Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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224 Formal Methods of Electrostatics

that is, the equation for a circle whose center is at the y-axis. Both sets of curves

consist of circles (Fig. 3.37). The thereby described field has many applications.

By appropriate choice of the parameters, one can obtain for example, the field of an

eccentric cylindrical capacitor (Fig. 3.38a) or that of a conductor made from two

cylinders (Fig. 3.38b). This resembles the previously discussed problem of

imaging a line charge on a cylinder (Sect. 2.6.3).

Example 4

Fig. 3.37

-Q

Fig. 3.38

wz()

2πε

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

3.12 The Complex Potential 225

If we start from Example 3 and take the limit to a line dipole ( ,

) then one finds

This is the complex potential of a line dipole.

From this one finds the equipotentials to be circles through the origin (centered on

the x-axis) and the flux lines are also circles through the origin (centered on y-axis)

as illustrated in Fig. 3.39, (which may also be regarded as being a result of taking

the limit of Fig. 3.37).

Example 5

i.e.,

d 0 q ∞→,→

qd finite

wz()

2πε

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

-----+

-----–

---------------ln

2πε

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

-----+





-----+





ln≈=

2πε

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

---+





2πε

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

.≈≈

wz()

2πε

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

qdz

2πε

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

qd x iy–()

2πε

+()

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

uxy,()

2πε

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

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

vxy,()

2πε

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

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

–=

Fig. 3.39

wCz

Cx iy+()

Cr iϕ()exp[]

ipϕ()exp== = =

uCr

pϕ()cos=

226 Formal Methods of Electrostatics

The lines , i.e.,

characterize special equipotential surfaces ( ), for example, for and

, this gives

The lines , i.e.,

are special flux lines ( ), for instance

When using surfaces of conductors, which are a possible implementation of

equipotential surfaces, one obtains fields like those illustrated in Fig. 3.40 (p > 1)

and Fig. 3.41 (p < 1).

For the case of Fig. 3.41 (p < 1), the derivative

diverges at the origin. Therefore, the electric field becomes infinitely large. This is

a typical characteristic of tips, and is also of great practical significance. Figs. 3.30

and 3.31 are special cases where p=1/2 and p=2, respectively.

vCr

pϕ()sin=

pϕcos 0=

2n 1–

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

π=

u 0= n 0=

n 1=

------

±=

pϕsin 0=

ϕ nπ=

v 0=

ϕ 0=

Cpz

p 1–

1 p–

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

p<1

ϕ=π/2p

ϕ= −π/2p

Fig. 3.41

Fig. 3.40

π/p

p>1

ϕ=π/2p

ϕ= −π/2p

3.12 The Complex Potential 227

Example 6

We may stage conformal mappings. For instance, we want to calculate the field of a

line charge inside a wedge-shaped area, like that of Fig. 3.40. As long as is an

even number multiple of the opening angle, we are able to solve this problem by

multiple imaging steps, as indicated for the for point charges in Figs. 2.34 and 2.35,

where the opening angle is and .

The imaging method can not be used for arbitrary angles. However, one may

approach this problem as follows. First, by writing

one obtains the field of two line charges in the -plane, which differ from

Example 3 only by a shift parallel to the -axis (Fig. 3.42), where the -axis has

to be seen as the equipotential surface (conductor surface) and -q is the image

charge. Now applying the mapping of Example 5 already solves the problem

(Fig. 3.43):

and

wz()

2πε

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

–

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







ln=

2π

ππ2⁄

2πε

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

ζζ

–

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







ln=

Fig. 3.42

– ξ

– iη

iη

-q

ζ z

228 Formal Methods of Electrostatics

where

is the location of the line charge in Fig. 3.42 and its location in Fig. 3.43.

Example 7

i.e.,

so that

Fig. 3.43

wz()

2πε

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

–

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







ln=

------cosh=

------cosh uiv+()cosh

---

uiv+()exp u– iv–()exp+[]==

---

u()exp vcos ivsin+[]u–()exp vcos ivsin–[]+{}=

vcos ucosh iv u ,sinhsin+=

zxiy+ avcos ucosh ia v usinhsin+==

xavcos ucosh=

yav u ,sinhsin=

vcos

vsin

ucosh

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

usinh

----------------------+==

3.12 The Complex Potential 229

and

This shows us that the equipotentials ( ) are elliptical cylinders and the

flux lines ( ) are hyperbolic cylinders. They are all confocal because for

an ellipse, we have the relation

and the relation for a hyperbola is

Their graph is given in Fig. 3.44. Special cases are, for example, the field between

two opposite edges (Fig. 3.45), or the field of an edge opposite a plane (Fig. 3.46).

Furthermore, the mapping

is remarkable because u and v, together with z constitute an orthogonal system in

the three-dimensional space, which allows for the separation of variables for

Laplace’s equation (coordinates of the elliptical cylinder).

