Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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184 Formal Methods of Electrostatics

(3.235)

By means of the relations of Section 3.7.2, we obtain

(3.236)

Multiply (3.235) with and , respectively. Then

integrate over z and ϕ, using the orthogonality relation to obtain

(3.237)

(3.238)

. (3.239)

Now we substitute for z, that is, the charge is located at . With

these coefficients the result is

(3.240)

We obtain for the inverse distance between the points and

(3.241)

This formula is more general than the previous result of eq. (3.204). which may be

obtained in the limit and . The rational behind this is that

and for , that is, . Now, only the bottom

expression is of interest since .

A different approach is also possible. Consider the plane carrying the

point charge Q at , which is represented by the surface charge density

(3.242)

The Ansatz for the potential is now

kkz()cos I

' kr

()K

()I

()K

' kr

()–[]

∞

∫

m 0=

∞

∑

k() mϕcos g

k() msin ϕ()+[]dk⋅

----

δ z()δϕ ϕ

–()=











' kr

()K

()I

()K

' kr

()–

-------=

k'zm'ϕcoscos k'zm'sin ϕcos

k()

2π

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

k()

-----------

mϕ

cos ,= m 1≥

k()

-----------

msin ϕ

– r

,,()

ia,

2π

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

kr()K

()

()K

kr()







∞

∫

m 0=

∞

∑

kz z

–()[]cos m ϕϕ

–()[]cos 2 δ

–()k . d⋅

r r ϕ z,,()r

,,()

–

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

---

kr()K

()

()K

kr()







∞

∫

m 0=

∞

∑

kz z

–()[]cos m ϕϕ

–()[]cos 2 δ

–()k . d⋅

0= z

0() 1= I

0() 0= m 1≥ I

0() δ

≥ 0=

z 0=

0,,()

σ r ϕ,()

----

δ rr

–()δϕϕ

–()=

185

(3.243)

This Ansatz was particularly chosen so that at the boundary , F and its

tangential derivatives are continuous. Furthermore we have

(3.244)

One uses the same procedure again: multiply with and

, respectively, then integrate over r and ϕ, apply the

orthogonality relation, and finally obtain

(3.245)

(3.246)

. (3.247)

All that remains now for a charge located at point , is to replace z by

. Applying the just obtained coefficients gives

(3.248)

and the inverse distance is

(3.249)

These equations represent a generalization of the previously found eqs. (3.230) and

(3.231). One obtains these equations by setting , and making use of

the fact that .

We were able to be rather brief here, as the procedure is similar to that of

Sections (3.7.3.1) and (3.7.3.4). Dividing the potentials (3.240) and (3.248) by Q

yields the respective Green’s functions . If there are arbitrary surface

charges

(3.250)

kr()f

k() mϕcos g

k() msin ϕ()+[]kz–()exp kd

∞

∫

m 0=

∞

∑

z 0=

z 0>()E

z 0<()–[]

z 0=

2kJ

kr()

∞

∫

m 0=

∞

∑

k() mϕcos g

k() msin ϕ()+[]kd⋅

----------

δ rr

–()δϕϕ

–()=

k'r() m'ϕ()cos⋅

k'r() m'ϕ()sin⋅

k()

4πε

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

()=

k()

2πε

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

()mϕ

cos ,= m 1≥

k()

2πε

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

()msin ϕ

,,()

–

F r()

4πε

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

2 δ

–()J

kr()J

()

∞

∫

m 0=

∞

∑

kz z

––[]m ϕϕ

–()[]cosexp k d⋅

–

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

2 δ

–()J

kr()J

() kz z

––[]m ϕϕ

–()[]cosexp k.d

∞

∫

m 0=

∞

∑

0 z

, 0==

() J

0() δ

G rr

,()

σσz ϕ,() =

3.7 Separation of Variables of Laplace’s Equation

186 Formal Methods of Electrostatics

(3.251)

prescribed on the cylinder at or on the plane surface , then the

related potentials are

(3.252)

where those integrals have to be evaluated over the cylinder surface or the plane

where

(3.253)

(3.254)

Notice that the rotationally symmetric charge distributions of Sections 3.7.3.1 and

3.7.3.4 are contained in this solution as special cases. Another special case shall be

discussed here as an example. Suppose that on the surface of a cylinder at ,

there exists a surface charge

(3.255)

