Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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154 Formal Methods of Electrostatics

renaming and yields

(3.108)

For now, we are only interested in a point charge Q at location , for

which

(3.109)

The boundary condition (3.105) can now be written in the following way:

i.e.,

(3.110)

This solves the problem:

(3.111)

Since also

kl,() kx x

–()[]cos ly y

–()[]cos kdld

∞

∫

∞

∫







∞–

+∞

∫

∞–

+∞

∫

k' xx

–()[]cos l ' yy

–()[]cos dxdy⋅

kl,() kx x

–()[]cos k' xx

–()[]xdcos

∞–

+∞

∫







∞

∫

∞

∫

ly y

–()[]cos l' yy

–()[]cos yd

∞–

+∞

∫







kdld⋅

kl,()π

δ kk'–()δkk'+()+[]δll'–()δll'+()+[]kdld

∞

∫

∞

∫

k' l ',() .=

k' k⇒ l' l⇒

kl,()

------

σ xy,() kx x

–()[]cos ly y

–()[]cos xdyd

∞–

+∞

∫

∞–

+∞

∫

kl,()

------

δ xx

–()δyy

–()kx x

–()[]cos ly y

–()[]cos xdyd

∞–

+∞

∫

∞–

+∞

∫

------

fkl,() kx x

–()[]cos ly y

–()[]k

+cos 2 kdld

∞

∫

∞

∫

-----------

kx x

–()[]cos ly y

–()[]cos kdld

∞

∫

∞

∫

kl,()

2π

-----------------------------------=

12,

2π

--------------

kx x

–()[]cos ly y

–()[]k

+ zz

––[]expcos

-----------------------------------------------------------------------------------------------------------------------------

dkdl

∞

∫

∞

∫

3.5 Separation of Laplace’s Equation in Cartesian Coordinates 155

a comparison reveals that

(3.112)

This is the Fourier integral for the inverse distance.

We have solved a quite simple problem with rather complicated means. We

had already known the potential of a point charge in the infinite space when we

calculated it earlier by much simpler means. The purpose of this example was to

illustrate the methodology. As a by-product did we obtain the important relation

(3.112) for which there may not exist a simpler derivation.

3.5.2.4 Appendix to Section 3.5: Fourier Series and Fourier Integrals

The topic of Fourier series and Fourier integrals with respect to one or two

dimensions automatically arose when separating Laplace’s equation in Cartesian

coordinates. This is a specific case of the more general expansion of functions by

certain orthogonal and complete systems of functions, whose occurrence is typical

for problems of this kind. We will begin the next section (3.6) with general

observations on such systems of functions, which will retroactively highlight the

role of Fourier series and Fourier integrals from a more general point of view.

Sections 3.7 and 3.8 will provide some examples for the expansion by orthogonal

and complete systems of functions. Here, we summarize the most important

formulas for Fourier series and Fourier integrals.

a) Fourier Series

A periodic function f(x) may be represented as a Fourier series. If c is the period,

then the series is represented by

(3.113)

It describes the value of f(x) everywhere, except where f is discontinuous. Where f

is discontinuous the series takes the value

(3.114)

That is, it assumes the average of the left and right sided limit at that point.

4πε

–()

---------------------------------------------------------------------------------------------=

–()

--------------------------------------------------------------------------------

---

kx x

–()[]cos ly y

–()[]k

+ zz

––[]dkdlexpcos

-----------------------------------------------------------------------------------------------------------------------------------------

∞

∫

∞

∫

fx() a

2πn

----------





cos

n 0=

∞

∑

2πn

----------





sin

n 1=

∞

∑

2πn

----------





cos

n 0=

∞

∑

2πn

----------





sin

n 1=

∞

∑

fx ε–()fx ε+()+

-------------------------------------------

ε 0→

lim=

156 Formal Methods of Electrostatics

The coefficients and of the expansion (3.113) can be found by means

of an orthogonality relation for trigonometric functions.

