Download 2 Integral form of Maxwell`s equations and time

EECS 530
Lecture 2
c Kamal Sarabandi
°
All rights reserved
Fall 2010
2 Integral form of Maxwell’s equations and time-varying
surfaces
As discussed before the general forms of Maxwell’s modified Ampère’s and Faraday’s
laws are given by
ZZ
I
d
D · ds
(1)
H · dℓ = I +
dt
C
I
Z ZS
d
E · dℓ = −
B · ds
(2)
dt
C
S
In situations where both the field quantities and s are varying with time explicit expressions for
ZZ
d
e
V =
B · ds
(3)
dt
S(t)
d
Ie =
dt
ZZ
D · ds
(4)
S(t)
are needed. To evaluate (3) or (4) we start with the definition of time derivative. For
example for (3)




ZZ
ZZ


1
e
B(t + ∆t) · ds −
B(t) · ds
V = lim

∆t→0 ∆t 


S(t+∆t)
S(t)
Noting that
∂B(t)
∆t
B(t + ∆t) ≃ B(t) +
¶ ∂t Z Z
ZZ µ
1
∂B
∂B
Ve1 = lim
∆t ds =
· ds
∆t→0 ∆t
∂t
∂t
(5)
S(t)
S(t+∆t)
The second component to be evaluated is




Z
Z
Z
Z

1 
e
V2 = lim
B(t) · ds −
B(t) · ds

∆t→0 ∆t 


S(t+∆t)
(6)
S(t)
Consider a moving surface geometry shown in Fig. 2.1, which shows progress of
S(t) in the time interval ∆t.
1
Figure 2.1. A moving surface is space. Shown is the surface change in a small time
interval ∆t.
Every point on the contour moves by
dℓ1 = V ∆t
where V is the local speed of the point. As the contour moves it traces a surface on the
side. We denote this surface by ∆S. Let us also denote the closed surface formed by
S(t), S(t + ∆t) and ∆S by S0 . Therefore the second component given by (6) can be
written as


ZZ
ZZ

1
B(t) · ds
(7)
Ve2 = lim
° B(t) · ds −

∆t→0 ∆t  S0
∆S
where the differential surface ds for the side surface ∆S is given by
ds = dℓ × ∆ℓ1 = dℓ × V ∆t
Thus
1
∆t→0 ∆t
lim
ZZ
B(t) · ds =
I
B(t) · dℓ × V(t)
I
V (t) × B(t)] · dℓ
[V
C
∆s
=
(8)
C
Also
ZZZ
ZZ
1
1
lim
∇ · B dv
° B · ds = lim
∆t→0 ∆t S0
∆t→0 ∆t
ZZ
1
V ∆t · ds
(∇ · B)V
= lim
∆t→0 ∆t
=
S(t)
ZZ
V · ds
(∇ · B)V
(9)
S(t)
Since ∇ · B = 0, the integral term given by (9) vanishes, and (2) now can be written as
I
ZZ
I
∂B
V × B) · dℓ
· ds + (V
−
(10)
E · dℓ =
∂t
C
C
S
2
The first term is referred to as the “transformer induction” and the second term is known
as the “motional induction.”
It is very important to note that in order to arrive at a non-vanishing motional induction term, the contour of the surface S has to be moving. That is, if the contour is
fixed and S(t) is still a function time there will be no motional induction. Physically the
contour of the integral (circuit) has to “cut” the flux lines of the magnetic flux density in
order to generate “electromotive potential.”
This phenomenon can physically be explained using the Lorentz force on a charged
particle q
V ×B
F = qE + qV
(11)
Since electric field is defined in terms of force per unit charge, the equivalent induced
electric field due to moving charges can be defined
Femf
=V ×B
q
I
I
V × B) · dℓ
= Eemf · dℓ = (V
Eemf =
Vemf
(12)
Equation (12) is consistent with (10).
q
(a) a positive moving charge in a
magnetic field
(b) a metallic rod moving in a magnetic
field which produces electromotive
voltage across the bar
In a similar manner (1) can be written as
I
ZZ
ZZ
I
∂D
V × D) · dℓ
V · ds − (V
H · dℓ = I +
· ds +
(∇ · D)V
∂t
C
S
S
Noting that ∇ · D = ρ
I
I
ZZ
ZZ
∂D
V × D) · dℓ
V · ds +
· ds − (V
H · dℓ = I +
ρV
∂t
C
C
S
(13)
S
V represents the drift current, ∂D/∂t
in which I represents the conduction current, ρV
V × D) represents the “motional current.”
represents the displacement current, and (V
3