Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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134 Formal Methods of Electrostatics

The -function is an idealization and useful mathematical tool. Nature does

not have -functions. It helps provide use with a formal analogue to the point

charge, which is also an idealization. A point charge may formally be described by

a charge density , which vanishes everywhere but at one particular location,

where it becomes infinitely large. We will generalize the one-dimensional

-function for this purpose.

(3.51)

This allows to describe a point charge Q at the location r’ by

Integrating over the entire space gives

as we had expected.

An important property of the -function that follows from above discussion

is:

(3.52)

Thus, the -function has the defining property of filtering a particular value from a

smooth function when integrated.

One may write as well

Similarly for a function , defined in the entire space

(3.53)

which results from multiple applications of (3.52).

The -function is symmetric

(3.54)

It is possible to differentiate it as well as integrate it. Its indefinite integral is

Heaviside’s step function.

(3.55)

Indeed,

δ xx'–() f

x()

a 0→

lim=

δ rr'–()δxx'–()δyy'–()δzz'–()=

ρ r() Qδ rr'–()=

ρ r()τd

∫

Qδ xx'–()δyy'–()δzz'–()xdydzd

∫

Q δ xx'–()xd δ yy'–()

∫

yd δ zz'–()

∫

Q 111⋅⋅⋅ Q ,==

fx()δxx'–()xd

∞–

+∞

∫

fx'()=

fx()δxx'–()xd

∞–

∞

∫

fx'()δxx'–()xd

∞–

∞

∫

fx'() δxx'–()xd

∞–

+∞

∫

fx'()1 .⋅==

f r()

f r()δrr'–()xd

entire

space

∫

f r'()=

δ xx'–()δxx'–[]–()δx' x–()==

Hx x'–()

0forxx'<

1forxx'>







135

conversely

i.e., differentiating the step function (which is not possible with ordinary functions)

results in the -function.

To avoid confusion about the dimensions, we note that by eq. (3.50), the

-function carries a dimension, namely the inverse of its argument, i.e., . When

the argument of the δ-function is a vector, then based on the definition of (3.51) its

dimensions are for example, for the three-dimensional case.

3.4.6 Point Charge and δ-Function

Poisson’s equation applies to the case of a point charge.

where

i.e.,

We know its solution already

i.e., we may now write for all locations, including the location of the point charge,

(which was previously impossible)

(3.56)

In Section 2.3, we had to exclude the locations of the point charges, because we

were unable to differentiate there. With the use of the -function, those difficulties

or restrictions are now removed.

δ x'' x'–()x''d

∞–

∫

0 for xx'<

1 for xx'>







δ x'' x'–()x''d

∞–

∫

Hx x'–()=

dH x x'–()

------------------------- δ xx'–()=

δ x

1–

3–

∇

-----–=

ρ r() Qδ rr'–()=

∇

Qδ rr'–()

-------------------------–=

4πε

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

rr'–

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

⋅–=

∇

4πε

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

rr'–

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

⋅

-----

δ rr'–()⋅–=

∇

rr'–

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

4πδ rr'–()–=

3.4 Some Properties of Poisson’s and Laplace’s Equations

136 Formal Methods of Electrostatics

The potential for an arbitrary charge distribution we previously found

by superposition to be

It is worthwhile to also formally prove this result. Applying the Laplace operator to

this equation gives

this shows that the expression for fulfills Poisson’s equation, proving its validity.

For completeness, we note that there exists a more general (nevertheless, not

the most general) solution for

which is given by

Only the boundary condition at infinity makes the solution unique and

forces the constant to vanish.

3.4.7 Potential in a Bounded Region

When considering the entire space with all its charges, then is given by the usual

integral

In contrast, if we just consider a finite space and only those charges inside this

region, then Green’s theorem (3.47) allows certain statements about this situation.

For this purpose we use (3.47) and substitute

where

and

ρ r()

ϕ r()

4πε

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

ρ r'()

rr'–

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

τ'd

∫

⋅=

∇

ϕ r()

4πε

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

∇

ρ r'()

rr'–

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

τ'd

∫

⋅=

4πε

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

ρ r'()∇

rr'–

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

τ'd

∫

⋅=

4πε

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

ρ r'() 4π–()δrr'–()τ'd

∫

⋅=

ρ r()

-----------– ,=

∇

Qδ rr'–()

-------------------------–=

4πε

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

rr'–

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

⋅– const+=

ϕ 0=

ϕ r()

4πε

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

ρ r'()

rr'–

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

τ'd

∫

⋅=

φϕ= ∇

ρ r()

-----------–=

137

This gives

This result assumes that the point r’ is inside the respective volume. If that point is

at its surface, then the factor needs to be replaced (in case of a “smooth

surface” as in Section 8.2.1) by the factor , and if that point is outside the

volume, then the factor becomes 0. Exchanging r and r’ gives

(3.57)

This result is peculiar and is sometimes called Kirchhoff theorem or Green’s

formula. It states that , besides its usual term

which results from charges inside a volume, gets additional contributions from the

boundary. Evidently, these represent charges that may possibly be located outside

the volume. These formally derived contributions also have a plausible

explanation.