The integration is carried out over a full period, but does not require to span from 0

to c, it could be any x

to x

+c. Multiply the expansion Ansatz (3.113) by

and , respectively, then integrate the resulting

equation over a full period while making use of (3.115) and (3.116). This gives

(3.117)

(3.118)

In particular, when f(x) is an “even” or “symmetric” function [ ]

then by (3.118) all , that is, the result is a pure “cosine series”. Conversely

when f(x) is an “odd” or “anti-symmetric” function [ ] then by

(3.118) all , that is, the result is a pure “sine series”.

Another option is to expand f(x) as a complex Fourier series

(3.121)

Because of

(3.115)

(3.116)

(3.119)

(3.120)

2πn

----------





cos

2πm

-----------





cos xd

∫

---

for n or m 1≥

c for nm0==











2πn

----------





sin

2πm

-----------





sin xd

∫

---

for n or m 1≥

c for nm0==











2πnx c⁄()cos 2πnx c⁄()sin

---

fx()

2πn

----------





cos xd

∫

for n 1≥

---

fx()xd

∫

for n 0=











---

fx()

2πn

----------





sin xd

∫

= for n 1≥

x() fx–()=

x() fx–()–=

fx() a

2πn

----------





cos

n 0=

∞

∑

= fx() even[]

fx() b

2πn

----------





sin

n 1=

∞

∑

= fx() odd[]

fx() d

2πn

----------





exp

n ∞–=

+∞

∑

3.5 Separation of Laplace’s Equation in Cartesian Coordinates 157

we may also write

This establishes the relationship between the expansions (3.113) and (3.121).

(3.122)

or the inverse relation

(3.123)

To immediately calculate the coefficients use Ansatz (3.121) and apply the

orthogonality relation

(3.124)

which yields

(3.125)

b) Fourier Integrals

Under certain, very general conditions which we will not discuss here, a function

may be represented as a Fourier integral:

(3.126)

We have introduced an arbitrary factor C because the Fourier integral is defined

differently by different authors. The inverse is the Fourier transform of f(x)

(3.127)

This is a result of (3.126) and the orthogonality relation

iα()exp αcos i αsin+=

fx() d

2πn

----------





cos id

2πn

----------sin x+

n ∞–=

+∞

∑

n–

+()

2πn

----------





cos id

n–

–()

2πn

----------sin x+

n 1=

∞

∑

n–

+ a

n–

–()b







n 1≥











n–

-------------------=

–

------------------- d

n–











n 1≥











2πn

----------





exp i

2πm

-----------

– x





exp xd

∫

cδ

---

fx() i

2πn

----------

– x





exp xd

∫

fx() Cf

k() ikx()exp kd

∞–

+∞

∫

k()

2πC

-----------

fx() ikx–()exp xd

∞–

+∞

∫

158 Formal Methods of Electrostatics

(3.128)

Multiplying (3.126) by and then integrate over x from through

just yields (3.127), when also considering (3.128).

When f(x) = f(-x) i.e., f(x) is an even function, then

that is

(3.129)

and

(3.130)

On the other hand, when f(x) = -f(-x) i.e., f(x) is an odd function, then

that is

(3.131)

and

(3.132)

Thus there are basically three Fourier integrals

1. the exponential Fourier integral (3.126) and its inverse (3.127)

2. the cosine Fourier integral (3.130) and its inverse (3.129)

3. the sine Fourier integral (3.132) and its inverse (3.131)

To directly derive eqs. (3.129) and (3.130), or (3.131) and (3.132) is possible when

using the orthogonality relations

(3.133)

(3.134)

The choice of the factor C is a mere matter of convenience. Frequently, it is set to

in (3.126), which causes a factor of in (3.127). Another frequent

choice is which makes (3.126) and (3.127) “symmetric”, that is, the

same factor occurs in (3.127). Similarly, the sine and cosine transforms are

by no means uniquely defined. Frequently the definition of yields the

factor in (3.130) to become one, or when defining , the factor in (3.132)

becomes one.

ikx()exp ik'– x()exp xd

∞–

+∞

∫

2πδ kk'–() =

ik'– x()exp ∞–

+∞

k()

2πC

-----------

fx() kx()cos xd

∞–

+∞

∫

k()

