Transcript Electromagnetism

Electromagnetics
Oana Mihaela Drosu
Dr. Eng., Lecturer
Politehnica University of Bucharest
Department of Electrical Engineering
LPP ERASMUS+
Contents
•
Review of Maxwell’s equations
•
Electromagnetic energy conservation
2
References
• J.D. Jackson: Classical Electrodynamics
• H.D. Young and R.A. Freedman: University Physics
(with Modern Physics)
• P.C. Clemmow: Electromagnetic Theory
• Feynmann Lectures on Physics
• W.K.H. Panofsky and M.N. Phillips: Classical
Electricity and Magnetism
• G.L. Pollack and D.R. Stump: Electromagnetism
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Basic Equations from Vector
Calculus
For a scalar function φx,y,z,t ,
 φ φ φ 
gradient : φ   , , 
 x y z 
Gradient is normal to surfaces
=constant

For a vector F  F1 , F2 , F3 ,
 F1 F2 F3
divergence :   F 


x
y
z
  F3 F2 F1 F3 F2 F1 
curl :   F  

,

,


z z
x x y 
 y
4
Basic Vector Calculus


 
 
  ( F  G)  G    F  F    G

    0,     F  0



2
  (  F )  (  F )   F
Stokes’ Theorem
 
 
   F  dS   F  d r
S
C
 
dS  n dS
Oriented
boundary C

n
Divergence or Gauss’
Theorem

 
   F dV   F  dS
V
S
Closed surface S, volume V,
outward pointing normal
5
What is Electromagnetism?
• The study of Maxwell’s equations, devised in 1863 to
represent the relationships between electric and magnetic
fields in the presence of electric charges and currents,
whether steady or rapidly fluctuating, in a vacuum or in
matter.
• The equations represent one of the most elegant and
concise way to describe the fundamentals of electricity and
magnetism. They pull together in a consistent way earlier
results known from the work of Gauss, Faraday, Ampère,
Biot, Savart and others.
• Remarkably, Maxwell’s equations are perfectly consistent
with the transformations of special relativity.
Maxwell’s Equations
Relate Electric and Magnetic fields generated by
charge and current distributions.
E = electric field
D = electric displacement
H = magnetic field
B = magnetic flux density
= charge density
j = current density
0 (permeability of free space) = 4 10-7 H/m
0 (permittivity of free space) = 8.854 10-12 F/m
c (speed of light) = 2.99792458 108 m/s
In vacuum


D   0 E,


B  0 H ,

D  

B  0


B
E  
t


 D
H  j 
t
 0 0c 2  1
 
E 
Maxwell’s 1st Equation
0
Equivalent to Gauss’ Flux Theorem:
 
 E 
0

  1
    E dV   E  d S 
V
0
S
Q
  dV  
V
0
The flux of electric field out of a closed region is proportional to
the total electric charge Q enclosed within the surface.
A point charge q generates an electric field

E
q
40 r
 
q
E  dS 

40
sphere

r
3
dS q

2

r
0
sphere
Area integral gives a measure of the net charge
enclosed; divergence of the electric field gives the density
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of the sources.

B  0
Maxwell’s 2nd Equation
Gauss’ law for magnetism:

B  0 
 
 B  dS  0
The net magnetic flux out of any
closed surface is zero. Surround a
magnetic dipole with a closed surface.
The magnetic flux directed inward
towards the south pole will equal the
flux outward from the north pole.
If there were a magnetic monopole
source, this would give a non-zero
integral.
Gauss’ law for magnetism is then a statement that
There are no magnetic monopoles


B
E 
t
Maxwell’s 3rd Equation
Equivalent to Faraday’sLaw of Induction:

 
B 
S   E  dS  S t  dS

 
d
d
E

d
l


B

d
S


C
dt 
dt
S
(for a fixed circuit C)
The electromotive
  force round a
circuit 
  E  dl
is proportional to the
rate of change of flux of magnetic
 
field,   B  dS through the circuit.

Faraday’s Law is the basis for electric
generators. It also forms the basis for
inductors and transformers.
N
S


 1 E
  B  0 j  2
c t
Maxwell’s 4th Equation


  B  0 j
Originates from Ampère’s (Circuital) Law :

C
Ampère
 
 
 
B  dl     B  dS  0  j  dS  0 I
S
S
Satisfied by the field for a steady line current (Biot-Savart Law,
 
1820):
 0 I
B
4

dl  r
r3
For a straight line current
Biot
0 I
B 
2 r
Need for Displacement Current
•
•
Faraday: vary B-field, generate E-field
Maxwell: varying E-field should then produce a B-field, but not covered by
Ampère’s Law.
 Apply Ampère to surface 1 (flat disk): line
integral of B = 0I
Surface 2
Surface 1
 Applied to surface 2, line integral is zero
since no current penetrates the deformed
surface.
Current I
dQ
dE
Q , so
 In capacitor, E 
I
 A
ε0 A
Closed loop

0
dt
dt


E
Displacement current density is j  
d
0
t


 

E
  B   0  j  jd    0 j   0 0
t
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Consistency with Charge Conservation
Charge conservation:
From Maxwell’s equations:
Total current flowing out of a region
equals the rate of decrease of charge
Take divergence of (modified) Ampère’s
equation
within the volume.

 1 
    B   0  j  2   E
c t


 0   0  j   0  0  
t   0 
 
 0   j 
t
 
d
j

d
S


 dV


dt


    j dV   
dV
t
 
  j 
0
t
 
Charge conservation is implicit in Maxwell’s Equations
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Maxwell’s Equations in Vacuum
In vacuum

 

1
D   0 E , B  0 H ,  0 0  2
c
Source-free equations:

B  0

 B
E
0
t
Source equations
 
E 
0

 1 E

B 2
 0 j
c t
Equivalent integral forms
(useful for simple
geometries)

  1
E  dS 

 
B  dS  0
0
  dV
 
 
d
d
 E  dl   dt  B  dS   dt
 
 
  1 d
 B  dl  0  j dS  c2 dt  E  dS
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Electromagnetic Energy
• Rate of doing work on unit volume of a system is


  
 
 

 
 v  f d  v   E  j  B   v  E   j  E
• Substitute for j from Maxwell’s equations and re-arrange into the form


 D  
  
  D
 

  E    E  H  H    E  E 
 j  E     H 
t 
t



  
 B  D
 S  H 
E
where S  E  H
t
t
1     
 S 
E  D  B  H Poynting vector
2 t


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


  


 
 1  
 j E 
 BH  E D  E  H
t  2


Integrated over a volume, have energy conservation law: rate
of doing work on system equals rate of increase of stored
electromagnetic energy+ rate of energy flow across
boundary.


  
dW
d
1    
  E  D  B  H dV   E  H  dS
dt
dt
2
electric +
magnetic energy
densities of the
fields
Poynting vector
gives flux of e/m
energy across
boundaries
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Poynting Vector
  
Poynting vector is S  E  H  E y H z ,0, E y H x 
Time-averaged:
Integrate over x:
 1
kA2 2 n x
S  0, 0,1
sin
2

a
1 akA2
Sz 
4 
So energy is transported at a rate:
Total e/m energy
density
1 2
W  A a
4
Sz
k

 vg
We  Wm 
Electromagnetic energy is transported down the waveguide
with the group velocity
17