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THE CERN ACCELERATOR SCHOOL
Theory of Electromagnetic Fields
Part I: Maxwell's Equations
Andy Wolski
The Cockcroft Institute, and the University of Liverpool, UKCAS Specialised Course on RF for Accelerators
Ebeltoft, Denmark, June 2010
Theory of Electromagnetic Fields
In these lectures, we shall discuss the theory of electromagnetic elds, with an emphasis on aspects relevant to RF systems in accelerators: 1.
Maxw ell'sequations
Maxwell's equations and their physical signicance
Electromagnetic potentials
Electromagnetic waves and their generation
Electromagnetic energy
2.
Standing W aves
Boundary conditions on electromagnetic elds
Modes in rectangular and cylindrical cavities
Energy stored in a cavity
3.
T ravellingW aves
Rectangular waveguides
Transmission linesTheory of EM Fields1Part 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 signicance; 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 Fields2Part I: Maxwell's Equations
Summary of relations in vector calculus
In cartesian coordinates:
gradf rf @f@x ;@f@y ;@f@z (1) div ~A r ~A@Ax@x +@Ay@y +@Az@z (2) curl ~A r ~A ^x^y^z @@x @@y @@z A xAyAz (3) r
2f@2f@x
2+@2f@y
2+@2f@z
2(4) Note that ^x, ^yand ^zare unit vectors parallel to thex,yandz axes, respectively.Theory of EM Fields3Part I: Maxwell's Equations
Summary of relations in vector calculus
Gauss' theorem:
Z V r ~AdV=I
S~Ad~S;(5)
for any smooth vector eld ~A, where the closed surfaceS bounds the volumeV.
Stokes' theorem:
Z S r ~Ad~S=I
C~Ad~`;(6)
for any smooth vector eld ~A, where the closed loopCbounds the surfaceS.
A useful identity:
r r ~A r(r ~A) r2~A:(7)Theory of EM Fields4Part I: Maxwell's Equations
Maxwell's equations
r ~D=r ~B= 0 r ~E=@~B@t r ~H=~J+@~D@t
James Clerk Maxwell
1831 { 1879
Note thatis the electric charge density; and~Jis the current density.
Theconstitutive relationsare:
D="~E;~B=~H;(8)
where"is the permittivity, andis the permeability of the material in which the elds exist.Theory of EM Fields5Part I: Maxwell's Equations Physical interpretation ofr ~B= 0Gauss' theorem tells us that foranysmooth vector eld~B: Z V r ~B dV=I
S~Bd~S;(9)
where the closed surfaceSbounds the regionV. Applied to Maxwell's equationr ~B= 0, Gauss' theorem tells us that the total ux entering a bounded region equals the total ux leaving the same region.Theory of EM Fields6Part I: Maxwell's Equations Physical interpretation ofr ~D=Applying Gauss' theorem to Maxwell's equationr ~D=, we nd that:Z V r ~D dV=I
S~Dd~S=Q;(10)
whereQ=R
VdVis the total charge within the regionV,
bounded by the closed surfaceS.
The total
ux of electric displacement crossing a closed surface equals the total electric charge enclosed by that surface.
In particular, at a distancerfrom the
centre of any spherically symmetric charge distribution, the electric displacement is:
D=Q4r2^r;(11)
whereQis the total charge within radiusr, and ^ris a unit vector in the radial direction.Theory of EM Fields7Part I: Maxwell's Equations
Physical interpretation ofr ~H=~J+@~D@t
Stokes' theorem tells us that foranysmooth vector eld~H: Z S r ~Hd~S=I
C~Hd~`;(12)
where the closed loopCbounds the surfaceS.
Applied to Maxwell's equation
r ~H=~J+@~D@t , Stokes' theorem tells us that the magnetic eld ~Hintegrated around a closed loop equals the total current passing through that loop. For the static case (constant currents and elds): I
C~Hd~`=Z
S~Jd~S=I:(13)Theory of EM Fields8Part I: Maxwell's Equations
The displacement current and charge conservation
The term
@~D@t in Maxwell's equationr ~H=~J+@~D@t is known as thedisplacement current density, and has an important physical consequence.
Since, for any smooth vector eld
~H: r r ~H0;(14) it follows that: r ~J+r @~D@t =r ~J+@@t = 0:(15) This is the continuity equation, that expresses the local conservation of electric charge. The signicance is perhaps clearer if we use Gauss' theorem to express the equation in integral form: I
S~Jd~S=dQdt
;(16) whereQis the total charge enclosed by the surfaceS.Theory of EM Fields9Part I: Maxwell's Equations
Physical interpretation ofr ~E=@~B@t
Applied to Maxwell's equationr ~E=@~B@t
, Stokes' theorem tells us that a time-dependent magnetic eld generates an electric eld. In particular, the total electric eld around a closed loop equals the rate of change of the total magnetic ux through that loop: Z S r ~Ed~S=I
C~Ed~`=@@t
Z
S~Bd~S:(17)
This is Faraday's law of electromagnetic
induction: E=@@t ;(18) whereEis the electromotive force (the integral of the electric eld) around a closed loop, and is the total magnetic ux through that loop.Theory of EM Fields10Part I: Maxwell's Equations
Solving Maxwell's equations
Maxwell's equations are of fundamental importance in electromagnetism, because they tell us the elds that exist in the presence of various charges and materials. In accelerator physics (and many other branches of applied physics), there are two basic problems: Find the electric and magnetic elds in a system of charges and materials of specied size, shape and electromagnetic characteristics. Find a system of charges and materials to generate electric and magnetic elds with specied properties.Theory of EM Fields11Part I: Maxwell's Equations Example: elds induced by a bunch in an accelerator
Theory of EM Fields12Part I: Maxwell's Equations
Linearity and superposition
Neither problem is particularly easy to solve in general; but fortunately, there are ways to decompose complex problems into simpler ones...
Maxwell's equations arelinear:
r ~B1+~B2=r ~B1+r ~B2;(19) and: r ~H1+~H2=r ~H1+r ~H2:(20)
This means that if two elds
~B1and~B2satisfy Maxwell's equations, so does their sum ~B1+~B2. As a result, we can apply theprinciple of superpositionto construct complicated electric and magnetic elds just by adding together sets of simpler elds.Theory of EM Fields13Part I: Maxwell's Equations Example: plane electromagnetic waves in free space Perhaps the simplest system is one in which there are no charges or materials at all: a perfect, unbounded vacuum.
Then, the constitutive relations are:
D="0~E;and~B=0~H;(21)
and Maxwell's equations take the form: r ~E= 0r ~B= 0 r ~E=@~B@t r ~B=1c
2@~E@t
quotesdbs_dbs4.pdfusesText_8
[PDF] Chapter 6 Maxwells Equations for Electromagnetic Waves