πC

-------

fx() kx()cos x d

+∞

∫

fx() 2Cf

k() kx()cos k d

+∞

∫

k()

2πC

-----------– fx() kx()sin xd

∞–

+∞

∫

k()

πC

-------

fx() kx()sin x d

+∞

∫

–=

fx() 2iC f

k() kx()sin k d

+∞

∫

kx()cos k'x()cos xd

+∞

∫

---

δ kk'–()

---

δ kk'+() +=

kx()sin k'x()sin xd

+∞

∫

---

δ kk'–()

---

δ kk'+() –=

C 1=12π⁄

C 12π⁄=

12π⁄

C 12⁄=

Ci–2⁄=

3.6 Complete Orthogonal Systems of Functions 159

As a practical matter, because of the different definitions, it is advisable not to

define this factor too early in the calculations. Errors can be excluded when

selecting the best fitting approach (while leaving the factor open), and then

calculate the coefficients by means of the orthogonality relation.

3.6 Complete Orthogonal Systems of Functions

Before starting Separation of Variables for Laplace’s equation in cylindrical and

spherical coordinates, we will generalize the terminology, derived previously from

rather specific examples. Furthermore, their rather hidden analogy to vector

calculus shall be illustrated. Solving boundary value problems, under suitable

conditions, leads to systems of functions which are orthogonal to each other and

are complete in the sense that all possible functions, defined in the given region can

be constructed from them by superposition. Our analysis be in one dimension, as

multiple ones will not change the principle nature of the relations which we are

interested in.

A function f(x), defined in an interval , can be thought of as a vector.

So to speak, x acts as continuous index which identifies the components of the

vector f(x). An integral

(3.135)

can be regarded as the scalar product of the two vectors f(x) and g(x). f* is the

conjugate complex function to f. Oftentimes we deal with real functions, in which

case it is of course

In Vector Calculus one introduces a number of basis vectors, depending on the

spatial dimensions, to construct every vector

(3.136)

The system of basis vectors is orthogonal and normalized if

(3.137)

The expansion of the vector a by its basis system e

is achieved by scalar

multiplication of eq. (3.136) by e

. This yields

i.e.,

(3.138)

and thus

cxd≤≤

x()gx()xd

∫

f g 〈〉=

x()gx()xd

∫

f g 〈〉 fx()gx()xd

∫

a a

i 1=

∑

•δ

• a

•

i 1=

∑

i 1=

∑

===

•=

160 Formal Methods of Electrostatics

(3.139)

It is worthwhile to analyze the parts of this sum.

is the projection of a onto the direction given by e

. One may think of this

undetermined product e

(frequently called dyadic product) to be an operator

which transforms a into another vector, namely the vector that results form

projecting it on e

. Thus e

is also called projection operator. The resulting vector

(3.140)

Of course, the sum of all projections has to yield the vector itself, if the system of

basis vectors is complete. In this case, the sum of all basis vectors has to result in

the identity operator (identity tensor):

(3.141)

This represents the completeness relation. Eqs. (3.139) and (3.141) are equivalent

and both express completeness of the basis system. The dyadic product is an

operator that has the form of a matrix when writing it in component form. The

dyadic product ab has the components (matrix elements)

The unit vectors of, for example, a three dimensional Cartesian coordinate

system are

The three projection operators result thereof

Their sum is indeed the identity operator

Applied to a vector

aae

•()e

i 1=

∑

()a•

i 1=

∑

()a•

• e

a•== =

i 1=

∑

ab()

()=

100,,〈〉=

010,,〈〉=

001,,〈〉=

100

000

010

000

001

i 1=

∑

100

010

001

() 1===

a a

,,〈〉=

3.6 Complete Orthogonal Systems of Functions 161

gives

which is the respective projection.

All this can be applied to functions treated as vectors of an infinite

dimensional space. We first build f(x) from a complete system of orthogonal

functions. There are two cases, depending on whether the eigenvalues are discrete

or continuous (or expressed in different words: whether they form a discrete or a

continuous spectrum). Examples for both were given in Section 3.5.

