supplement - elektromagnetic fields local laws

8/17/2019 Supplement - Elektromagnetic Fields Local Laws

1/55


2/55

c2007 Laboratory of Electromagnetic Research

All rights reserved. No part of this publication may be reproduced, stored

in a retrieval system, or transmitted, in any form or by any means, elec-

tronic, mechanical, photocopying, recording, or otherwise, without the prior

permission of the Laboratory of Electromagnetic Research.


3/55

PrefaceThis booklet forms a supplement to Physics for Scientists and Engineers by

Paul A. Tipler and is especially written for Electrical Engineering students.

You may ask: ‘Why do I need a supplement to a book that already counts

more than 1,100 pages ?’ I shall briefly answer this. Tipler’s book aims at

all kinds of students that have to acquire a general knowledge of different

parts of physics. In this respect, Tipler has written a great book. However,

Electrical Engineering students need a more than average knowledge of Elec-

tricity and Magnetism. In addition to the items treated in Part IV of the

book, in the early stage of their educational career they must get acquainted

with the following topics:

• local laws of the electric, magnetic and electromagnetic field;• the boundary conditions for these fields;

• alternative field quantities that appear in the electrotechnical litera-

ture.

These topics are not treated in Tipler’s book. This supplement fills the gap.

The best way to use this supplement is first to finish an entire subject

(such as electrostatics) from the book, and then to study the associated

sections from this booklet, according to the following scheme:

Suggested use of the supplement

Subject Book chapters Supplement sections

Electrostatics 21, 22, 23, 24 1, 2, 3Electric current 25 4, 5Magnetostatics 26, 27 6Electromagnetics 28, 30 7

In this supplement two types of numbers are used for referencing. A

number that starts with the character ‘S’ refers to an object (equation,

figure or table) in this supplement, while a number that starts with a digit

refers to an object in Tipler’s book.

Finally, I acknowledge my colleagues Robert van Amerongen, Dirk Quak

and Johan Smit for their valuable comments.

Delft Martin Verweij

December 2000


4/55

Preface to the second edition

In this edition, problems have been added at the end of each section. More-

over, references to equations and chapter numbers in Physics for Scientists

and Engineers have been updated to be compatible with the fifth edition

of this book. The author acknowledges Koos Huijssen for making these

improvements.


September 2005

Preface to the third edition

In this edition, references to equations and chapter numbers in Physics for

Scientists and Engineers have been updated to be compatible with the sixth

edition of this book. For the same reason, some equations have been slightly

modified.


November 2007


5/55

Contents

1 Global versus local laws 1

2 Integral theorems of Gauss and Stokes 2

Gauss’s integral theorem 4

Gauss’s integral theorem using the nabla operator 5

Stokes’s integral theorem 5

3 Local laws for the electrostatic field 8

Conservation of energy in the electrostatic field 8

Local electrostatic field equations 9

The electric flux density D 11

Boundary conditions for E and D 14Equations of Poisson and Laplace 18

4 Conservation of charge 20

Global and local laws for the transport of charge 20

Boundary condition for J 22

5 Ohm’s law 24

6 Local laws for the magnetostatic field 27

Local magnetostatic field equations 27

The magnetic field strength H 29

Boundary conditions for H and B 33

7 Local laws for the electromagnetic field 37

Local Maxwell’s equations 37

Boundary conditions for E , D, H and B 40

A Proof of integral theorems 43

Proof of Gauss’s integral theorem 43

Interpretation of Gauss’s integral theorem 45

Proof of Stokes’s integral theorem 45

Interpretation of Stokes’s integral theorem 47

B Answers to problems 49


6/55


7/55

SECTION 1 Global versus local laws 1

1 Global versus local lawsPhysical experiments show that electromagnetic quantities behave in a pre-

dictable and reproducible way. We can use mathematical equations to de-

scribe this behavior. For example, from numerous experiments it has become

clear that the electric field of a system of point charges can be written as

E iP = kq ir2iP

r̂iP . S-1

This is Equation 21-7, which describes how the electric field strength E iP at position P depends on such things as:

• other electromagnetic quantities: the strength of the point charge q icausing the electric field,

• properties of the configuration: the length riP and direction r̂iP of the

vector from the point charge to the field point P at which the electric

field strength is measured,

• constants: the Coulomb constant k = 8.99 × 109 N · m2/C2.

The electric field strength is a vector. As a consequence, the right-hand

side of the equation above is also a vector, and the equation is called a

vector equation. A vector equation simultaneously describes the threescalar equations for the three vector components.

The example shows that a physical law may be expressed in the language

of mathematics. But there is more. The tools of mathematics may be used

to cast a physical law in several forms. In case of electromagnetic field

quantities, two large classes of laws may be distinguished: global laws and

local laws.

An example of a global law is Equation 22-16

φnet = S

E · n̂ dA = S

E n dA = Qinside

ε0. S-2

The law states that the total electric flux through any closed surface S is

equal to 1/ε0 times the total electric charge enclosed by the surface. Both the

flux and the enclosed charge are quantities that are related to the surface S .

In general, quantities related to a volume, surface, or line are called global


8/55

2 SUPPLEMENT Electromagnetic fields — Lo cal relations

quantities, and the laws connecting global quantities are called global laws.

In view of this it is not surprising that in many global laws we encounterintegrations with respect to the spatial coordinates. Since all classical elec-

tromagnetic experiments have been performed with devices having a certain

size, the observed quantities are in principle global quantities, and the re-

sulting basic laws are primarily global laws. Thus a benefit of global laws is

that these yield a direct description of physical observations. On the other

hand many global laws contain integrals, which often give rise to integral

equations that are hard to solve by hand. Therefore a drawback of global

laws is that these are less suitable for manual mathematical calculations.

An example of a local law is Equation 23-17

E = − ∇V = −

∂V

∂xî +

∂V

∂yˆ j +

∂ V

∂zk̂

. S-3

This law gives the connection between the electric field strength E and the

potential V for each single point in space. Both the electric field strength

and the gradient of the potential are quantities that are related to individ-

ual points. In general, quantities that are related to single points in space

are called local quantities, and the laws connecting local quantities are

called local laws. Many local quantities depend on some spatial derivative

of another local quantity. This explains why in many local laws we find dif-

ferentiations with respect to the spatial coordinates. Local laws often resultin differential equations that can be solved by hand. Thus a benefit of local

laws is that these are suitable for manual mathematical calculations. How-

ever, in classical electromagnetics, observations always involve the collective

effect of all points in a certain measurement area, not only that of a single

point. Therefore a drawback of local laws is that these do not yield a direct

description of physical observations.

2 Integral theorems of Gauss and StokesIn the theory of electric and magnetic fields we encounter scalar quanti-ties like the electric potential V , and vector quantities like the electric field

strength E . Moreover, these quantities may depend on a scalar quantity

such as the time t, and a vector quantity such as the position vector r. In


9/55

SECTION 2 Integral theorems of Gauss and Stokes 3

view of this, the mathematical functions that describe the behavior of elec-

tromagnetic quantities may belong to some of the four classes presented inTable S-1.

The electrostatic field strength E is a vector that is a function of the po-

sition vector r. We use the notation E ( r) to indicate the value of the electric

field strength E as a function of the position r. Since E = E xî + E yˆ j + E zk̂

we may write

E ( r) = E x( r)î + E y( r)ˆ j + E z( r)k̂. S-4

Moreover, r = xî+ yˆ j+ zk̂, so E x, E y , E z depend on the coordinates x, y,z.

To show this explicitly, we may also write

E ( r) = E x(x,y,z)î + E y(x,y ,z)ˆ j + E z(x,y ,z)k̂. S-5

Thus E ( r) is nothing more than a shorthand notation indicating that the

components of E depend on the components of r, i.e. the spatial coordinates

x,y ,z.

