = ∫ E ⋅ dl
The total flux through a circuit is equal to the integral of the normal
component of the flux density B over the surface bounded by the
circuit, i.e
ψ m = ∫∫ B ⋅ ds
(2)
s
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where
s is the surface bounded by the closed circuit . From (1)
V = −N
and (2),
d
dt
∫∫ B ⋅ ds
s
ds = surface element (m2)
B = Flux density in Tesla
t = time in seconds
If the closed circuit is stationary or fixed, i.e (velocity v = 0 ), then
V = − N ∫∫
s
dB
⋅ ds
dt
Moving Conductor in a Magnetic Field
y
Lorentz force F on a particle of
x
1
charge Q moving with a velocity v in
a magnetic field B is
v
F = Q (v × B )
Ec
F
= E = (v × B )
Q
2
2
1
1
emf = V = ∫ E ⋅ dl = ∫ (v × B ) ⋅ dl
•
B
2
For closed circuit moving in a magnetic field with velocity v ,
emf = V = ∫ E ⋅ dl = ∫ (v × B ) ⋅ dl
General Case of Induction
The total emf include in a closed circuit due to time varying
magnetic field and moving by velocity v is given by
emf = ∫ (v × B ) ⋅ dl − ∫∫
s
emf due to
motion
dB
⋅ ds
dt
emf due to time
var ying magntic field
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Sign Convention
a- B increases Î ψ i increases
B
b- the induced current produces a
Ee
flux ψ i trying to compensate for
_ 1
V
the increase of the flux due to B .
+
ψi
2
I
c- Ee is the induced potential
1
V = ∫ Ee ⋅ dl = − A
2
∂B
∂t
a- ψ i is increasing due to the
movement of the bar.
b- the induced current produces a
flux ψ i trying to compensate for
the increase of the flux due to B .
c- the induced potential
1
V = ∫ Ee ⋅ dl = v × B l = − B
2
_
V
+
ψi
1
2
I
v
Ee
B
∂A
∂t
A = area of the circuit
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