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6.013/ESD.013J Electromagnetics and Applications, Fall 2005
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Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.013 Electromagnetics and Applications
Lecture 1, Sept. 8, 2005
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
Current
of H
MQS form:
∫ H i ds
C
6.013 Electromagnetics and Applications
Prof. Markus Zahn
=
∫ε
0
E i da
S
Displacement
Current
∫ J i da
S
Lecture 1
Page 1 of 7
dv
(capacitor)
dt
3. Gauss’ Law for Electric Field
EQS circuit form: i = C
∫ 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
6.013 Electromagnetics and Applications
Prof. Markus Zahn
Lecture 1
Page 2 of 7
∫ 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
∫ ε E i da = ε E 4π r
0
0 r
2
=q
S
Er =
q
4π ε0r2
T sin θ = fc =
q2
4π ε0r2
T cos θ = Mg
tan θ =
6.013 Electromagnetics and Applications
Prof. Markus Zahn
q2
r
=
4π ε0r2Mg 2l
Lecture 1
Page 3 of 7
⎡ 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
IV. Edgerton’s Boomer
1. Magnetic Field, Current, and Inductance
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
6.013 Electromagnetics and Applications
Prof. Markus Zahn
Lecture 1
Page 4 of 7
∫ H i ds ≈ H 2 π a = N
1
i ⇒ H1 ≈
1 1
Cb
(
)
λ ≈ N1 π a2 µ0 H1 =
L=
N12 π a 2 µ0
2 π
a
N1 i1
2πa
i1 ≈
N12 a µ0
i1
2
2
1
λ N a µ
0
≈
i1
2
ω =
1
LC
1
1
L ip2 ≈ C vp2 ⇒ ip ≈ vp C
L
2
2
C = 25 µ f, vp = 4 k V, N1 = 50, a ≈ 7 c m
L1 ≈ 0.1 mH
ip ≈ 2000 A, ω ≈ 20 x 103 / s ⇒ f =
ω
≈ 3k Hz
2π
Hp ≈ 2.3 x 105 A / m ⇒ Bp = µ0 Hp ≈ 0.3 Teslas ≈ 3000 Gauss
2. Electrical Breakdown in Single Turn Coil with Small Gap
R
Bp
∆
⎧0
E≈ ⎨
⎩E0
Inside Metal Coil
Small Gap ∆
6.013 Electromagnetics and Applications
Prof. Markus Zahn
Lecture 1
Page 5 of 7
d
∫ Eids = E ∆ = − dt ( B πR
0
2
p
)
C
Bp = Bm cos ωt
E0 =
BmωπR 2
sin ωt
∆
Take: Bm ≈ 0.3 Tesla, ω ≈ 20, 000 radians/second, R ≈ 0.07 m, ∆ = 0.01 mm
Bm ωπR 2 0.3(20, 000)π(0.07)2
=
= 9 × 106 Volts/meter
∆
10−5
Breakdown strength of air ≈ 3 × 106 Volts/meter.
Em =
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
3. Force on Metal Disk
∫ E i ds ≈ 2 π aE
φ
=−
Cb
dBp
d
B i da ≈ −π a2
= πa2Bmω sin ωt
∫
dt Sa
dt
Jφ = σ Eφ = −
F = J x µ0 H ,
σ a dBp
σa
Bm ω sin ωt
=
2
dt
2
f = ∫ F dV
=
Force per unit
volume
∫ Jx µ
0
H dV
V
V
total force K φ ≈ Jφ ∆ = −Hr ⇒ Hr = − Jφ ∆
F = J × µ0 H = Jφ iφ × µ0Hr ir = −µ0 JφHr iz = µ0 Jφ2 ∆iz
2
⎛ σa
⎞
Fz = µ0 J2φ ∆ = µ0 ∆ ⎜
Bmω ⎟ sin2 ωt
2
⎝
⎠
fz = Fz πa2 ∆ = π
µ0 ∆2 σ2 a4 2 2
Bmω sin2 ωt
4
6.013 Electromagnetics and Applications
Prof. Markus Zahn
Lecture 1
Page 6 of 7
σaluminum ≈ 3.7 × 107 Siemens/meter, a=0.07 m,
∆ =2 mm, ω = 20, 000
radians/second, Bm ≈ 0.3 Tesla, M=0.08 kg
fz =
µ0
π∆σa2 ωBm
4π
(
)
(
2
sin2 ωt
)(
2
)
2
= 10−7 ⎡ π 2 × 10−3 3.7 × 107 (.07 ) 20, 000 ( 0.3) ⎤ sin2 ωt
⎣
⎦
6
2
= 4.7 × 10 sin ωt
Mg = (0.08)9.8 ≈ 0.8 Newtons
fmax
4.7 × 106
≈
≈ 5.9 × 106
Mg
0.8
Neglecting losses:
1
1
CV2 = Mv2 (t = 0+ ) = Mgh
2
2
C
V
M
C = 25µf , M = .08 kg, Vp = 4000 volts
v(t = 0+ ) =
v(t = 0+ ) = 70.7 meters/second
h=
(Initial velocity)
2
v
( t = 0+ ) = 255 meters
2g
(Maximum height)
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
6.013 Electromagnetics and Applications
Prof. Markus Zahn
Lecture 1
Page 7 of 7