Basic Theoretical Physics: A Concise Overview P14 pps

132 17 Electrostatics and Magnetostatics17.2 Electrostatic and Magnetostatic Fields in Polarizable Matter 17.2.1 Dielectric Behavior For a polarizable medium, such as water, the permitti

130 17 Electrostatics and Magnetostatics

Proof of the above statements is again based on Gauss’s integral theorem,

according to which (with E n:=E · n)

Q1+ Q2=





∂V1+∂V2

However

∂S = −∂V1− ∂V2, where the change of sign reflects that, e.g., the outer normal of V1is an inner normal of S But according to Gauss’s integral theorem:





∂S

End2A =



S

Since

divE = 0 , we have Q1+ Q2= 0 ,

as stated

The solution of the corresponding Neumann problem (i.e., not U , but Q is

given) is also essentially unique for the capacitor Here the proof of uniqueness

is somewhat more subtle (N.B we have used the term essentially above) For a given Q j two different solutions φ k (k = 1, 2) must both satisfy

−ε0·





∂V j

(∇φk · n) d2A = Qj (for j = 1, 2) ;

thus their difference w := φ2− φ1 satisfies





∂V j

∇w · nd2A = 0 ,

and since w is constant on ∂V j, one even has





∂(Is) (w ∇w) · nd2A = 0

Moreover, according to Gauss’s integral theorem it follows that



Is∇ · (w∇w)dV = 0 , i.e.,



Is

( (∇w)2+ w ∇2w)

dV = 0

Therefore, since

∇2w = 0 we have



Is(∇w)2dV = 0 ;

as a consequence

∇w ≡ 0

everywhere (apart from a set of zero measure) Therefore, it is not now the

potential that is unique but its essence, the gradient, i.e., the electric field

itself

The capacity of a condenser is defined as the ratio of charge to voltage across the plates, or C := Q U Elementary calculations contained in most

textbooks give the following results: For a plate capacitor (with plate area F and plate separation d), a spherical capacitor (with inner radius R and outer radius R + ΔR), and for a cylindrical capacitor (with length L and inner radius R ⊥) one obtains, respectively:

C = ε0

F

d , C = 4πε0

R · (R + ΔR)

ΔR , C = 2πε0

L

lnR ⊥ +ΔR ⊥

R ⊥

. (17.30)

In the limit ΔR  R (ΔR⊥  R⊥) these results are all identical

17.1.7 Numerical Calculation of Electric Fields

The uniqueness theorem for the Dirichlet problem is useful, amongst other reasons, because it allows one immediately to accept a solution (found or guessed by any means available) as the only solution One should not

under-estimate the practical importance of this possibility!

If all analytical methods fail, numerical methods always remain These methods are not necessarily as complicated as one might think, and they can often be used in the context of school physics If one considers, for example,

a simple-cubic grid of edge length a between lattice points, then one has (up

to a small discretization error∝ a2):

∇2φ n ≡

6



j=1 (φ n+Δ j − φ n )/a2. (17.31)

Here the six vectorsn + Δj denote the nearest neighbours (right – left, backwards – forwards, up – down) of the lattice point considered, and

φ n := φ( r n )

To obtain a harmonic function, it is thus only necessary to iterate (to the

desired accuracy) until at any lattice point one has obtained

φ n ≡1

6

6



j=1

φ n+Δ j ,

i.e., the l.h.s must agree with the r.h.s

132 17 Electrostatics and Magnetostatics

17.2 Electrostatic and Magnetostatic Fields

in Polarizable Matter

17.2.1 Dielectric Behavior

For a polarizable medium, such as water, the permittivity of free space ε0

must be replaced by the product ε0 · ε, where ε is the relative dielectric constant of the medium, which is assumed to be isotropic here (this applies

to gases, liquids, polycrystalline solids and crystals with cubic symmetry8; ε

is a dimensionless constant of the order of magnitude O(ε) = 10 to 100 (e.g., for water ε ∼= 81)9

When a dielectric (or polarizable material) is placed between the plates

of a condenser, the capacity is increased by a factor ε For a parallel plate

condenser, for example, the capacity becomes:

C = εε0

F

d .