Example 8

This rather simple looking mapping exhibits peculiar properties and shall serve

here as an example for many of similar kind.

Squaring gives

i.e.,

and

ucosh

usinh

–1

vcos

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

vsin

-------------------–==

u const=

v const=

ucosh

usinh

– a

vcos

vsin

+ a

Fig. 3.44

-a

Fig. 3.46

Fig. 3.45

-a

uuxy,()=

vvxy,()=

wz() z

– i2uv+ z

+ x

– B

i2xy++===

– x

– B

uv xy=

230 Formal Methods of Electrostatics

If (u-axis) then

Thus, it has to be either or . The case yields

and the case yields

which is always the case. This now, results in a peculiar behavior of the u-axis in

the z-plane. The range , (i.e., that part of the negative u-axis), maps to

the negative x-axis. Similarly, for the range , (i.e., that part of the

positive u-axis), maps to the positive x-axis ( ). Both points

map onto the origin of the z-plane. The section of the u-axis between -B and +B

maps onto the points between -B and +B on the y-axis. For , the image

points on the y-axis move from the origin towards the points +B and -B. For

, the image points on the y-axis move back towards the origin

Conversely, we find for (v-axis)

Here, y = 0 is impossible because this would require

which is not possible since v has to be real. Hence, and

i.e.,

This means either

v 0=

0 u

≤ x

– B

xy 0=







x 0= y 0= x 0=

0 y

– B

+≤

≤ x 0=()

y 0=

0 x

+≤ y 0=()

∞– uB–≤<

Bu≤ +∞<

0 x +∞<≤ uB±=

Bu0≤≤–

0 u +B≤≤

Fig. 3.47

-B

u=-B

u<-B

u>+B

u=+B

u=0

-B

u 0=

–≥ x

– B

xy 0=







– x

x 0=

– y

– B

+ B

≥=

yB>

3.12 The Complex Potential 231

Now, the v-axis maps onto the part of the y-axis that lacks the range from -B to +B.

The current conformal mapping is an example of an interesting category of

mappings, the Schwarz-Christoffel mappings.

Fig. 3.48 illustrates this configuration, which is suitable to determine the

fields for a more or less sharp edge (Fig. 3.49).

yB–<

Fig. 3.48

v=v

u=u

Fig. 3.49

4 The Stationary Current Density Field

The following shall discuss the field of the stationary electric currents, or more

precisely, the field generated by the current density g. By certain simplifying

assumptions, we may reduce the current density problems to electrostatic problems

of Chapter 2 and 3.

4.1 The Basic Equations

For electric currents inside metallic conductors to exist, Ohm’s law requires that

there be an electric field present, such that

(4.1)

where is the specific electric conductivity. It shall be emphasized that this simple

form of Ohm’s law is not always valid. Oftentimes, it has to be replaced by much

more complicated relations. Even if eq. (4.1) is applicable in the given form,

may still depend on the location, i.e., for some heterogeneous material, or even in a

homogeneous material, if a non-uniform magnetic field is applied, which is a

property that is frequently exploited to measure magnetic fields. This shall not be

our concern here, however. We assume that is at least piece wise constant.

We have already derived the continuity equation, or the principle of charge

conservation (1.58):

(4.2)

which reduces in the stationary case to

(4.3)

From (4.1) and (4.2) follows

Letting be constant and using

gives

(4.4)

For an arbitrary volume, this results in

(4.5)

g κE=

g∇•

t∂

∂ρ

+0=

g∇• 0=

κE()∇• 0=

E ϕ∇–=

κ E∇• κ ϕ∇()∇•–0==

∇

ϕ 0=

g∇• τd

∫

g dA•

∫

0==

G. Lehner, Electromagnetic Field Theory for Engineers and Physicists,

4.1 The Basic Equations 233

This means that for the stationary case, the sum of all currents emerging from the

volume have to vanish. A special case of this is Kirchhoff’s first theorem (Fig. 4.1).

(4.6)

This is a fundamental relation in electric circuits theory.

The question whether stationary currents actually exist arises. Based on our

previous knowledge, one might get the impression that they did not. To find the

answer, consider a charged capacitor within which one introduces a conductor

(Fig. 4.2). Initially, there is a field inside the conductor that causes currents. This

represents moving charges, which will cease moving only, when the field inside the

conductor has vanished, that is, when influence charges appear. The current in the

conductor decays. The time for this to occur is the relaxation time and depends on

the conductivity (see Sect. 4.2). Similar considerations lead to decay of currents in

every “electrostatic” situation. Since

(4.7)

and therefore

(4.8)

In a stationary, closed circuit, this can only be satisfied if

Fig. 4.1

i 1=

∑

Fig. 4.2

E∇× 0=

E ds•

∫

g ds•

∫