1. The term

can be regarded as the potential of a distribution of surface charges (see

(2.26)).

with

2. The term

rr'–

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

rr'–

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

∇

ϕϕ∇

rr'–

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

–





τd

∫

ρ r()–

rr'–

-------------------- ϕ r()4πδ rr'–()+ τd

∫

ρ r()

rr'–

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

τd

∫

–4πϕ r'()+=

rr'–

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

n∂

∂ϕ

n∂

∂ 1

rr'–

---------------–





∫

4π

2π

ϕ r()

4πε

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

ρ r'()τ'd

rr'–

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

∫

4π

------

n'∂

∂

ϕ r'()

rr'–

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

∫

dA'

4π

------

ϕ r'()

n'∂

∂ 1

rr'–

---------------dA'

∫

–+=

4πε

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

ρ r'()τ'd

rr'–

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

∫

4π

------

n'∂

∂

ϕ r'()

rr'–

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

∫

dA'

4πε

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

σ r'()

rr'–

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

A'd

∫

σε

n∂

∂ϕ

3.4 Some Properties of Poisson’s and Laplace’s Equations

138 Formal Methods of Electrostatics

can be regarded as the potential of a dipole layer (see (2.69)),

with

Thus, by choosing the proper combination of dipole layer and surface charges, one

can simulate the effects that all possible charges outside the volume have on the

inside. This result then does not describe the field outside the region of

consideration. On the contrary, the dipole layer causes the boundary to have a

potential , while the surface charge causes the boundary to exhibit

. When we label the quantities at the inside with the index i and

those at the outside with the index a, then

and because of

follows the claim that

Furthermore, it is

Because of

follows our claim that

The surface charge and the dipole layer cause discontinuities of and

, respectively, which is of paramount importance in field theory. This point will

be discussed in more detail (Section 8.2.1 and 8.2.2).

The quantities mentioned are closely related to the method of image charges

which we have already discussed (Section 2.6.1). Our previous discussion used the

reverse arguments, where the effects of surface charges were replaced by

appropriate image charges.

In closing, we add a note of caution with regards to a potential

misinterpretation of eq. (3.57). It does by no means allow, when inside the

4π

------

ϕ r'()

n'∂

∂ 1

rr'–

---------------dA'

∫

–

4πε

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

n'∂

∂ 1

rr'–

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

dA'

∫

–

τε

ϕ=

∂ϕ ∂n⁄()

– σε

n∂

∂ϕ





– ε

n∂

∂ϕ





+==

σε

n∂

∂ϕ





n∂

∂ϕ





–

-----=

τε

–=

∂ϕ ∂n⁄

ρ r()

3.5 Separation of Laplace’s Equation in Cartesian Coordinates 139

region was already given, to arbitrarily choose and on the surface, and

from there calculate inside by (3.57). If we were to independently prescribe

and on the surface, then the problem would be overdetermined. To specify

one of the two quantities, already makes the problem unique This means that

eq. (3.57) merely expresses that is of this general form if the values for and

are compatible. Nevertheless, it is possible to use this relation as starting

point for a solution, if by means of suitable Green functions, either one or the other

of these two terms is eliminated. We will omit the details of this discussion, but

will return to apply Green functions for concrete situations. Eq. (3.57) represents

an important basis for analytical and numerical methods to solve boundary value

problems, in particular the boundary element method (Sections 8.2 and 8.8).

Eq. (3.57) is also related to the Helmholtz theorem. We will come back to this

in Appendix A.5, where the significance of these two surface integrals will become

even more apparent.

3.5 Separation of Laplace’s Equation in Cartesian

Coordinates

3.5.1 Separation of Variables

In the following few sections, we will solve Laplace’s equation by separation of

variables for a number of different coordinate systems. We demonstrate the

methodology by applying it to the simplest example, that is, to the Cartesian

coordinate system. The equation to solve is

(3.58)

Writing as

(3.59)

and inserting this in (3.58), gives

Divide this equation by to obtain

(3.60)

It is important to observe that the first term depends only on x, the second only on

y, and the third only on z. The sum vanishes. This is possible only, if each of the

three terms is constant. This enables one to write:

ϕ∂ϕ∂n⁄

ϕϕ

∂ϕ ∂n⁄

ϕϕ

∂ϕ ∂n⁄

∇

∂





ϕ 0==

ϕ Xx()Yy()Zz()=

∂

Xx() XZ

∂

Yy() XY

∂

Zz()++ 0=

ϕ XYZ=

---

∂

Xx()

---

∂

Yy()

---

∂

Zz()++ 0=

140 Formal Methods of Electrostatics

(3.61)

This introduces two arbitrary constants, the so-called constants of separation.

Slightly rewriting this yields

(3.62)

These are ordinary differential equations. Their general solutions are:

(3.63)

This gives

(3.64)

The separation constants k and l may be chosen at will. The resulting solution for

is thus only a very specific one. The general solution is, however, obtained by

the superposition of all possible solutions (i.e., using all possible values of k and l).