This enhancement is often considerable, and in practice a dielectric material, such as an insulating plastic, is inserted between condenser plates in order to

increase the amount of charge that can be stored for a given U

In addition, the electric displacement vectorD (see later)10is defined for such materials as D := ε0ε E Then, from Gauss’s law we obtain the first of

Maxwell’s equations (i) in integral form



∂V D · nd2A = Q(V ) (Maxwell I) , (17.32)

and (ii) in differential form

What are the atomistic reasons for dielectric behavior ? The answer to this

short question requires several sections (see below)

17.2.2 Dipole Fields; Quadrupoles

An electric dipole at a position r  in vacuo can be generated by the following elementary dumbbell approach.

8 Generalizations to non-cubic crystals will be treated below in the context of crystal optics

9

The product ε0 · ε is sometimes called the absolute dielectric constant of the material and also written as ε; however this convention is not followed below,

i.e., by ε we always understand the relative dielectric constant, especially since

ε0 does not appear in the cgs system whereas the relative dielectric constant is

directly taken over into that system In addition it should be mentioned that the

dielectric constant is not always constant but can be frequency dependent.

10

So-called for historical reasons

Firstly consider two exactly opposite point charges each of strength±q,

placed at the ends of a small dumbbell, r  + (a/2) and r  − (a/2) Then

take the limit (i) a → 0 while (ii) q → ∞, in such a way that qa → p(= 0),

whereas the limit qa2→ 0 The result of this procedure (the so-called “dipole

limit”) is the vectorp, called the dipole moment of the charge array Similarly

to Dirac’s δ-function the final result is largely independent of intermediate

configurations

The electrostatic potential φ (before performing the limit) is given by

φ( r) = q

4πε0

 1

|r − r  − a

2| −

1

|r − r +a

2|



Using a Taylor expansion w.r.t.r and neglecting quadratic terms in |a|, one

then obtains:

φ( r) ∼=− q a

4πε0 · grad r |r − r1  | = q a

4πε0 · |r − r r − r   |3 . (17.35)

The electrostatic potential φ Dpdue to an electric dipole with dipole vector

p at position r  is thus given by

φDp(r) = p

4πε0 · |r − r r − r   |3 , (17.36) and the corresponding electric field,EDp=−gradφDp, is (→ exercises):

E(r)Dp=(3(r − r )· p) (r − r )− |r − r  |2p

4πε0|r − r  |5 . (17.37)

In particular one should keep in mind the characteristic kidney-shaped

appear-ance of the field lines (once more a sketch by the reader is recommended)

A similar calculation can be performed for an electric quadrupole The

corresponding array of charges consists of two opposite dipoles shifted by

a vector b.11 The Taylor expansion must now be performed up to second order Monopole, dipole and quadrupole potentials thus decay∼ r −1,∼ r −2

and∼ r −3, whereas the fields decay∝ r −2,∝ r −3 and∝ r −4, respectively.

17.2.3 Electric Polarization

In a fluid (i.e., a gas or a liquid) of dielectric molecules, an external electric fieldE polarizes these molecules, which means that the charge center of the

(negatively charged) electron shells of the molecules shifts relative to the (positively charged) nuclei of the molecule As a consequence, a molecular electric dipole moment is induced, given by

11Another configuration corresponds to a sequence of equal charges of alternating sign at the four vertices of a parallelogram

134 17 Electrostatics and Magnetostatics

p = αε0E ,

where α is the so-called molecular polarizability, which is calculated by

quan-tum mechanics

Now, a volume element ΔV of the fluid contains

ΔN := nV ΔV

molecules of the type considered Moreover, the electric moment of this vol-ume element is simply the sum of the electric moments of the molecules

contained in ΔV , i.e.,

Δ p ≡ P ΔV

This is the definition of the so-called electric polarization:

P ≡ Δ p

ΔV , where Δ p := 

r i ∈ΔV

p(ri) ;

i.e., the polarization P is the (vector) density of the electric moment (dipole

density) Under the present conditions this definition leads to the result

P = nV ε0α E

Furthermore, the displacementD, is generally defined via the dipole

den-sity as

For dielectric material we obtain

D = ε0· (1 + χ) · E where χ = nV α

is the electric susceptibility, i.e., ε = 1 + χ.