This allows one to calculate based on (3.240) to obtain

(3.256)

Since the charge distribution (3.255) is independent of , integration over

yields

(3.257)

and one needs to investigate the Bessel functions for vanishing arguments. Using

eqs. (3.178) through (3.180) gives

(3.258)

and

(3.259)

where C, although it becomes infinitely large, is still insignificant and may be

arbitrarily chosen. Finally, one obtains

σσr ϕ,() =

= zz

F r() G rr

,()σr

() A

∫

dϕ

σσ

nϕ cos=

ia,

r()

ia,

2π

∫

∞–

+∞

∫

+∞

∫

2π

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

kr()K

()

()K

kr()







m 0=

∞

∑

kz z

–()[]cos m ϕϕ

–()[]cos 2 δ

–()r

nϕ

cos⋅

kz z

–()[]cos z

∞–

+∞

∫

2πδ k()=

kr()K

()

k 0→

lim

------

----





()K

kr()

k 0→

lim

------

----















for n 0≠

kr()K

()

k 0→

lim Cr

ln–=

()K

kr()

k 0→

lim Crln–=











for n 0=

187

(3.260)

and

(3.261)

where has been chosen. For the current case, it is easier to derive this

result by not using the general formulas. The independence of z, makes it possible

to start from the differential equation

(3.262)

which results from eqs. (3.156) and (3.160) with . We can immediately write

its solutions

(3.263)

. (3.264)

Thus, for a cylinder of radius (carrying arbitrary surface charges) we obtain

(3.265)

. (3.266)

Again, the coefficients were chosen such that the potential and thereby the

tangential components of the electric field are continuous at the surface of the

cylinder . If the surface charge (3.255) is prescribed, then the following

must hold

(3.267)

Obviously, except for A

, all terms A

and B

vanish:

ia,

2nε

-----------

----





----















nϕ cos= for n 0≠

ia,

-----------

----ln











–= for n 0=

ln=

r∂

∂

r∂

∂

Rr() m

Rr()=

k 0=

RAr

------

+=for m 0≠

RABrln+= for m 0=

----





mϕ cos B

msin ϕ +()

m 1=

∞

∑

0+=

----





mϕ cos B

msin ϕ +()

m 1=

∞

∑

----ln+=

–()

r∂

∂F

r∂

∂F

–





-------

mϕ cos B

msin ϕ +()

------–

m 1=

∞

∑

------

nϕ cos=

3.7 Separation of Variables of Laplace’s Equation

188 Formal Methods of Electrostatics

(3.268)

, (3.269)

which just leads to the potentials provided above (3.260) and (3.261). Thus it

becomes now obvious that the direct calculation of the potentials is indeed simpler

than by using the general formulas. The reason for this is that here, the modified

Bessel functions converge onto the elementary solutions (3.263) for and

(3.264) for , which is obvious from eqs. (3.178) through (3.180). For

, we obtain the solutions we already know, i.e., the logarithmic potential

outside and a constant potential inside, where the reference point was chosen such

that the potential vanishes at the cylinder surface . Since,

(3.270)

thus

(3.271)

3.8 Separation of Variables for Laplace’s Equation in

Spherical Coordinates

3.8.1 Separation of Variables

By eq. (3.43), the potential equation for the potential F in spherical polar

coordinates is

(3.272)

We attempt to separate this equation via the product solution

(3.273)

Multiplying eq. (3.272) by

yields

(3.274)

The first two terms solely depend on r and

, the last solely on ϕ. This allows us to

separate the equations, for example, by substituting

2nε

-----------= for n 0≠

-----------–= for n 0=

m 0≠

m 0=

n 0=

2πr

-----------–

----ln

2πε

------------–

----ln==

-----

r∂

∂

r∂

∂ 1

θsin

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

θ∂

∂

θ∂

∂

sin

θsin

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

∂

∂ϕ

---------





F 0=

Frθϕ,,()Rr()D θ()φϕ()=

θsin

Rr()D θ()φϕ()

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

θsin

Rr()

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

r∂

∂

r∂

∂

Rr()

θsin

D θ()

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

θ∂

∂

θ∂

∂

D θ()sin

φϕ()

-----------

∂

∂ϕ

---------

φϕ()++0=

189

(3.275)

with its general solution

(3.276)

or equivalently

(3.277)