Any vector that is a function of another quantity is called a vector

function. A quantity that is a function of the position vector describes a

so-called field. Examples of fields are the scalar field V ( r) and the vector

field E ( r). For vector fields there are some special theorems for turning one

type of integral into another type. Two integral theorems are particularly

Table S-1

Classification of functions

Class of functionFunctionvalue Argument Example of quantity

Scalar function of scalar argument

Scalar Scalar I (t) - Electric currentversus time

Scalar function of vectorial argument

Scalar Vector V ( r) - Electric potentialversus position

Vector function of scalar argument

Vector Scalar τ (t) - Torque on adipole versus time

Vector function of vectorial argument

Vector Vector E ( r) - Electric fieldstrength versus position


10/55


n̂

D

S

Figure S-1 Domain D with closed boundary surface S for the application of Gauss’s integral theorem.

important, since these are used in electromagnetics to derive local laws from

global laws. These integral theorems will now be presented.

Gauss’s integral theorem

Suppose we have a vector function v( r) and a volume D with a closed

boundary surface S . See for example Figure S-1. In that case Gauss’s

integral theorem states that

S

v · n̂ dA =

S

vn dA =

D

∂vx∂x

+ ∂ vy

∂y +

∂vz∂z

dV . S-6

Gauss’s integral theorem

The proof and the interpretation of this this theorem are given in the Ap-

pendix. Some requirements must be met before the theorem may be em-

ployed. The theorem only makes sense if v( r) is defined on a domain that at

least consists of the volume D. Moreover, all the components of v( r) must

be continuously differentiable with respect to all the coordinates x,y ,z on adomain that at least consists of the volume D. Finally, the normal vector

n̂ on S must point away from D. Thus, the value of vn is positive at those

locations on S where v is pointing to the outside of D, and the value of vnis negative at those locations on S where v is pointing to the inside of D .


11/55


Gauss’s integral theorem using the nabla operator

Gauss theorem may be written in a slightly different form when we introduce

a new mathematical object called the nabla operator. The nabla operator

is indicated by the symbol ∇ and defined as

∇ = ∂

∂xî +

∂

∂yˆ j +

∂

∂zk̂. S-7

Nabla operator

Like any operator, the nabla operator is only useful if it works on something

suitable. In the present case it makes sense to use the dot product to let ∇

work on a vector function like v. By applying the dot product from Equation

6-15 and assembling the derivatives in the obvious way, we get

∇ · v = ∂vx

∂x +

∂ vy∂y

+ ∂vz

∂z . S-8

Divergence of a vector field

The quantity ∇ · v is called the divergence of the vector field v. An

alternative notation for ∇ · v is div v. By applying these new items, Gauss’s

integral theorem may be written as

S v · n̂ dA =

D

∇ · v dV =

D div v dV.

S-9

Gauss’s integral theorem — Alternative notations

Stokes’s integral theorem

Suppose we have a vector function v( r) and a surface S with a closed bound-

ary curve C . A possible configuration is given in Figure S-2. In this case

Stokes’s integral theorem states that

C

v · d = S

( ∇× v) · n̂ dA. S-10

Stokes’s integral theorem

The proof and the interpretation of this this theorem are given in the Ap-

pendix. Again some conditions must be satisfied before this theorem may be


12/55


n̂

S

d

C

Figure S-2 Surface S with closed boundary curve C for the application of Stokes’s integral theorem.

applied. The theorem only makes sense if v( r) is defined on a domain that

at least consists of the surface S . Moreover, all the components of v( r) must

be continuously differentiable with respect to all the coordinates x,y ,z on

a domain that at least consists of the surface S . Finally, the normal vector

n̂ on S points in the direction given by a (right-handed) corkscrew that isturning in the direction of d .

In Stokes’s integral theorem the cross product of the nabla operator ∇

and a vector function v appears. The cross product of two ordinary vectors

a and b may be found with the aid of Equations 10-2, 10-7a and 10-7b. An

easier way to remember is to write this product using a determinant

a× b =

î ˆ j k̂ax ay azbx by bz

= (aybz − azby)î + (azbx − axbz)ˆ j + (axby − aybx)k̂. S-11

Cross product using determinant notation

Applying this recipe for the cross product and assembling the derivatives in

the obvious way, we get


13/55


∇× v =

∂vz∂y

− ∂ vy∂z

î+

∂vx∂z

− ∂ vz∂x

ˆ j+

∂vy∂x

− ∂ vx∂y

k̂. S-12

Curl of a vector field

The quantity ∇ × v is called the curl of the vector field v. An alternative

notation for ∇× v is curl v or rot v.

Problems

2-1 Four vector fields are given as a mathematical expression together with

a graphical representation. The arrows in the figures indicate the direction

and magnitude of the field in any given point. For each vector field, calculatethe divergence ∇ · v and curl ∇× v, and explain the results qualitatively

from the figures.

a)

v( r) = −x î

z

x

y b)

v( r) = y î

z

x

y

c)

v( r) = −x î− z k̂

y

z

x d)

v(r) = x k̂ − z î

y

z

x


14/55


2-2 Given is a scalar quantity V (x,y ,z) = x2 + yx2 + 3z2.

a) Calculate E = − ∇V .

b) Calculate the divergence of E .

c) Calculate the curl of E .

3 Local laws for the electrostatic fieldIn the previous section the integral theorems of Gauss and Stokes have been

presented as purely mathematical rules. In this section we will show how

these tools enable us to find the basic local laws for the static electric field.

Conservation of energy in the electrostatic field

In Section 23-2 a potential function V for the electric field of a point charge

is found. This potential function gives the potential energy of a unit test

charge in the field of a point charge. The potential function only depends

on the spatial coordinates through the distance r between the test charge

and the point charge. This implies that only the position of the test charge

relative to the point charge is important.

More complex situations may be described by either adding discretelydistributed charges q i or by integrating continuously distributed charges dq .

The potential functions of these complex charge distributions are obtained

by adding the potential functions of the discrete point charges q i as in Equa-

tion 23-10, or by integrating the potential functions of the continuously

distributed point charges dq as in Equation 23-18. Just like the potential

function of a single point charge, any composed potential function only de-

pends on the position of the test charge relative to a fixed point in the charge

distribution. This is an important property of the electrostatic field, as will

soon become clear.

Now consider two points a and b in an arbitrary electrostatic field. Thework we deliver when we carry a test charge q 0 from a to b is proportional to

the potential difference V b−V a. Since the potential function only depends on

the position relative to the corresponding charge distribution, this difference

only depends on the positions of a and b with respect to the charge distribu-


15/55

SECTION 3 Local laws for the electrostatic field 9

tion. It doesn’t matter along which path the test charge is carried from a to

b. As a result, a round-trip from a to b along one path and from b to a alonganother path will always cost us q 0(V b − V a) − q 0(V a − V b) = 0 work. So the

fact that any potential function only depends on the position relative to the

charge distribution causes the work for a trip around any closed contour C

to be zero. This may be stated mathematically as

C

F · d = 0. S-13

Conservative property for F — Global law

We have just proven that the force F in any electrostatic field is conserva-

tive. For a description of conservative forces see Section 7-1. Since F = q 0 E ,the electrostatic field is also conservative. This property may be stated as

C

E · d = 0. S-14

Conservative property for E — Global law

This is a basic property of the electrostatic field. Without this property,

Equation 23-2b would not make sense.

Local electrostatic field equations

There are two basic global laws that fully describe the electrostatic field.

From these we will now derive the two basic local laws.

