THE CERN ACCELERATOR SCHOOL
Theory of Electromagnetic Fields
Part I: Maxwell’s Equations
Andy Wolski
The Cockcroft Institute, and the University of Liverpool, UK
CAS Specialised Course on RF for Accelerators
Ebeltoft, Denmark, June 2010
Theory of Electromagnetic Fields
In these lectures, we shall discuss the theory of electromagnetic
fields, with an emphasis on aspects relevant to RF systems in
accelerators:
1. Maxwell’s equations
•
Maxwell’s equations and their physical significance
•
Electromagnetic potentials
•
Electromagnetic waves and their generation
•
Electromagnetic energy
2. Standing Waves
•
Boundary conditions on electromagnetic fields
•
Modes in rectangular and cylindrical cavities
•
Energy stored in a cavity
3. Travelling Waves
•
Rectangular waveguides
•
Transmission lines
Theory of EM Fields
1
Part I: Maxwell’s Equations
Theory of Electromagnetic Fields
I shall assume some familiarity with the following topics:
•
vector calculus in Cartesian and polar coordinate systems;
•
Stokes’ and Gauss’ theorems;
•
Maxwell’s equations and their physical significance;
•
types of cavities and waveguides commonly used in
accelerators.
The fundamental physics and mathematics is presented in
many textbooks; for example:
I.S. Grant and W.R. Phillips, “Electromagnetism,”
2nd Edition (1990), Wiley.
Theory of EM Fields
2
Part I: Maxwell’s Equations
Summary of relations in vector calculus
In cartesian coordinates:
grad
f
≡ ∇
f
≡
∂f
∂x
,
∂f
∂y
,
∂f
∂z
!
(1)
div
~
A
≡ ∇ ·
~
A
≡
∂A
x
∂x
+
∂A
y
∂y
+
∂A
z
∂z
(2)
curl
~
A
≡ ∇ ×
~
A
≡
ˆ
x
ˆ
y
ˆ
z
∂
∂x
∂
∂y
∂
∂z
A
x
A
y
A
z
(3)
∇
2
f
≡
∂
2
f
∂x
2
+
∂
2
f
∂y
2
+
∂
2
f
∂z
2
(4)
Note that ˆ
x
, ˆ
y
and ˆ
z
are unit vectors parallel to the
x
,
y
and
z
axes, respectively.
Theory of EM Fields
3
Part I: Maxwell’s Equations
Summary of relations in vector calculus
Gauss’ theorem:
Z
V
∇ ·
~
A dV
=
I
S
~
A
·
d
~
S,
(5)
for any smooth vector field
~
A
, where the closed surface
S
bounds the volume
V
.
Stokes’ theorem:
Z
S
∇ ×
~
A
·
d
~
S
=
I
C
~
A
·
d
~
‘,
(6)
for any smooth vector field
~
A
, where the closed loop
C
bounds
the surface
S
.
A useful identity:
∇ × ∇ ×
~
A
≡ ∇
(
∇ ·
~
A
)
 ∇
2
~
A.
(7)
Theory of EM Fields
4
Part I: Maxwell’s Equations
Maxwell’s equations
∇ ·
~
D
=
ρ
∇ ·
~
B
= 0
∇ ×
~
E
=

∂
~
B
∂t
∇ ×
~
H
=
~
J
+
∂
~
D
∂t
James Clerk Maxwell
1831 – 1879
Note that
ρ
is the electric charge density; and
~
J
is the current
density.
The
constitutive relations
are:
~
D
=
ε
~
E,
~
B
=
μ
~
H,
(8)
where
ε
is the permittivity, and
μ
is the permeability of the
material in which the fields exist.
Theory of EM Fields
5
Part I: Maxwell’s Equations
Physical interpretation of
∇ ·
~
B
= 0
Gauss’ theorem tells us that for
any
smooth vector field
~
B
:
Z
V
∇ ·
~
B dV
=
I
S
~
B
·
d
~
S,
(9)
where the closed surface
S
bounds the region
V
.
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