Here, m is an integer if the dependency on ϕ is periodic with the period . Then,

after dividing eq. (3.274) by we obtain

(3.278)

Now the first term solely depends on r, while the others solely depend on

. This

permits further substitution

(3.279)

from which we obtain the general solution

(3.280)

which may instantly be verified by substitution. Substituting (3.279) into (3.278)

and multiplying by gives

(3.281)

This important differential equation is identified as the generalized Legendre

equation, and its solutions are the Associated Legendre functions. For the special

case of , they are called Legendre functions. Since this is a differential

equation of second order, there must be two linearly independent solutions, i.e., the

associated Legendre functions of the first and second kind. Only functions of the

first kind are finite everywhere on the sphere, i.e., for all values of

, while the ones

of the second kind have singularities at the poles (at and ).

Therefore, when solving a problem that includes the poles, one has to exclude the

associated Legendre functions of the second kind, since they would cause the

potential to diverge. For this reason, we will not study them further, realizing that

we exclude certain boundary value problems, namely those which do not contain

the poles or . (Notice also that our choice of integer values for m

represents another restriction on the generality, as we always take the whole space

of angles and not just a fraction thereof). So, we only study the

associated Legendre functions of the first kind. These are finite only for n being

whole numbers. They are labelled by , and we have

(3.282)

φϕ()

-----------

∂

∂ϕ

---------

φϕ() m

–=

φ A

imϕ()exp A

imϕ–()exp+=

φ A

mϕ()cos A

mϕ()sin+=

2π

θsin

Rr()

-----------

r∂

∂

r∂

∂

Rr()

θsin D θ()

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

θ∂

∂

θ∂

∂

D θ()sin

θsin

-------------–+0=

Rr()

-----------

r∂

∂

r∂

∂

Rr() nn 1+()=

Rr() B

n 1+

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

D θ()

θsin

-----------

θ∂

∂

θ∂

∂

D θ()sin nn 1+()

θsin

-------------– D θ()+0=

m 0=

θ 0= θπ=

0 ϕ 2π≤≤

D θ() P

θcos()=

3. Separation of Variables of Laplace’s Equation8

190 Formal Methods of Electrostatics

Specifically for

(3.283)

Frequently, the variable

(3.284)

is introduced. This changes the differential equation (3.281) into

(3.285)

and

(3.286)

The functions

(3.287)

(3.288)

are called spherical harmonics, which is the same name as for the functions

(3.289)

The name reflects on the fact that when holding r fixed, the angles address all

points on the sphere with . The various sorts of spherical harmonics are

all solutions of the differential equation

(3.290)

which emerges from (3.274), when one separates only the radial part.

The can be expressed in the following way:

(3.291)

This makes the simplest spherical harmonics (Legendre polynomials)

(3.292)

etc. The simplest associated Legendre functions are

m 0=

D θ() P

θcos()P

θcos()==

ξθcos dξ,θsin dθ–==

ξ∂

∂

1 ξ

–()

ξ∂

∂

D ξ() nn 1+()

1 ξ

–

--------------– D ξ()+0=

D ξ() P

ξ()=

θcos()mϕ()cos

θcos()mϕ()sin

θcos()imϕ()exp=

θϕ,

r const=

θsin

-----------

θ∂

∂

θ∂

∂

sin nn 1+()

θsin

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

∂

∂ϕ

---------

++ F θϕ,()0=

ξ()

1 ξ

–()

m 2⁄

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

nm+

dξ

nm+

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

1–()

1==

ξθcos===

---

3ξ

1–()

---

3 θcos

1–()

---

32θcos 1+()== = =











191

(3.293)

etc. It is only necessary to evaluate m-values which are less or equal to n, as

(3.294)

To expand an arbitrary function within the interval by is possible.

For this purpose, use the orthogonality relation

(3.295)

As we have already seen before, the angular functions are orthogonal. For the

given interval , the orthogonality relations are

(3.296)

The exponential functions are also orthogonal

(3.297)

Note that according to the definition (3.135), the scalar product of complex

functions requires the use of the complex conjugate, that is, . The

relations (3.296) and (3.297) are equivalent because of

Of course, there is a continuum analogue for (3.297), which shall be given for

comparison:

(3.298)

This formula is the basis for the exponential Fourier transformation. It also reveals

an important expression for the δ-function.