The first basic global law is the description of the conservative property

for E given by Equation S-14. Since E is a vector function of the position

vector r, we may apply Stokes’s integral theorem from Equation S-10. This

results in

C

E · d = S

( ∇× E ) · n̂ dA = 0. S-15

Equation S-15 must hold for any surface S . This can only be achieved when

the integrand of the surface integral is zero. This results in


16/55


∇× E = 0. S-16

Conservative property for E — Local law

We have now found the local law that corresponds to the global law in

Equation S-14. From the conditions of Stokes’s integral theorem it follows

that Equation S-16 is only valid on a domain for which all the components of E are continuously differentiable with respect to all the coordinates x, y, z.

The second basic global law is Gauss’s law from Equation 22-16

φnet = S

E · n̂dA = S

E n dA = Qinside

ε0 . S-17

Gauss’s law for E — Global form

The total charge Qinside enclosed by a surface S depends on the volume

charge density ρ introduced on page 728

Qinside =

D

ρdV. S-18

Here D is the domain enclosed by S . Substitution into Equation S-17 yields

S E · n̂dA =

D

ρ

ε0 dV. S-19

Now we may apply Gauss’s integral theorem from Equation S-9. This gives S

E · n̂dA =

D

∇ · E dV =

D

ρ

ε0dV. S-20

Equation S-20 must hold for any domain D. This can only be achieved when

the integrands of both volume integrals are equal. This gives

∇ · E = ρ

ε0. S-21

Gauss’s law for E — Local form

We have now obtained the local Gauss’s law that corresponds to its global

counterpart in Equation S-19. From the conditions of Gauss’s integral the-

orem it follows that Equation S-21 is only valid on a domain for which all


17/55


the components of E are continuously differentiable with respect to all the

coordinates x, y, z. In the next part we will see that it is customary to castboth the global and the local form of Gauss’s law in a slightly different form.

The electric flux density D

In Tipler’s book the electric field is described by means of only one quantity

called ‘the electric field’ E . In the electromagnetic literature E is usually

called the electric field strength. Moreover, it is common practice to

introduce the electric flux density D as a second quantity.

The distinction between the electric field strength and the electric flux

density becomes important when there is both free charge (from free conduc-tion electrons or ions) and bound charge (from dipolar charges of electrically

polarized molecules). This occurs for example in a capacitor with a dielec-

tric, see Section 24-5. In this case, Equation 22-16 may be written as

φnet =

S

E · n̂ dA =

S

E n dA = 1

ε0(Qf + Qb). S-22

Here Qf is the amount of enclosed free charge, and Qb is the amount of

enclosed bound charge. Equation S-22 shows that E is related to the total

amount of charge. On the other hand, the electric flux density satisfies

S

D · n̂ dA = S

Dn dA = Qf . S-23

Gauss’s law for D — Global form

This is the alternative form of Gauss’s law that is commonly used in the

literature. Equation S-23 shows that D only depends on the amount of

free charge. Moreover, the factor 1/ε0 is absent here. From the equation it

also follows that the product of electric flux density and area yields electric

charge, so the unit of D is C/m2. Just like Equation S-17, we may take the

global law in Equation S-23 and derive a local law from it. Using the same

analysis as before, its follows that

∇ · D = ρf . S-24

Gauss’s law for D — Local form


18/55


Here ρf is the volume density of the free charge. This is the local Gauss’s

law for D. Since Gauss’s integral theorem is employed in the derivation,Equation S-24 is only valid on a domain for which all the components of D

are continuously differentiable with respect to all the coordinates x, y, z.

In vacuum there is no bound charge. Comparison of Equation S-22 and

Equation S-23 shows that the electric field strength and the electric flux

density are then related by

D = ε0 E . S-25

Relation between

D and

E in vacuum

Inside a dielectric the polarized molecules enhance the electric flux density

of vacuum by an amount P

D = ε0 E + P . S-26

Relation between D, E and P in a dielectric

The quantity P is called the polarization of the dielectric. Normally, D

and P depend on E in a manner that depends on the type of dielectric

material.

There are two approaches to analyze a capacitor with a dielectric. One

of these is to disconnect the voltage source from the capacitor before the

dielectric is put in place. This is the procedure described at the beginning

of Section 24-4. Let us derive the dependence of D and P on E using this

procedure∗. Since in this case the free charge Qf = Q on the capacitor plates

remains constant, the electric flux density D has a constant value

D = σf = Q

A = ε0E 0. S-27

Here E 0 is the electric field strength before the dielectric is inserted. Afterinsertion of the dielectric, the induced bound charge at the surface of the

∗Following Section 24-4, for the analysis of the parallel plate capacitor we only consider

the scalar magnitude of the field quantities D, P and E . This is allowed since these field

quantities are normal to the capacitor plates.


19/55


Figure S-3 The electric field between the plates of a capacitor with a dielec-tric. The surface charge on the dielectric weakens the original field betweenthe plates.

dielectric opposes the free charge at the plates, see Section 24-5 and Fig-

ure S-3. According to Equation 24-18, this causes the electric field strength

E to weaken to a value

E = σf + σb

ε0=

E 0κ

. S-28

Removing E 0 from both equations above givesD = κε0E = εE. S-29

Substitution of Equation S-29 into Equation S-26 further shows that

P = (κ − 1)ε0E = χeε0E. S-30

The quantity χe = κ − 1 is called the electric susceptibility of the dielec-

tric. In this paragraph and in Sections 24-4 and 24-5 it is assumed that the

dielectric is isotropic. The word isotropic indicates that P and E have the

same direction. In such materials, the latter two equations may be general-

ized to their vector forms

D = κε0 E = ε E , S-31

P = (κ − 1)ε0 E = χeε0 E . S-32

D and P expressed in terms of E


20/55


The notion of dielectric constant and the symbol κ are commonly used in

physics. In the electrotechnical literature this quantity is usually indicatedby the name relative permittivity and the symbol εr.

A benefit of using two separate quantities E and D is that we are now

able te analyze∗ a capacitor with a dielectric without using the electric field

strength E 0 or the charge Q0 that is present before placing the dielectric. To

show this, we choose to keep the voltage source connected to the capacitor

when the dielectric is inserted. This is the other approach, which is shortly

described in Section 24-4 just above Practice Problem 24-13. Under this

circumstance the potential difference V between the capacitor plates remains

at the constant value

V = Ed. S-33

Assuming we know ε of the dielectric, the free charge Qf = Q of the capacitor

is simply

Q = σf A = DA = εEA. S-34

Dividing Equation S-34 by S-33 yields for the capacitance

C = εA

d . S-35

Boundary conditions for E and D

Equations S-16 and S-24 are the basic local laws of electrostatics. Unlike

their global counterparts, these local laws are only valid when the relevant

field quantities are continuously differentiable with respect to the spatial

coordinates. This need not always be the case. For example, at a boundary

between two media with different ε, some part of the field quantities will

jump and the local laws will in general cease to hold. To fully describe the

behavior of the electrostatic field in configurations with jumps in the medium

parameters, the basic local equations must be supplemented by boundary

conditions. The purpose of these boundary conditions is to link the fieldquantities at both sides of a boundary.

∗Following Section 24-4, for the analysis of the parallel plate capacitor we only consider

the scalar magnitude of the field quantities D and E . This is allowed since these field

quantities are normal to the capacitor plates.


21/55


Medium 1ε1

ε2Medium 2

C 4

C 1

C 3

C 2

E t2

E t1

a

b

Figure S-4 Rectangular loop around a part of the boundary between twodifferent media.

First let us investigate the behavior of E at a boundary where ε jumps.