We have therefore the following expansions for f(x):

For the continuous case, the integration has to be over all possible values of k. The

expansions are based on the orthogonality relations for the basis functions which

are assumed to be normalized.

where

Discrete spectrum Continuous spectrum

(3.142)

(3.143)

(3.144)

a•

100

000

00,,〈〉==

fx() a

x()

n 1=

∞

∑

= fx() ak()ϕkx;()kd

∫

x()ϕ

x()xd

∫

= ϕ kx;()ϕ

kx;()xd

∫

δ kk'–()=

fx()ϕ

x()xd

∫

= fx()ϕ

kx;()xd

∫

x()ϕ

x()xd

∫

n 1=

∞

∑

n 1=

∞

∑

ak()ϕkx;()ϕ

k' x;()xdkd

∫∫

ak()δkk'–()kd

∫

ak'()==

fx()ϕ

x()xd

∫

x() fx()〈〉=

ak() fx()ϕ

kx;()xd

∫

ϕ kx;() fx()〈〉=

162 Formal Methods of Electrostatics

This determines the coefficients of the expansion, the components of the vector so

to speak, where relation (3.144) is entirely equivalent to (3.138). The projection

onto the “direction” of and , respectively is then

For completeness, the sum of all those projections has to result in the function

itself, i.e.,

At the same time, it also is

By comparison, we obtain the completeness relations

On the other hand, to calculate the expansion coefficients by means of the

completeness relation is also possible. First, one may apply the identity operator,

that is in our case basically the δ-function, to any function. So we start with

Substituting gives

what immediately leads to the coefficients of the expansion

Let us now consider two functions f(x) and g(x) whose scalar product we aim to

find. f(x) shall have the components

(3.145)

x() ϕkx;()

()

x()ϕ

x'()fx'()x'd

∫

x()ϕ

x'()fx'()〈〉=

k()

()

ϕ kx;()ϕ

kx';()fx'()x'd

∫

ϕ kx;()ϕkx';()fx'()〈〉=

fx() a

x()

n 1=

∞

∑

x()ϕ

x'()fx'()x'd

n 1=

∞

∑

∫

fx() ak()ϕkx;()kd

∫

ϕ kx;()ϕ

kx';()dk f x'()x' .d⋅

∫∫

x() δxx'–()fx'()x'd

∫

x()ϕ

x'()

n 1=

∞

∑

δ xx'–()=

ϕ kx;()ϕ

kx';()kd

∫

δ xx'–()=

x() δxx'–()fx'()x'd

∫

fx() ϕ

x()ϕ

x'()

n 1=

∞

∑

fx'()x'd

∫

fx() ϕkx;()ϕ

kx';()kd

∫

fx'()x'd

∫

x'()fx'()x'd

∫

= ak() ϕ

kx';()fx'()x'd

∫

ak()

3.6 Complete Orthogonal Systems of Functions 163

while the components of g(x) shall be

Then

Therefore

which is in complete analogy to Vector Calculus, especially for the case of the

discrete spectrum.

In closing this section we shall provide a few explanatory words on integral

operators. We have already met some of those integral operators when studying the

various Green’s functions in the previous Section 3.5. They cause a transformation

of a function into another function in the form of some integral

transform:

(3.147)

The δ-function is just one such integral operator. It projects a function onto itself.

The image function is then equal to

(3.146)

bk()

fx() a

x()

n 1=

∞

∑

fx() ak()ϕkx;()kd

∫

gx() b

x()

n 1=

∞

∑

gx() bk()ϕkx;()kd

∫

gf 〈〉 fx()g

x()xd

∫

gf 〈〉 fx()g

x()xd

∫

x()ϕ

x()xd

∫

nm, 1=

∞

∑

ak()b

k'()ϕkx;()ϕ

k' x;()xdkddk'

∫∫∫

n 1=

∞

∑

ak()b

k'()δkk'–()kddk'

∫∫

n 1=

∞

∑

ak()b

k()kd

∫

fx() f

x()

x() Oxx',()fx'()x'd

∫

x()

x() δxx'–()fx'()x'd

∫

fx()==