(3.299)

Because of the symmetry of the cosine function and the anti-symmetry of the sine

function, this allows one to write

1 ξ

– θsin==

3ξ 1 ξ

–3θsin θcos 3 2⁄()2θsin== =

31 ξ

–()3 θsin

---

12θcos–()===











0= for mn>

1 ξ 1≤≤– P

ξ()P

ξ()ξd

1–

∫

2 nm+()!

2n 1+()nm–()!

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

nn'

0 ϕ 2π≤≤

mϕ()cos m'ϕ()cos ϕd

2π

∫

mm'

or m

πδ

mm'

for m 12…,,=







mϕ()sin m'ϕ()sin ϕd

2π

∫

πδ

mm'

for m 12…,,==

mϕ()cos m'ϕ()sin ϕd

2π

∫

imϕ()exp

imϕ()exp im'ϕ–()exp ϕd

∫

2πδ

mm'

im'ϕ–()exp

imϕ±()exp mϕ()cos imϕ()sin±=

ikx()exp ik'x–()exp xd

∞–

+∞

∫

ik k'–()x[]exp xd

∞–

+∞

∫

2πδ kk'–()==

δ kk'–()

2π

------

ik k'–()x[]exp xd

∞–

+∞

∫

3. Separation of Variables of Laplace’s Equation8

192 Formal Methods of Electrostatics

This represents a slightly different form from the previously used eq. (3.198).

The orthogonality of the respective product functions, i.e. of the spherical

harmonics, is a consequence of eqs. (3.295) through (3.297). For instance, it is

(3.300)

where we used the solid angle on the sphere’s surface

(3.301)

Furthermore, integrating over the entire solid angle gives

(3.302)

For the general solution of the potential we may try the form

(3.303)

as well as this form

(3.304)

The properties of the spherical harmonics are compiled in the previously

mentioned books [3-7] (see Section 3.7.2).

δ kk'–()

2π

------

kk'–()x[]cos ikk'–()x[]sin+{}xd

∞–

+∞

∫

2π

------

kk'–()x[]cos xd

∞–

+∞

∫

---

kk'–()x[]cos xd

+∞

∫

mϕ()cos m'ϕ()cos ϕd

∫

ξ()P

ξ()ξd

1–

∫

mϕ()cos P

θcos()m'ϕ()cos P

θcos() θcos()ϕdd

1–

∫

mϕ()cos P

θcos()m'ϕ()cos P

θcos()θsin θϕdd

1–

∫

mϕ()cos P

θcos()m'ϕ()cos P

θcos()dΩ

∫

2π

2n 1+

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

nm+()!

nm–()!

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

nn'

mm'

for m 1≥

4π

2n 1+

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

nm+()!

nm–()!

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

nn'

mm'

for m 0=











2π 1 δ

+()

2n 1+

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

nm+()!

nm–()!

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

nn'

mm'

θsin θϕdddΩ=

dΩ

∫

4π

2n 1+

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

nm+()!

nm–()!

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

nn'

mm'

⋅⋅=

n 1+

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





θcos()C

mϕ()cos D

mϕ()sin+[]

m 0=

∑

n 0=

∞

∑

n 1+

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





θϕ,()

mn–=

∞

∑

n 0=

∞

∑

193

3.8.2 Examples

3.8.2.1 Dielectric Sphere in a Uniform Electric Field

Let us start with a simple problem, whose solution is already known (Sect. 2.12),

namely the problem of a sphere in a uniform electric field (Fig. 3.21). The potential

of the outside applied electric field is

(3.305)

The potential due to the bound charges at the sphere’s surface has to be added. For

the inside potential we write

(3.306)

This is a result of the Ansatz (3.303) and the rotational symmetry . has

to be zero to avoid the singularity at the center of the sphere. Conversely, for the

outside space we write

Comparison with (3.292) reveals that , i.e., one might as well write:

(3.307)

F has to be continuous at so that the tangential components of E will be

continuous there. This requires that

Fig. 3.21

o ∞,

a ∞,

z– E

a ∞,

r θcos–==

θcos()

n 0=

∞

∑

m 0= B

n 1+

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

θcos()

n 0=

∞

∑

a ∞,

θcos P

n 1+

------------ δ

a ∞,

r–





θcos()

n 0=

∞

∑

3. Separation of Variables of Laplace’s Equation8