This is done by considering a small, rectangular loop around a part of the

boundary, as depicted in Figure S-4. The boundary is assumed to be locally

flat. The loop extends into the media on both sides of the boundary. It

has the sides a and b, and may be subdivided into parts C 1 through C 4.

According to Equation S-14 we find for the loop

C 1

E · d +

C 3

E · d +

C 2

E · d +

C 4

E · d = 0. S-36

Since we want to know what happens really close to the boundary, we take

the limit b → 0. This means that we let the loop shrink around the con-sidered part of the boundary. Since E remains finite near the boundary, we

find that

limb→0

C 3

E · d = 0, S-37

limb→0

C 4

E · d = 0. S-38

So near the boundary we have

C 1 E · d +

C 2 E · d = 0. S-39

Since this must hold for any length a, the contribution coming from each

part of C 1 must exactly be cancelled by a contribution coming from the

corresponding part of C 2 on the opposite side of the boundary. Because


22/55


Medium 1ε1

ε2

Medium 2

S 3

S 1

S 2

Dn1

Dn2

r

h

Figure S-5 Circular cylindrical box around a part of the boundary betweentwo different media.

C 1 and C 2 are directed in opposite directions, this is only possible if at the

boundary

E t1 = E t2. S-40

Boundary condition for E

This must hold for any direction of the loop, so it may be concluded that

at a boundary the tangential component of E is continuous. Warning: the

analysis does not provide a statement about the normal component of E ,

which will in general jump.

Second we will investigate the behavior of D at a boundary where ε

jumps. This is done by considering a small, circular cylindrical box (‘pillbox’)

around a part of the boundary, as depicted in Figure S-5. The boundary is

assumed to be locally flat. At the boundary, a free surface charge density

σf may be present. The box extends into the media on both sides of the

boundary. It has a radius r and a height h, and its surface may be subdivided

into the top S 1, the bottom S 2 and the rim S 3. According to Equation S-23we find for the box

S 1

Dn dA +

S 2

Dn dA +

S 3

Dn dA = Qf . S-41


23/55


Next we shrink the box around the considered part of the boundary by taking

the limit h → 0. Since D remains finite near the boundary, we find that

limh→0

S 3

Dn dA = 0. S-42

Near the boundary we now have

S 1

Dn dA +

S 2

Dn dA = Qf . S-43

This must hold for any radius r, so also in the limit r → 0. But in this limit

S 1 and S 2 become infinitesimally small and D and σf may be considered

constant over these surfaces. In that case, Equation S-43 may be replaced

by

πr2Dn1 − πr2Dn2 = πr

2σf . S-44

The minus sign comes from the fact that on S 2 the normal component Dn2points to the inside of the box, while the normal component Dn in Equation

S-41 points to the outside. Dividing by πr2 shows that at the boundary

Dn1 − Dn2 = σf . S-45

Boundary condition for D

It may be concluded that at a boundary the normal component of D jumps

by an amount equal to the surface charge density σf . If there is no surface

charge, the normal component of D is continuous at the boundary. Warning:

the analysis does not provide a statement about the tangential component

of D

, which will in general jump.For a charged surface that is surrounded by vacuum or air, Equation S-45

is equivalent to Equation 22-20 since in this case Dn = ε0E n for both medium

1 and medium 2. When the charged surface is surrounded by a dielectric,

Equation S-45 remains valid, while Equation 22-20 no longer holds.


24/55


Equations of Poisson and Laplace

In Section 23-3 it is shown that in the static case the electric field strength E may be found from the potential V . The recipe given in Equation 23-17

is

E = − grad V = −

∂V

∂xî +

∂ V

∂yˆ j +

∂ V

∂zk̂

. S-46

Using the nabla operator from Equation S-7, this may be written as

E = − ∇V. S-47

Substitution of Equation S-31 yields

1

ε D = − ∇V. S-48

Dot-multiplying both sides of this equation with the nabla operator and

using the local version of Gauss’s law from Equation S-24 finally gives

( ∇ · ∇)V = −ρf ε

. S-49

This local law between the potential and the charge density is called Poisson’s

equation. The operator combination ( ∇ · ∇) is usually written as ∇2 and is

called the Laplace operator. Application of the dot product from Equation

6-15 shows that the Laplace operator is in fact

∇2 = ∂ 2

∂x2 +

∂ 2

∂y2 +

∂ 2

∂z2. S-50

Laplace operator

With the Laplace operator, Poisson’s equation becomes

∇2V = −ρf ε

. S-51

Poisson’s equation


25/55


When the charge density is zero, Poisson’s equation turns into what is known

as Laplace’s equation

∇2V = 0 S-52

Laplace’s equation

There exist powerful numerical packages for solving Poisson’s equation and

Laplace’s equation in almost every kind of situation.

Problems

3-1 In a certain region in space the electric field potential is given by

V (x,y ,z) = −12

x2y2 V.

a) Determine the electric field strength E .

b) Is this electric field a conservative field? Explain your answer.

c) Obtain the volume charge density ρ.

3-2 Given is an electric field E = (kq/r3) r V/m, where r = xî + yˆ j + zk̂

is the position vector and r =

x2 + y2 + z2 is its length.

a) Determine the divergence ∇ · E outside the origin r = 0.

b) Is there any volume charge in the region outside the origin?

3-3 A uniform dielectric medium (r = 9) of large extent has an electric

flux density D = 15 pC m−2 applied.

a) Find D inside a thin disk-shaped air cavity cut in the dielectric with flat

sides normal to D.

b) Find D inside a slender needle-shaped air cavity with axis parallel to D.

3-4 Find the total free charge Qf in a cube (x,y,z) ∈ {1 ≤ x ≤ 2,

2 ≤ y ≤ 3, 3 ≤ z ≤ 4} m, in which D = 4xî + 3y2ˆ j + 2z3k̂ C/m2. Ob-

tain Qf bya) integrating ρf = ∇ · D throughout the volume of the cube.

b) integrating D over the surface of the cube.


26/55


4 Conservation of chargeIt is impossible to create or destroy an amount of electric charge without

creating or destroying an equal amout of opposite charge. This is the law of

conservation of charge as discussed in Section 21-1. Here we will see what

consequences this law has for electric currents.

Global and local laws for the transport of charge

Suppose we have a fixed domain D with a closed surface S as in Figure

S-1. In view of the conservation of charge no net charge can be created

or destroyed inside D. Consequently, the amount of charge Qinside that is

present inside D can only change if charge is transported through S . Let

−∂Qinside/∂t be the rate at which the charge inside the domain decreases.

Then the electric current flowing through the surface out of the domain is

I = −∂Qinside

∂t . S-53

Conservation of charge — Global law

This is the global law relating current and charge for an entire object. It

is tempting to suggest that this is simply a modified version of Equation

25-1. This is not true, however, since Equation S-53 is a consequence of theconservation of charge while Equation 25-1 is just the definition of electric

current and does not imply the conservation of charge.

