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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

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