(N.B.: In a cgs system, instead of (17.38) we have: D  := E  + 4π P ,

where P  = n

v α E  , for unchanged α Hence ε = 1 + 4πχ  = 1 + χ, i.e.,

χ = 4πχ  Unfortunately the prime is usually omitted from tables of data,

i.e., the authors of the table rely on the ability of the reader to recognize

whether given data correspond to χ or χ  Often this can only be decided if

one knows which system of units is being used.)

17.2.4 Multipole Moments and Multipole Expansion

The starting point for this rather general subsection is the formula for the

potential φ( r) of a charge distribution, which is concentrated in a region of

vacuum G :

φ( r) =



G 

r  )dV  4πε0|r − r  | .

We assume that for allr  ∈ G  the following inequality holds:  |, i.e.,

that the sampling pointsr are very far away from the sources r .

A Taylor expansion w.r.t.r  can then be performed, leading to

1

|r − r  | ∼=

1

r+

3



i=1

(−x 

i) ∂

∂xi

1

r+

1 2!

3



i,k=1

(−x 

i) (−x 

k) ∂

2

∂xi∂xk

1

r + (17.39) Substitution of (17.39) into the formula for φ( r) finally results in

φ( r) = 1

4πε0

⎧

⎨

⎩

Q

r +

p · r

r3 +1 2

3



i,k=1

qi,k 3x i x k − r2δ i,k

r5 +

⎫

⎬

⎭ , (17.40) with

a) the total charge of the charge distribution,

Q :=



G  r  )dV  ; b) the dipole moment of the charge distribution, a vector with the three

components

pi:=



G  r  )x 

i dV ;

and

c) the quadrupole moment of the distribution, a symmetric second-order

tensor, with the components

qi,k=



G  r  )x 

i x 

k dV  .

Higher multipole moments q i1, ,i lare calculated analogously, i.e., in terms

of order l (l = 0, 1, 2, ) one obtains

φ( r) ∼= Q

4πε0r+

p · r

4πε0r3

+

∞



l=2

3



i1, ,i l=1

(−1) l l!

1

4πε0

qi1, ,i l · ∂ l

∂xi1 ∂xi l

 1

r



. (17.41)

However, only a minor fraction of the many 2l -pole moments q i1, ,i l

ac-tually influence the result; in every order l, there are only 2l + 1 linear

in-dependent terms, and one can easily convince oneself that the quadrupole

136 17 Electrostatics and Magnetostatics

tensor can be changed by the addition of an arbitrary diagonal tensor,

q i,k → qi,k + a · δi,k ,

without any change in the potential

As a result, for l = 2 there are not six, but only 2l + 1 = 5 linearly independent quadrupole moments q i,k = q k,i Analogous results also apply

for l > 2, involving so-called spherical harmonics Y lm (θ, ϕ), which are listed

in (almost) all relevant books or collections of formulae, especially those on quantum mechanics One can write with suitable complex expansion

coeffi-cients c l,m:

φ( r) = Q

4πε0r+

p · r

4πε0r3 +

∞



l=2

1

4πε0r l+1 ·

+l



m= −l cl,mYl,m (θ, ϕ) (17.42)

Due to the orthogonality properties of the spherical harmonics, which are

described elsewhere (see Part III), the so-called spherical multipole moments cl,m, which appear in (17.42), can be calculated using the following integral

involving the complex-conjugates Y ∗

l,m of the spherical harmonics:

c l,m= 4π

2l + 1

∞



0

drr2 π



0

dθ sin θ

2π



0

r)r l Y ∗ l,m (θ, ϕ) (17.43)

Dielectric, Paraelectric and Ferroelectric Systems;

True and Effective Charges

a) In dielectric systems, the molecular dipole moment is only induced,

p molec. = ε0α E.

b) In contrast, for paraelectric systems one has a permanent molecular dipole

moment, which, however, forE = 0 vanishes on average, i.e., by

perform-ing an average w.r.t to space and/or time (This is the case, e.g., for

a dilute gas of HCl molecules.)

c) Finally, for ferroelectric systems, e.g., BaTiO3crystals, below a so-called

critical temperature Tc a spontaneous long-range order of the electric

po-larizationP exists, i.e., even in the case of infinitesimally small external

fields (e.g E → 0+) everywhere within the crystal a finite expectation value of the vectorP exists, i.e., for T < Tc one hasP  = 0.

