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6.641 Electromagnetic Fields, Forces, and Motion, Spring 2005
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6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 1: Integral Form of Maxwell’s Equations
I. Maxwell’s Equations in Integral Form in Free Space
1. Faraday’s Law d
∫ E i ds = - dt ∫ µ
C
H i da
0
S
Circulation
of E
Magnetic Flux
µ0 = 4π ×10-7 henries/meter
[magnetic permeability of
free space]
EQS form:
∫ E i ds = 0
(Kirchoff’s Voltage Law, conservative electric
C
field)
MQS circuit form: v = L
di
(Inductor)
dt
2. Ampère’s Law (with displacement current)
∫ H i ds
=
C
∫ J i da
+
S
d
dt
Circulation Conduction
of H
Current
MQS form:
∫ H i ds
=
C
EQS circuit form: i = C
∫ε
0
E i da
S
Displacement
Current
∫ J i da
S
dv
(capacitor)
dt
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 1
Page 1 of 6
3. Gauss’ Law for Electric Field
∫ ε E i da
0
=
S
∫ ρ dV
V
-9
10
≈ 8.854 ×10-12 farads/meter
36π
1
c=
≈ 3 × 108 meters/second (Speed of electromagnetic waves in
ε0 µ 0
ε0 ≈
free space)
4. Gauss’ Law for Magnetic Field
∫ µ H i da
0
= 0
S
In free space:
B = µ0 H
magnetic
flux
density
(Teslas)
magnetic
field intensity
(amperes/meter)
5. Conservation of Charge
Take Ampère’s Law with displacement current and let contour C → 0
lim
C →0
d
∫ H i ds = 0 = ∫ J i da + dt ∫ ε E i da
0
C
S
S
∫ ρ dV
V
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 1
Page 2 of 6
∫ J i da
S
+
d
dt
∫ ρ dV = 0
V
Total current
Total charge
leaving volume inside volume
through surface
6. Lorentz Force Law
(
f = q E + v × µ0 H
)
II. Electric Field from Point Charge
∫
ε0 E i da = ε0Er 4π r2 = q
S
Er =
q
4π ε0r2
T sin θ = fc =
q2
4π ε0r2
T cos θ = Mg
tan θ =
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
q2
r
=
4π ε0r2Mg 2l
Lecture 1
Page 3 of 6
⎡ 2π ε0r3Mg ⎤
q= ⎢
⎥
l
⎣
⎦
1
2
III. Faraday Cage
d
d
dq
∫ J i da = i = - dt ∫ ρ dV = - dt (-q) = dt
S
∫ idt = q
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 1
Page 4 of 6
IV. Boundary Conditions
1. Gauss’ Continuity Condition
Courtesy of Krieger Publishing. Used with permission.
∫ ε E i da = ∫ σ dS ⇒ ε (E
0
s
S
S
0
2n
- E1n ) dS = σ sdS
ε0 (E2n - E1n ) = σ s ⇒ n i ⎡⎣ε0 (E2 - E1 ) ⎤⎦ = σ s
2. Continuity of Tangential E
Courtesy of Krieger Publishing. Used with permission.
∫ E i ds = (E
1t
- E2t ) dl = 0 ⇒ E1t - E2t = 0
C
(
)
n× E1 - E2 = 0
Equivalent to Φ1 = Φ2 along boundary
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 1
Page 5 of 6
3. Normal H
∇ i µ0 H = 0 ⇒
∫µ
0
H i da = 0
S
µ 0 (H a n - Hbn ) A = 0
H an = Hbn
n i ⎣⎡H a - H b ⎤⎦ = 0
4. Tangential H
∇ ×H = J ⇒
∫ H i ds = ∫ J i da
C
S
Hbt ds - H at ds = Kds
Hbt - H at = K
n × ⎡⎣H a - H b ⎤⎦ = K
5. Conservation of Charge Boundary Condition
∇ i J+
∂ρ
=0
∂t
d
∫ J i da + dt ∫ ρdV = 0
S
V
∂
n i ⎡⎣ J a - J b ⎤⎦ +
σs = 0
∂t
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 1
Page 6 of 6