It is easy to derive a local form of the above global law. First we introduce

the current density as the current per unit area. The vectorial current

density J is related to the vectorial drift velocity vd of the charge carriers

through

J = qn vd = ρf vd, S-54

Relation between current density and drift velocity

which corresponds to Equation 25-4. Here q is the charge of each of the

charge carriers, n is their number density, and ρf the density of the free

charge they represent. If we know the current density J in each point of a

surface S , according to Equation 25-5 the current I that flows through the


27/55

SECTION 4 Conservation of charge 21

surface is

I =

S

J · d A =

S

J · n̂dA. S-55

Current through a surface

Here J · n̂ is the normal component of J in the direction of the unit normal

vector n̂. Of course I may be negative. This indicates that actually more

charge crosses S in the direction opposite to n̂ than in the direction parallel

to n̂. Combining Equation S-55 with Equation S-53 yields

S

J n dA = −∂Qinside

∂t , S-56

where as usual n̂ points away from D. Further the charge Qinside may be

expressed in terms of the volume charge density ρ

Qinside =

D

ρdV. S-57

Substitution into Equation S-56 gives

S

J n dA = −

D

∂ρ

∂t dV. S-58

Next we may apply Gauss’s law from Equation S-9 and obtain S

J n dA =

D

∇ · J dV = −

D

∂ρ

∂t dV. S-59

Since this must hold for any domain D , it must be concluded that the inte-

grands of both volume integrals are equal. This yields

∇ · J = −∂ρ

∂t. S-60

Conservation of charge — Local law

We have now obtained the local law relating current density and volume

charge density. From the conditions of Gauss’s integral theorem it follows

that Equation S-60 is only valid on a domain for which all the components of J are continuously differentiable with respect to all the coordinates x,y ,z.


28/55


Medium 1ρ1

ρ2

Medium 2

S 3

S 1

S 2

J n1

J n2

r

h


Boundary condition for J

The requirement that J is continuously differentiable with respect to the

spatial coordinates will not always be met. At a boundary between two

media with different resistivities∗ ρ, some part of J will jump and the local

law in Equation S-60 will in general cease to hold. To fully describe the

behavior of the current density in configurations with jumps in the resistivity,

the basic local equation must be supplemented by a boundary condition that

links the current density at both sides of a boundary.

To investigate the behavior of J at a boundary where ρ jumps, we employ

the same method as for the electric flux density D. We start by taking a

small, circular cylindrical box (‘pillbox’) around a part of the boundary,

as depicted in Figure S-6. The boundary is assumed to be locally flat. It

may be able to store surface charge, so a surface charge density σ may be

present. The box extends into the media on both sides of the boundary. It

has a radius r and a height h, and its surface may be subdivided into the

top S 1, the bottom S 2 and the rim S 3. According to Equation S-56 we find

∗The symbol ρ used here for the resistivity was used in previous sections for volume

charge density. Care must be taken to distinguish which quantity ρ refers to. Usually this

will be clear from the context.


29/55

SECTION 4 Conservation of charge 23

for the box

S 1

J n dA +

S 2

J n dA +

S 3


∂t . S-61


the limit h → 0. Since J remains finite near the boundary, we find that

limh→0

S 3

J n dA = 0. S-62


S 1

J n dA +

S 2


∂t . S-63

This must hold for any radius r, so also in the limit r → 0. But in this limit

S 1 and S 2 become infinitesimally small and J and σ may be considered

constant over these surfaces. In that case, Equation S-63 may be replaced

by

πr2J n1 − πr2J n2 = −πr

2 ∂ σ

∂t

. S-64

The minus sign comes from the fact that on S 2 the normal component J n2points to the inside of the box, while the normal component J n in Equation


J n1 − J n2 = −∂σ

∂t . S-65

Boundary condition for J

It may be concluded that at a boundary the normal component of J jumps

by an amount −∂σ/∂t. If there is no changing surface charge, the normalcomponent of J is continuous at the boundary. Warning: the analysis does

not provide a statement about the tangential component of J , which will

in general jump. The size of this jump may be determined by combining

Equation S-40 and Equation S-74 from the next section.


30/55


Problems

4-1 A circular disc with thickness d and radius R rotates around its axis

with angular velocity ω. The disc is situated on the xy-plane and its axis

coincides with the z-axis. The disc is charged with a uniform volume charge

density ρ.

a) Give an expression for the current density J (x) on the x-axis.

b) Obtain the total current through the plane described by {x > 0, y = 0}.

c) Show by equation S-60 that the rate of change of ρ on the x-axis is zero.

5 Ohm’s law

In Section 25-2 the electric current flowing in a piece of wire is considered.

The charge carriers inside the wire are set in motion by the force caused by

a time-independent electric field inside the wire. After a very short time, the

average velocity of the charge carriers (drift velocity) stabilizes at a constant

value because of collisions of the charge carriers with the lattice ions. From

this moment on, a steady current is present in the wire.

As in Section 25-2, let us consider the segment of wire in Figure S-7.

When it is assumed that the electric field strength E is the same everywherein the segment, the potential difference V over the distance ∆L is given by

Equation 25-6 as

V = V a − V b = E ∆L. S-66

Figure S-7 A segment of wire carrying a current I . The potential differenceis V = V a − V b = E ∆L.


31/55

SECTION 5 Ohm’s law 25

Suppose that A is the area of the cross section of the wire. If we assume that

the contribution of the moving charge carriers to the current I is distributeduniformly over the cross section, the current density J is the same everywhere

on the cross section and we may write

I = J A. S-67

The ratio of V and I is the resistance of the segment. For ohmic materials the

resistance is independent of V and I . The relation between these quantities

is then given by

V = IR, R constant. S-68

Ohm’s law — Global form

This is Equation 25-9, which is Ohm’s law in global form. To find its local

counterpart, we apply the fact that according to Equation 25-10

R = ρ L

A. S-69

Here ρ is the resistivity∗ of the conducting material. The inverse of the

resistivity is the conductivity†

σ = 1

ρ . S-70

Conductivity

The unit of conductivity is Ω−1 · m−1 or S · m−1, where the symbol S is

the abbreviation for the unit siemens. In terms of σ , Equation S-69 may be

written as

R = L

σA. S-71

Combination of Equations S-66, S-67, S-68 and S-71 leads to

EL = J A

L

σA . S-72

∗Here the symbol ρ does not indicate volume charge density.†The symbol σ used here for the conductivity was used in previous sections for surface

charge density. Care must be taken to distinguish which quantity σ refers to. Usually this

will be clear from the context.


32/55


Figure S-8 Conducting object with an elementary domain at point P .

Elimination of L and A shows that J and E are related through

J = σE. S-73

This is the local version of Ohm’s law, although still in scalar form. This

equation not only applies to a wire segment. Suppose we have an arbitrarily

shaped object in which E and J change in a continuous fashion with position

(no jumps). In this case we may look at an infinitesimally small elementary

domain inside the object, as depicted in Figure S-8. Due to its small size,

inside the elementary domain E and J may be considered constant. Perform-

ing the same analysis as above again gives Equation S-73. The elementary

domain is infinitesimally small, so this equation in fact holds in the point

P at which the elementary domain is located. Since an elementary domain

may be located everywhere in the object, Equation S-73 applies to every

point of the object.

In isotropically conducting materials ( J and E in the same direction),

the scalar Equation S-73 may be generalized to

J = σ E . S-74

Combining this result with Equation S-70, we find Equation 25-11, which is

the local version of Ohm’s law in vector form.


33/55

SECTION 6 Local laws for the magnetostatic field 27

Problems

5-1 At the plane boundary between two conductors the normal electric

field is 2 V/m in medium 1, and the normal current density in medium 2 is

12 A/m2. The charge at the boundary can be considered constant. Find the

conductivity of medium 1.

6 Local laws for the magnetostatic fieldIn this section we will show how the integral theorems of Gauss and Stokes

enable us to find the basic local laws for the static magnetic field. The

derivation of these local laws will reveal a close mathematical correspondence

between the magnetostatic and the electrostatic field, despite the physical

differences.

Local magnetostatic field equations

There are two basic global laws that fully describe the magnetostatic field.

From these we will now derive the two basic local laws.

