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8/17/2019 Supplement - Elektromagnetic Fields Local Laws
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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.
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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
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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.
Delft Martin Verweij
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.
Delft Martin Verweij
November 2007
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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
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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
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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
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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
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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 .
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SECTION 2 Integral theorems of Gauss and Stokes 5
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
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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
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SECTION 2 Integral theorems of Gauss and Stokes 7
∇× 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
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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-
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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
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10 SUPPLEMENT Electromagnetic fields — Lo cal relations
∇× 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
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SECTION 3 Local laws for the electrostatic field 11
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
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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.
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SECTION 3 Local laws for the electrostatic field 13
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
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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.
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SECTION 3 Local laws for the electrostatic field 15
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
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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
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SECTION 3 Local laws for the electrostatic field 17
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.
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18 SUPPLEMENT Electromagnetic fields — Lo cal relations
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
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SECTION 3 Local laws for the electrostatic field 19
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.
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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
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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.
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Medium 1ρ1
ρ2
Medium 2
S 3
S 1
S 2
J n1
J n2
r
h
Figure S-6 Circular cylindrical box around a part of the boundary betweentwo different media.
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.
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SECTION 4 Conservation of charge 23
for the box
S 1
J n dA +
S 2
J n dA +
S 3
J n dA = −∂Qinside
∂t . S-61
Next we shrink the box around the considered part of the boundary by taking
the limit h → 0. Since J remains finite near the boundary, we find that
limh→0
S 3
J n dA = 0. S-62
Near the boundary we now have
S 1
J n dA +
S 2
J n dA = −∂Qinside
∂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
S-61 points to the outside. Dividing by πr2 shows that at the boundary
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.
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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.
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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.
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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.
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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
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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
Equation S-78 must hold for any surface S . This can only be achieved when
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
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SECTION 6 Local laws for the magnetostatic field 29
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.
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30 SUPPLEMENT Electromagnetic fields — Lo cal relations
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.
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SECTION 6 Local laws for the magnetostatic field 31
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
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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.
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SECTION 6 Local laws for the magnetostatic field 33
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.
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Medium 1µ1
µ2Medium 2
C 4
C 1
C 3
C 2
H t2
H t1
a
b
Figure S-10 Rectangular loop around a part of the boundary between twodifferent media.
has the sides a and b, and may be subdivided into parts C 1 through C 4.
According to Equation S-84 we find for the loop
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
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SECTION 6 Local laws for the magnetostatic field 35
Medium 1µ1
µ2
Medium 2
S 3
S 1
S 2
Bn1
Bn2
r
h
Figure S-11 Circular cylindrical box around a part of the boundary betweentwo different media.
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
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36 SUPPLEMENT Electromagnetic fields — Lo cal relations
Next we shrink the box around the considered part of the boundary by taking
the limit h → 0. Since B remains finite near the boundary, we find that
limh→0
S 3
Bn dA = 0. S-110
Near the boundary we now have
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
S-109 points to the outside. Dividing by πr2 shows that at the boundary
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.
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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
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38 SUPPLEMENT Electromagnetic fields — Lo cal relations
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-
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SECTION 7 Local laws for the electromagnetic field 39
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
Equation S-119 must hold for any surface S . This can only be achieved when
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
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40 SUPPLEMENT Electromagnetic fields — Lo cal relations
Medium 1ε1, σ1
ε2, σ2Medium 2
C 4
C 1
C 3
C 2
E t2
E t1
a
b
Figure S-12 Rectangular loop around a part of the boundary between twodifferent media.
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.
According to Equation S-115 we find for the loop
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
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SECTION 7 Local laws for the electromagnetic field 41
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.
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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.
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44 SUPPLEMENT Electromagnetic fields — Lo cal relations
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
+ higher order terms. S-134
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
+ higher order terms. S-135
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
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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
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .