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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

E  da   = - dt  H   ds
0 C S

d

Circulation of E

Magnetic Flux

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

EQS form: field)

E  = 0  ds
C

(Kirchoff’s Voltage Law, conservative electric

MQS circuit form: v = L

di dt

(Inductor)

2. Ampère’s Law (with displacement current)

H   ds
C

=

J da 
S



d dt

 da E
0 S

Circulation Conduction of H Current

Displacement Current

MQS form:

H   ds
C

=

J da 
S

6.013 Electromagnetics and Applications Prof. Markus Zahn

Lecture 1 Page 1 of 7

dv (capacitor) dt 3. Gauss’ Law for Electric Field
EQS circuit form: i = C

 E   da
0 S

=

V

  dV

     8.854 ×10-12 farads/meter 0
c=

10-9 36 1

0  0

3  10 meters/second (Speed of electromagnetic waves in

8

free space) 4. Gauss’ Law for Magnetic Field

 H  da
0 S

= 0

In free space:

B = H 0
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

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

d

  dV
V

6.013 Electromagnetics and Applications Prof. Markus Zahn

Lecture 1 Page 2 of 7

J   da
S

+

d dt

  dV = 0
V

Total current Total charge leaving volume inside volume through surface 6. Lorentz Force Law

f = q E + v × H 0





II. Electric Field from Point Charge

r E  =  4  = q  da E 
0 0 r 2 S

Er =

q 4 0r  
2

T sin θ= fc =

q2 4  0r  
2

T cos θ= Mg
tan θ= q2 4 0 r Mg  
2

=

r 2l

6.013 Electromagnetics and Applications Prof. Markus Zahn

Lecture 1 Page 3 of 7

  r 3Mg 2 2   q=  0  l  

1

J   -q =  = i = - dt dV = - dt   dt  da
S

d

d

dq

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

Cb

H   H 2 a = N  ds
1

1 1

N i i  H1  1 1 2 a N2 a  0 i1  1 i1 2

N1  2  H1 = a 0

 

N2 a2  1 0 2 a

2  N a 0 L=  1 i1 2

=

1 LC

1 2

1 2 2 L ip  C vp  ip vp C L 2

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 =
5

 3k Hz 2

Hp 2.3 x 10 A / m  Bp =  Hp 0.3 Teslas  3000 Gauss 0
2. Electrical Breakdown in Single Turn Coil with Small Gap

R

C

p B
E0



 0 E  E 0

Inside Metal Coil Small Gap  d  2) R

E  E dt (B ds
0 C

p

6.013 Electromagnetics and Applications Prof. Markus Zahn

Lecture 1 Page 5 of 7

Bp Bm cos  t

B R  E0  m sin  t 
Take: Bm 0.3 Tesla, 20, 000 radians/second, R 0.07 m, 0.01 mm

2

B  R2  0.3(20, 000) (0.07)2 Em  m  9  6 Volts/meter 10 5  10 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

Ca

E   2 aE  ds



dBp d 2 2 =    a B da   Bm sin  a t dt Sa dt

 dBp a  a J =   =  E  Bm sin  t 2 dt 2 F = J x H , 0
f = dV F
V

=

Jx   H dV
0 V

Force per unit volume

total force

K J r  Hr  H J

F J  H J Hr ir  0 J r iz  J2  i  H iz 0 0 0 a   Fz  J2  Bm  sin2  t 0  0  2 
 a 2 2 fz Fz  2  0 a Bm  sin2  t 4
2 2 4

2

6.013 Electromagnetics and Applications Prof. Markus Zahn

Lecture 1 Page 6 of 7

 minum  3.7  7 Siemens/meter, a=0.07 m, =2 mm, 20, 000 10 alu radians/second, Bm 0.3 Tesla, M=0.08 kg
fz   0   a B sin 4
2 2 m  7 3  2   10  2

 t
7 2 2 2

10



0.3  3.7  .07 20, 000   sin  10  

 t

4.7  6 sin2  10 t Mg (0.08)9.8 0.8 Newtons fmax Mg  4.7  10 6 5.9  10 0.8
6

Neglecting losses:

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

v h    255 meters t 0  2g

2

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

```
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