The first basic global law is Ampère’s law given by Equation 27-16

C

Btd = C

B · d = µ0I C , for any closed curve C. S-75

Ampère’s law for B — Global form

The current I C enclosed by a curve C depends on the current density J

introduced in Equation S-55

I C =

S

J · n̂dA. S-76

Here S is any surface that has C as its boundary curve. Substitution intoEquation S-75 yields

C

B · d = µ0

S

J · n̂dA. S-77


34/55


Now we may apply Stokes’s integral theorem from Equation S-10. This gives C

B · d =

S

( ∇× B) · n̂ dA = µ0

S

J · n̂dA. S-78


the integrands of both surface integrals are equal. This gives

∇× B = µ0 J . S-79

Ampère’s law for B — Local form

We have now obtained the local Ampère’s law that corresponds to its global

counterpart in Equation S-77. From the conditions of Stokes’s integral the-orem it follows that Equation S-79 is only valid on a domain for which all

the components of B are continuously differentiable with respect to all the

coordinates x, y, z. In the next part we will see that it is customary to cast

both the global and the local form of Ampère’s law in a slightly different

form.

The second basic global law is Gauss’s law for magnetism from Equation

27-15

φm,net = S

B · n̂ dA = S

Bn dA = 0. S-80

Gauss’s law for B — Global form

When we apply Gauss’s theorem from Equation S-9, this results in S

Bn dA =

D

∇ · B dV = 0. S-81

Equation S-81 must hold for any volume V . This can only be achieved when

the integrand of the volume integral is zero. This results in

∇ · B = 0. S-82

Gauss’s law for B — Local law

We have now found the local law that corresponds to the global law in

Equation S-80. From the conditions of Gauss’s integral theorem it follows


35/55


that Equation S-82 is only valid on a domain for which all the components of

B are continuously differentiable with respect to all the coordinates x, y,z.

The magnetic field strength H

In Tipler’s book the magnetic field is described by means of only one quantity

called ‘the magnetic field’ B. In the electromagnetic literature B is usually

called the magnetic flux density∗. Moreover, it is common practice to

introduce the magnetic field strength H as a second quantity.

The distinction between the magnetic field strength and the magnetic

flux density becomes important when there is both a conduction current

and an amperian current. A conduction current is due to freely movingconduction electrons or ions. This is the ‘ordinary’ current that forms the

subject of Chapter 25, and Sections S-4 and S-5. An amperian current is

due to microscopic current loops of moving bound atomic charges. This is

the hypothetical current that accounts for the magnetization of a material,

as discussed in Section 27-5. Both currents occur in a solenoid with a core of

magnetic material, see page 938 and pages 944-946. In this case, Equation

27-16 may be written as C

B · d = µ0(I f + I a), for any closed curve C. S-83

Here I f is the enclosed conduction current and I a is the enclosed amperiancurrent. Equation S-83 shows that B is related to the total current. On the

other hand, the magnetic field strength satisfies

C

H · d = I f , for any closed curve C. S-84

Ampère’s law for H — Global form

This is the alternative form of Ampère’s law that is commonly used in the

literature. Equation S-84 shows that H only depends on the conduction

current. Moreover, the factor µ0 is absent here. From the equation it alsofollows that the product of magnetic field strength and length yields electric

current, so the unit of H is A/m. Just like Equation S-75, we may take the

∗This is consistent with the electrostatic case since B in Equation S-80 plays the same

role as the electric flux density D in Equation S-23.


36/55


global law in Equation S-84 and derive a local law from it. Using the same

analysis as before, its follows that

∇× H = J f . S-85

Ampère’s law for H — Local form

Here J f is the conduction current density. This is the local Ampère’s law

for H . Since Stokes’s integral theorem is employed in the derivation, Equa-

tion S-85 is only valid on a domain for which all the components of H are

continuously differentiable with respect to all the coordinates x, y, z.

In vacuum there are no magnetic dipoles, so the magnetization is zero.This implies that a curve C that is entirely located in vacuum does never

enclose an amperian current. Comparison of Equation S-83 and Equation

S-84 shows that the magnetic field strength and the magnetic flux density

are then related by

B = µ0 H . S-86

Relation between B and H in vacuum

Inside a magnetic material the magnetic dipoles change the magnetic flux

density of vacuum by an amount µ0 M

B = µ0 H + µ0 M . S-87

Relation between B, H and M in a dielectric

The quantity M is called the magnetization of the magnetic material.

Normally, B and M depend on H in a manner that depends on the type of

magnetic material.

Let us derive the dependence of B and M on H using a thin∗ solenoid

∗This means that the radius of the solenoid is much smaller than its length. In this

case we may neglect the influence of both ends on the inside magnetic field, and assume

that this field is parallel to the axis of the solenoid. This fact allows us to only consider

the scalar magnitude of the field quantities B, M and H inside the solenoid.


37/55


M

H

I a

C 1

C 2

C 3

C 4

a

Figure S-9 Solenoid with a magnetic core.

with a magnetic core. Consider a coil of length with n turns per unit length,

where the wire carries a conduction current I . Application of Equation S-84

to the loop in Figure S-9 gives C 1

H · d +

C 3

H · d +

C 2

H · d +

C 4

H · d = anI. S-88

The field outside the solenoid (except near the ends) is small and may be

neglected, so

C 1

H · d = 0. S-89

The field inside the solenoid (except near the ends) is constant and in the

direction of the path C 2, so C 2

H · d = aH. S-90

Along C 3 and C 4 the field is perpendicular to the direction of the path, so C 3

H · d = 0, S-91

C 4

H · d = 0. S-92

Substitution of these results in Equation S-88 gives for the field inside the

solenoid

H = nI = Bapp

µ0. S-93


38/55


Here Bapp is the magnetic flux density of the magnetizing field, see pages

938-939. This is the flux density of the field that would exist inside theempty coil. In the presence of the core, the amperian current at the surface

accounts for its magnetization. According to Equation 27-20, the value of

the amperian current per unit length is

J a = di

d = M. S-94

The quantity J a is a surface current density with the unit A/m. When

the wire of the solenoid is thin and closely wound, the conduction current

in the individual turns may be replaced by a uniformly distributed surface

current as well. The value of this surface conduction current density is

J f = nI . S-95

This surface conduction current flows very close to the amperian surface

current. Application of Equation S-83 to the curve C in Figure S-9 then

yields

B = µ0(J f + J a) = Bapp + µ0M = K mBapp. S-96

Removing Bapp from Equation S-93 and Equation S-96 gives

B = K mµ0H = µH. S-97

Substitution of Equation S-97 into Equation S-87 further shows that

M = (K m − 1)H = χmH. S-98

The quantity χm = K m − 1 is the magnetic susceptibility of the mate-

rial. In this paragraph and in Section 27-5 it is assumed that the magnetic

material is isotropic ( M and H in the same direction). In such materials,

the latter two equations may be generalized to their vector forms

B = K mµ0 H = µ H , S-99

M = (K m − 1) H = χm H . S-100

B and M expressed in terms of H

In the electrotechnical literature the relative permeability K m is usually

indicated by the symbol µr.


39/55


A benefit of using two separate quantities H and B is that it is now

relatively easy to extend the analysis at the beginning of Section 28-6 anddetermine the self-inductance of a thin solenoid∗ with a magnetic core. To

show this, we again assume that a conduction current I flows through a

solenoid with n turns per unit length. Then the magnetic field strength H

inside the core has the constant value

H = nI . S-101

Assuming we know µ of the magnetic material, the magnetic flux φm through

the N = n turns of the coil is simply

φm = BAN = µHAN = µn2IA. S-102

Dividing this equation by I yields for the self-inductance

L = µn2A. S-103

Boundary conditions for H and B

Equations S-82 and S-85 are the basic local laws of magnetostatics. Unlike

their global counterparts, these local laws are only valid when the relevant

field quantities are continuously differentiable with respect to the spatialcoordinates. This need not always be the case. For example, at a boundary

between two media with different µ, some part of the field quantities will

jump and the local laws will in general cease to hold. To fully describe

the behavior of the magnetostatic field in configurations with jumps in the

medium parameters, the basic local equations must be supplemented by

boundary conditions. The purpose of these b oundary conditions is to link

the field quantities at both sides of a boundary.