In each case where P = 0 Gauss’s law does not state that





∂V

ε0E · nd2A = Q(V ) ,

but, instead (as already mentioned, withD = ε0E + P ):





∂V

D · nd2

Alternatively, in differential form:

and not simply

divε0

For a parallel plate condenser filled with a dielectric, for given charge Q,

the electric fieldE between the plates is smaller than in vacuo, since part of

the charge is compensated by the induced polarization The expression

E := divε0E

represents only the remaining non-compensated (i.e effective) charge density,

E true− divP ,

(see below)

true) In contrast, the expression

E := ε0divE

will be called the effective charge density These names are semantically

some-what arbitrary In each case the following equation applies:

E − divP

Calculating the Electric Field in the “Polarization

Representation” and the “Effective Charge Representation”

a) We shall begin with the representation in terms of polarization, i.e., with the existence of true electric charges plus true electric dipoles, and then

apply the superposition principle In the case of a continous charge and dipole distribution, one has the following sum:

φ( r) =

)dV  4πε0|r − r  | +



dV  P (r )· (r − r ) 4πε0|r − r  |3 . (17.46)

This dipole representation is simply the superposition of the Coulomb

potentials of the true charges plus the contribution of the dipole potentials

of the true dipoles

138 17 Electrostatics and Magnetostatics

b) Integrating by parts (see below), equation (17.46) can be directly

trans-formed into the equivalent effective-charge representation, i.e., with

one obtains:

φ( r) =



E(r  )dV  4πε0|r − r  | . (17.47)

The electric polarization does not appear in equation (17.47), but instead

of effective (i.e., not compensated) charges (see above) For simplicity we

have assumed that at the boundary ∂K of the integration volume K the

charge density and the polarization do not jump discontinously to zero but that, instead, the transition is smooth Otherwise one would have to add to equation (17.47) the Coulomb potential of effective surface charges

σEd2A := (σ + P · n)d2A ,

i.e., one would obtain

φ( r) =



K

E(r  )dV  4πε0|r − r  |+





∂K

σE(r )d2A  4πε0|r − r  | . (17.48) Later we shall deal separately and in more detail with such boundary divergences and similar boundary rotations Essentially they are related to

the (more or less elementary) fact that the formal derivative of the unit step function (Heaviside function)

Θ(x)(= 1 for x > 0 ; = 0 for x < 0 ; =1

2 for x = 0)

is Dirac’s δ-function:

dΘ(x)

dx = δ(x)

Proof of the equivalence of (17.47) and (17.46) would be simple: the equiv-alence follows by partial integration of the relation

r − r 

|r − r  |3 = +∇ r  1

|r − r  | .

One shifts the differentiation to the left and obtains from the second term in (17.46):

−

 ∇

r  · P (r ) 4πε0|r − r  | dV  .

Together with the first term this yields the required result

17.2.5 Magnetostatics

Even the ancient Chinese were acquainted with magnetic fields such as that due to the earth, and magnetic dipoles (e.g., magnet needles) were used by

mariners as compasses for navigation purposes In particular it was known that magnetic dipoles exert forces and torques on each other, which are analo-gous to those of electric dipoles

For example, the torque D, which a magnetic field H exerts on a magnetic

dipole, is given by

D = m × H ,

and the forceF on the same dipole, if the magnetic field is inhomogeneous,

is also analogous to the electric case, i.e.,

F = (m · grad)H

(see below)

On the other hand a magnetic dipole at r  itself generates a magnetic

fieldH, according to12

H(r) = −gradφm , with φ m= m · (r − r )

4πμ0|r − r  |3 .

All this is completely analogous to the electric case, i.e., according to this

convention one only has to replace ε0by μ0,E by H and p by m.

However, apparently there are no individual magnetic charges, or

mag-netic monopoles, although one has searched diligently for them Thus, if one

introduces analogously to the electric polarization P a so-called magnetic

po-larization J 13, which is given analogously to the electric case by the relation

Δ m = JΔV ,

then one can define a quantity

B := μ0H + J ,

the so-called “magnetic induction”, which is analogous to the “dielectric dis-placement”

D := ε0E + P ;

but instead of

div

12

Unfortunately there are different, although equivalent, conventions: many au-thors writeD = m B ×B, where in vacuo B = μ0H, and define m B as magnetic

moment, i.e., m B =m/μ0, where (unfortunately) the indexB is omitted

13All these definitions, including those forP and J, are prescribed by international

committees and should not be changed