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


C
E ds = -  H da
dt  S
0

Circulation Magnetic Flux

of E

0 = 4 ×10 -7 henries/meter
[magnetic permeability of
free space]

EQS form: 

C
E ds = 0 (Kirchoff’s Voltage Law, conservative electric

field)

di
MQS circuit form: v = L (Inductor)
dt

2. Ampère’s Law (with displacement current)

d

H ds

C
= J da

S

dt
E da

S
0

Circulation Conduction Displacement

of H Current Current

MQS form: 
H ds
C
= J da

S

6.013 Electromagnetics and Applications Lecture 1

Prof. Markus Zahn Page 1 of 7
 dv
EQS circuit form: i = C (capacitor)
dt
3. Gauss’ Law for Electric Field


E da0
= 

dV
S V

10-9

0 
 

8.854 ×10-12 farads/meter

36
1 8
c= 3 10 meters/second (Speed of electromagnetic waves in

0 0

free space)

4. Gauss’ Law for Magnetic Field



S
H da0
= 0

In free space:

B = 0 H

magnetic magnetic
flux field
density intensity
(Teslas) (amperes/meter)

5. Conservation of Charge

Take Ampère’s Law with displacement current and let contour C  0

d
lim
C 0

H ds = 0 = 
J da +
dt

E da 0
C S S



V
dV

6.013 Electromagnetics and Applications Lecture 1

Prof. Markus Zahn Page 2 of 7
 d


S
J da +
dt 

V
dV = 0

Total current Total charge

leaving volume inside volume
through surface

6. Lorentz Force Law


f = q E + v × 0 H 
II. Electric Field from Point Charge


E da = E 4 
r =q 2
0 0 r
S

q
Er =
0r
4
2

q2
T sin θ= fc =
4
0r
2

T cos θ= Mg

q2 r
tan θ= =

40
r Mg
2
2l

6.013 Electromagnetics and Applications Lecture 1

Prof. Markus Zahn Page 3 of 7
 1
r 3Mg 2
2
q=  0 
 l 

III. Faraday Cage

d d dq

J da = i = -
dt 

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 Lecture 1

Prof. Markus Zahn Page 4 of 7
 N i

H
Cb
ds H 2 a = N
1 1 1
i  H1  1 1
2a

N21 a2 0 N2 a 0
 
N1 a2 0 H1 =
2 a
i1  1
2
i1

 N a 0
2
L=  1
i1 2

1
=
LC

1 1
L ip  C vp  ip vp C
2 2

2 2 L

C = 25 f, v p 4 k V, N1 = 50, a 7 c m

L1 0.1 mH


ip 2000 A, 20 x 103 / s  f = 3k Hz
2

5
Hp 2.3 x 10 A / m  Bp = 0 Hp 0.3 Teslas 3000 Gauss

2. Electrical Breakdown in Single Turn Coil with Small Gap


Bp

E0


0 Inside Metal Coil

E 
E0 Small Gap 

d

E

C
ds E  (B
dt
0 p

R 2)

6.013 Electromagnetics and Applications Lecture 1

Prof. Markus Zahn Page 5 of 7
 Bp Bm cos t
2
B R
E0  m sin t


Take: Bm 0.3 Tesla, 20, 000 radians/second, R 0.07 m, 0.01 mm

B R2 0.3(20, 000)(0.07)2
Em  m  9 106 Volts/meter
 105
6
Breakdown strength of air 3 10 Volts/meter.

Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.

3. Force on Metal Disk

d dBp

E =  B da a
2 2
ds 2 aE 
a Bm sin t
Ca
dt Sa dt

a dBp a
J = E =   Bm sin t
2 dt 2

F = J x 0 H , f=
F dV = 
J x  H dV
0
V V

Force per unit total force

volume

K JHr  Hr J

F J 0 H J
i 0 Hr ir 0 JHr iz 0 J2 iz

2
a  2
Fz 0 J2
0  Bm  sin t
2 

2 2 4
 a 2 2
fz Fz a2  0 Bm  sin2 t
4

6.013 Electromagnetics and Applications Lecture 1

Prof. Markus Zahn Page 6 of 7
 alu minum 3.7 107 Siemens/meter, a=0.07 m, =2 mm, 20, 000
radians/second, Bm 0.3 Tesla, M=0.08 kg

0
a B sin
2
2 2
fz  m
t
4
2
10
7
 
2 10 3 3.7 10 .07 20, 000 0.3 
7 2


sin
2
t

4.7 106 sin2 t

Mg (0.08)9.8 0.8 Newtons

fmax 4.7 10
6
6
 5.9 10
Mg 0.8

Neglecting losses:

1 2 1 2
CV  Mv (t 0) Mgh
2 2
C
v(t 0 )  V
M
C 25f , M .08 kg, Vp 4000 volts
v(t 0 ) 70.7 meters/second (Initial velocity)
2
v
h  t 0 255 meters (Maximum height)
2g

Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.

6.013 Electromagnetics and Applications Lecture 1

Prof. Markus Zahn Page 7 of 7