First let us investigate the behavior of H at a boundary where µ jumps.

This is done by considering a small, rectangular loop around a part of the

boundary, as depicted in Figure S-10. The boundary is assumed to be locallyflat. The loop extends into the media on both sides of the boundary. It

∗As before, for the analysis of the thin solenoid we only consider the scalar magnitude of

the field quantities B and H inside the solenoid. This is allowed since these field quantities

are parallel to the axis of the solenoid.


40/55


Medium 1µ1

µ2Medium 2

C 4

C 1

C 3

C 2

H t2

H t1

a

b


has the sides a and b, and may be subdivided into parts C 1 through C 4.


C 1

H · d +

C 3

H · d +

C 2

H · d +

C 4

H · d = I f . S-104

Since we want to know what happens really close to the boundary, we take

the limit b → 0. This means that we let the loop shrink around the consid-

ered part of the boundary. Since H remains finite near the boundary, we

find that

limb→0

C 3

H · d = 0, S-105

limb→0

C 4

H · d = 0. S-106

Let us assume there is no surface conduction current. Then the enclosed

current I f becomes zero in this limit. Near the boundary we have

C 1

H · d +

C 2

H · d = 0. S-107

Since this must hold for any length a, the contribution coming from eachpart of C 1 must exactly be cancelled by a contribution coming from the

corresponding part of C 2 on the opposite side of the boundary. Because

C 1 and C 2 are directed in opposite directions, this is only possible if at the

boundary


41/55


Medium 1µ1

µ2

Medium 2

S 3

S 1

S 2

Bn1

Bn2

r

h


H t1 = H t2. S-108

Boundary condition for H

This must hold for any direction of the loop, so it may be concluded that at

a boundary the tangential component of H is continuous. However, if there

is a surface conduction current, the above derivation is no longer valid andthe tangential component of H will jump at the boundary. Warning: the

analysis does not provide a statement about the normal component of H ,

which will in general jump.

Second we will investigate the behavior of B at a boundary where µ

jumps. This is done by considering a small, circular cylindrical box (‘pillbox’)

around a part of the boundary, as depicted in Figure S-11. The boundary

is assumed to be locally flat. The box extends into the media on both sides

of the boundary. It has a radius r and a height h, and its surface may be

subdivided into the top S 1, the bottom S 2 and the rim S 3. According to

Equation S-80 we find for the box S 1

Bn dA +

S 2

Bn dA +

S 3

Bn dA = 0. S-109


42/55



the limit h → 0. Since B remains finite near the boundary, we find that

limh→0

S 3

Bn dA = 0. S-110


S 1

Bn dA +

S 2

Bn dA = 0. S-111

This must hold for any radius r, so also in the limit r → 0. But in this limitS 1 and S 2 become infinitesimally small and B may be considered constant

over these surfaces. In that case, Equation S-111 may be replaced by

πr2Bn1 − πr2Bn2 = 0. S-112

The minus sign comes from the fact that on S 2 the normal component Bn2points to the inside of the box, while the normal component Bn in Equation


Bn1 = Bn2. S-113

Boundary condition for B

It may be concluded that at a boundary the normal component of B is

continuous. Warning: the analysis does not provide a statement about the

tangential component of B, which will in general jump.

Problems

6-1 Find the free current I f through a square area 5 m on a side with

corners at (0, 3, 0), (0, 8, 0), (5, 8, 0) and (5, 3, 0) m, where we have a magneticfield H = 3y3î A/m. Do this by

a) using J = ∇× H and I f = A

J · d s.

b) using I f = C

H · d l.


43/55

SECTION 7 Local laws for the electromagnetic field 37

6-2 A ferromagnetic medium (µr = 175) of large extent has a uniform flux

density B = 2 T.

a) Find H inside a thin disk-shaped air cavity with flat sides perpendicular

to B.

b) Find H inside a long needle-shaped air cavity with axis parallel to B.

6-3 The xy-plane forms the plane boundary between two isotropic media,

denoted by medium 1 (with µr1 = 2, located at z > 0) and medium 2 (with

µr2 = 6, located at z < 0). Just above the boundary, in medium 1, the

magnetic field strength is given as H = 3î + 2ˆ j − 6k̂ A/m. Calculate the

magnetic flux density B just below the boundary.

7 Local laws for the electromagnetic fieldIn this section we will show how the integral theorems of Gauss and Stokes

enable us to find the local Maxwell equations for the electromagnetic field.

In essence, the approach is the same as for the electrostatic and the magne-

tostatic case.

Local Maxwell’s equations

When a magnetic field changes in time, it causes an electric field. This

phenomenon is called magnetic induction and is described by Faraday’s

law in Equation 28-5. Reversely, when an electric field changes in time,

it causes a magnetic field. This is due to the fact that a time-dependent

electric field causes Maxwell’s displacement current, which leads to the

generalized form of Ampère’s law in equation 30-4. Due to the occurence

of magnetic induction and Maxwell’s displacement current, the electric field

and the magnetic field mutually influence each other as soon as they are no

longer static. The combination of mutually coupled electric and magnetic

fields is called the electromagnetic field. The basic equations of the elec-

tromagnetic field are called Maxwell’s equations, which form the theoreticalbasis of all electrotechnical applications.

Let us first consider the global form of Maxwell’s equations. One version

may be found in Section 30-2 of Tipler’s book. In these equations, only


44/55


the electric field strength E and the magnetic flux density B occur. In the

electromagnetic literature, it is common practice to use an alternative versionin which also the electric flux density D and the magnetic field strength H

show up. Traditionally the equations are presented in a sequence that differs

from Tipler’s. Therefore we present Maxwell’s equations in global form as

C

H · d =

S

J n dA + d

dt

S

Dn dA, S-114

Ampère’s law or Maxwell’s first equation — Global form

C

E · d = −d

dt S

Bn dA, S-115

Faraday’s law or Maxwell’s second equation — Global form

S

Dn dA =

D

ρf dV, S-116

Gauss’s law or Maxwell’s third equation — Global form

S

Bn dA = 0. S-117

Gauss’s law for magnetism or Maxwell’s fourth equation — Global form

A benefit of this version is that all these equations are valid for all kinds

of material, while for example Equation 30-6d is only valid for vacuum.

The influence of a material shows up in the relations between D and E

(Equation S-31), J and E (Equation S-74), and B and H (Equation S-

99). Such equations give the material dependent relations between the field

quantities and are called constitutive equations.