We can choose other functions, i.e., instead of cos and sin, we could have chosen

exp(ikx) and exp(-ikx), or exp(kx) and exp(-kx), respectively, which resulted in

writing X, Y, Z in a form different from above:

(3.65)

This does not actually introduce anything new, because of

---

∂

Xx() k

–=

---

∂

Yy() l

–=

---

∂

Zz() k











∂

Xx() k

Xx()–=

∂

Yy() l

Yy()–=

∂

Zz() k

+()Zz()=











XA kxcos⋅ Bksin x⋅+=

YC lycos⋅ Dlsin y⋅+=

ZE k

+ zcosh⋅ Fk

+sinh z⋅+=











ϕ Akxcos⋅ Bksin x⋅+()Clycos⋅ Dlsin y⋅+()

+ zcosh⋅ Fk

+sinh z⋅+()

XA ikx()exp⋅ Bikx–()exp⋅+=

Y C ily()exp⋅ D ily–()exp⋅+=

ZE k

+ z()exp⋅ Fk

+–()z()exp⋅+=











3.5 Separation of Laplace’s Equation in Cartesian Coordinates 141

(3.66)

Whether one finds by superposing functions of type (3.63) or (3.65) is of lesser

importance. For a specific case, however, there may be reasons to select one over

the other. Depending on the type of problem, one needs to use functions with all

values of k and l, or just specific values of k and l. This will be clarified when some

specific examples are considered.

In very simple cases, one would find a quicker solution and not use the

method of separation of variables. Let us consider the problem of finding the

potential between two, infinite, parallel plates whose potentials are given constants

(Fig. 3.9).

One may proceed as follows: First, observe that can only depend on x. Therefore

with the general solution

The integration constants A and B are determined by

i.e.,

and

ikx±()exp kxcos iksin x±=

kx±()exp kzcosh ksinh z±=







Fig. 3.9

x=0 x=d

∇

∂

∂x

--------- 0==

ϕ ABx+=

AB0⋅+=

ABd⋅+=

A ϕ

–

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

ϕϕ

–

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

x+=

142 Formal Methods of Electrostatics

which already solves this simple problem. Of course the formal route using

separation is open as well, i.e., using the Ansatz

The reason why k has to approach zero is, that because of the independence of y

and z, both l, as well as k

, have to vanish. This results in

when substituting and . Note that even for , B may be

finite, as may assume any, even infinitely large values.

We will apply this method of separation of variables to other coordinate

systems as well. It shall be noted, however, that the method of separation of

variables is not generally applicable, but rather is a specific property of certain

orthogonal coordinate systems which allow for the separation of certain equations.

Besides Cartesian, cylindrical, and spherical coordinates, there are 8 more, for a

total of 11 coordinate systems which permit separation of the three-dimensional

Laplace equation and the Helmholtz equation, yet to be introduced. Besides these,

there are arbitrarily many coordinates systems that permit separation of the two-

dimensional or plane Laplace equation. Finally, there is the possibility to expand

the meaning of the term separability (now R separability, to distinguish it from

simple separability), which then allows for the separation of Laplace’s equation in

a few more coordinate systems. A very useful summary of all these problems is

provided by [2].

3.5.2 Examples

3.5.2.1 Dirichlet Boundary Value Problem without Charges Inside

The problem is to find the electric potential inside a cuboid whose sides are of

lengths a, b, c (in x, y, z direction, as shown in Fig. 3.10). The boundary conditions

are

x∂

∂ϕ

–

------------------–==

ϕ A

kxcos⋅ B

ksin x⋅+= k 0→

ϕ A

kx+ ABx+==

= BB

k= k 0→

Fig. 3.10

3.5 Separation of Laplace’s Equation in Cartesian Coordinates 143

on all faces except on one (for example, the top one where

on this one face for .

There shall be no charges inside the volume. For the solution, according to

eq. (3.63), we make the Ansatz for the x-dependency

Since for , it must be and , which leads to

(3.67)

Since also for

Consequently

(3.68)

and

where n is a whole number. This means that the boundary conditions force k to take

on very specific values:

(3.69)

Those values are often called eigenvalues of the problem. Putting everything

together, gives

(3.70)

In similar manner, we find

(3.71)

To find the z-dependency, we start with the Ansatz

for so that and therefore

(3.72)

ϕ 0=

zc=

ϕϕ

xy,()= zc=

kxcos B

ksin x+= for k 0≠

x+= for k 0=

ϕ 0= x 0= A

0= A

ksin x= for k 0≠

x= for k 0=

ϕ 0= xa=

ksin x 0=

x 0=

ka nπ,= k

nπ

------=

nπ

------==

nπx

---------sin=,n 123…,,,=

mπx

-----------sin=, m 123…,,,=

ZE k

+ zcosh⋅ Fk

+sinh z⋅+=

ϕ 0= z 0= E 0=

nπ

------





mπ

-------





+ zsinh=