Let us next derive the local counterparts of Maxwell’s equation in global

form. First we take Equation S-114 and substitute Stokes’ integral theorem

from Equation S-10. This gives

C

H ·d =

S

( ∇× H ) ·n̂ dA =

S

J ·n̂ dA+ d

dt

S

D ·n̂ dA. S-118

Now we state that we only want to derive the local relations for a non-


45/55


moving situation. This implies that C and S do not move. Then there is

no motional emf (see Section 28-4), so the time differentiation may be putinside the surface integral and we arrive at

C

H · d =

S

( ∇× H ) · n̂ dA =

S

J · n̂ dA +

S

d D

dt · n̂ dA. S-119


the integrands of all the surface integrals are equal. This gives

∇× H = J + d D

dt . S-120

Ampère’s law or Maxwell’s first equation — Local form

Applying the same kind of analysis to Equation S-115, we find

∇× E = −d B

dt . S-121

Faraday’s law or Maxwell’s second equation — Local form

Subsequently we may apply Gauss’s integral theorem from Equation S-9

to Equation S-116 and Equation S-117. But the global form of Maxwell’s

third and fourth equation is equal to the global form of Gauss’s law for D and B, respectively. Clearly, the same will apply to the corresponding

local equations. Therefore we give these equations without repeating their

derivation

∇ · D = ρf , S-122

Gauss’s law or Maxwell’s third equation — Local form

∇ · B = 0. S-123

Gauss’s law for magnetism or Maxwell’s fourth equation — Local form


46/55


Medium 1ε1, σ1

ε2, σ2Medium 2

C 4

C 1

C 3

C 2

E t2

E t1

a

b


Boundary conditions for E , D, H and B

The Maxwell’s equations S-120 through S-123 are the basic local laws of

electromagetics. Unlike their global counterparts, these local laws are only

valid when the relevant field quantities are continuously differentiable with

respect to the spatial coordinates. This need not always be the case. For ex-

ample, at a boundary between two media with different ε, σ or µ, some part

of the field quantities will jump and the local laws will in general cease to

hold. To fully describe the behavior of the electromagnetic field in configura-

tions with jumps in the medium parameters, the basic local equations must

be supplemented by boundary conditions. The purpose of these boundaryconditions is to link the field quantities at both sides of a boundary.

First let us investigate the behavior of E at a boundary where ε or σ

jumps. This is done by considering a small, rectangular loop around a part

of the boundary, as depicted in Figure S-12. The boundary is assumed to be

locally flat. The loop extends into the media on both sides of the boundary.

It has the sides a and b, and may be subdivided into parts C 1 through C 4.


C 1

E ·d + C 3

E ·d + C 2

E ·d + C 4

E ·d = −d

dt S

Bn dA. S-124

Here S is a surface that has the loop as its boundary curve. Since we want

to know what happens really close to the boundary, we take the limit b → 0.

This means that we let the loop shrink around the considered part of the


47/55


boundary. Since E remains finite near the boundary, we find that

limb→0

C 3

E · d = 0, S-125

limb→0

C 4

E · d = 0. S-126

Since B remains finite near the boundary, we also find that

limb→0

S

Bn dA = 0, S-127

This means that the enclosed magnetic flux becomes zero in this limit. This

is to be expected since the area A of surface S will become zero when b → 0.

Near the boundary we then have

C 1

E · d +

C 2

E · d = 0. S-128

Since this must hold for any length a, the contribution coming from each

part of C 1 must exactly be cancelled by a contribution coming from the

corresponding part of C 2 on the opposite side of the boundary. BecauseC 1 and C 2 are directed in opposite directions, this is only possible if at the

boundary

E t1 = E t2. S-129

Boundary condition for E — General case

This must hold for any direction of the loop, so it may be concluded that

at a boundary the tangential component of E is continuous. Comparison

with Equation S-40 shows that this boundary condition is the same as forthe static case. The analysis does not provide a statement about the normal

component of E , which will in general jump.

The boundary condition for D at a jump in ε or σ follows from Equation

S-116. As observed before, this equation is identical to Equation S-23.


48/55


49/55

APPENDIX A Proof of integral theorems 43

APPENDIX A Proof of integral theoremsIn Section 2 the integral theorems of Gauss and Stokes have been in-

troduced without proof. Here the proofs of both theorems will be given.

Moreover, the physical meaning of these theorems will be explained.

Proof of Gauss’s integral theorem

To prove Gauss’s integral theorem and to understand what it means, it is

best to first interpret ∇ · v as a scalar quantity indicating the outflow of

the vector field. It can be obtained from the limiting behavior of the net

outflow integral for a vanishing small elementary domain. To show this

we first compute the net outflow of the vector field v over the infinitesimally

small elementary domain dD in Figure S-13. The center of this elementary

domain is given by rc = 12

dx î + 12

dy ˆ j + 12

dz k̂. By Taylor’s theorem, the

x

y

z

O

d x

d y

d z

d S

d D

s

~

r

c

Figure S-13 Elementary domain dD with closed boundary surface dS inthree-dimensional space.


50/55


component vx is

vx(x,y ,z) = vx( rc) + ∂ vx

∂x (x− 1

2dx) +

∂ vx∂y

(y− 12

dy) + ∂ vx

∂z (z − 1

2dz)

+ higher order terms. S-133

The surface integral of the normal component vn (in the direction of the

outward normal) over the top surface {x = dx, 0 < y < dy, 0 < z < dz} of

the elementary domain is

dyy=0

dzz=0

vx(dx,y,z) dA =

vx( rc) +

12

∂vx∂x

dx

dy dz


The surface integral of the normal component vn (in the direction of the

outward normal) over the bottom surface {x = 0, 0 < y < dy, 0 < z < dz}

of the elementary domain is

−

dyy=0

dzz=0

vx(0, y , z) dA = −

vx( rc) −

12

∂vx∂x

dx

dy dz


The negative sign in front of the integral is coming in because the outwardpointing component vn for the bottom surface is −vx. The sum of the surface

integrals over these two faces is therefore simply (∂vx/∂x) dx dy dz, to the

order of approximation considered here. The contributions to the other faces

depend on vy and vz and can be computed in a similar way. The net outflow

integral from the elementary domain is therefore

dS

vn dA =

∂vx∂x

+ ∂ vy

∂y +

∂ vz∂z

dx dy dz = ( ∇ · v) dx dy dz,

S-136

in which dS denotes the boundary surface of the elementary domain dD.The net outflow integral per unit volume at an arbitrary point r is

limdV →0

dS

vn dA

dV =

∂vx∂x

+ ∂ vy

∂y +

∂ vz∂z

= ∇ · v. S-137


51/55

APPENDIX A Proof of integral theorems 45

Here dV = dx dy dz is the volume of an elementary domain dD around r. In

general the integral dS vn dA is called the flux of the vector field v through

the surface dS .

Knowing all this, the proof of Gauss’s integral theorem is easy. Consider

the domain D with closed boundary S . All we have to do is to subdivide

D in elementary domains dD described above. Now in the interior of D the

outflow through one side of an elementary domain is the inflow (or nega-

tive outflow) through the corresponding side of the neighbouring elementary

domain, so both contributions cancel each other. As a result of all these

cancellations, the total contribution from the inner boundaries between the

elementary domains is zero. Consequently, the total outflow through all

boundaries dS is just equal to the total outflow through the outer boundaryS . Adding Equation S-136 for all elementary domains dD in D then results

in Gauss’s integral theorem as stated in Equation S-9.

Interpretation of Gauss’s integral theorem

After the foregoing analysis it is not difficult to see what Gauss’s integral

theorem means. On one hand, the expression S

vn dA is the total outflow of

a vector field through the closed boundary S of a domain D. On the other

hand, D

( ∇ · v) dV is the integral of the outflow of the vector field per unit

volume of D. According to Gauss’s integral theorem, both expressions are

equal. Thus, loosely stated, the theorem says that the outflow of a vectorfield through the closed surface of a volume must be generated in an equal

amount inside that volume.

Proof of Stokes’s integral theorem

To prove Stokes’s integral theorem and to understand what it means, it is

best to first interpret ∇× v as a vectorial quantity indicating the circulation

of the vector field. It can be obtained from the limiting behavior of the

net circulation integral around a vanishing small elementary surface dS .

To show this we first consider the infinitesimally small elementary surface

{x = 12

dx, 0 < y < dy , 0 < z < d z}, which is perpendicular to î. The

situation is depicted in Figure S-14. The center point of this elementary

surface is given by rc = 12

dx î + 12

dy ˆ j + 12

dz k̂. The circulation integral for


52/55


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

supplement - elektromagnetic fields local laws

Documents