Engineering Field Theory

247 Pages·1973·13.733 MB·English

by  A. J. Baden Fuller and P. Hammon (Auth.)

Engineering Field Theory
A.J.BADEN FULLER, M.A.,C.ENG.,M.I.E.E.
Lecturer, Department of Engineering
University of Leicester
PERGAMON PRESS
OXFORD · NEW YORK . TORONTO
SYDNEY · BRAUNSCHWEIG
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Copyright © 1973 A.J.Baden Fuller
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, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of
Pergamon Press Ltd.
First edition 1973
Library of Congress Cataloging in Publication Data
Baden Fuller, A J
Engineering field théorie.
(Commonwealth and international library. Applied electricity and electronics division)
Bibliography: p.
1. Field theory (Physics) I. Title.
QC 661." B 22 1973 530.1'4 72-13071
ISBN 0-08-017033-1
ISBN0-08-017034-X (pbk).
Printed in Germany
This book is sold subject to the condition
that is shall not, by way of trade, be lent,
resold, hired out, or otherwise disposed
of without the publisher's consent,
in any form of binding or cover
other than that in which
it is published.
Preface
FIELD theory gives a unified mathematical theory which can be used in a number of
different physical situations. This work presents the theory in the context of all its many
applications, gravitation, electrostatics, magnetism, electric current flow, conductive heat
transfer, fluid flow, and seepage.
The contents arise from lectures given by the author to first-year students in the Depart-
ment of Engineering in the University of Leicester, where there is a unified approach to
all branches of engineering. There has not been found to be any comparable student textbook
which adequately introduces the wide engineering applications of field theory whilst only
assuming the elementary mathematical knowledge of the principles of differentiation and
integration. Some evidence for the experimental basis of field theory is valuable to students,
but in many situations it will be found that this experimental background will have been
gained by students in physics courses at school. This prior experience arising from school
physics is why the initial concepts of field theory are developed from electrostatics and
gravitation rather than from fluid flow. A novel approach is the development of the concept
of flux in its many applications before proceeding to the complementary concept of poten-
tial.
The contents of the book fall naturally into four parts. The first part is a single chapter
of introduction which as well as introducing field theory as a subject also contains sections
on units and dimensions and on vector quantities. The second part develops the concept
of flux, starting with electric flux and proceeding to applications in gravitation, ideal fluid
flow, and magnetism. The third part introduces the concept of potential, again starting
with the electric potential, and proceeds to applications in gravitation, electric conduction,
conductive heat transfer, fluid flow through permeable media, ideal fluid flow, and mag-
netism. Attention is confined to static fields, and, although there is a chapter on electro-
magnetic induction, time-varying fields are not included. The fourth part discusses techni-
ques of field plotting by free-hand sketching, by numerical solution of Laplace's equation,
and by experiment. There are a large number of worked examples in the body of the text,
and each chapter concludes with a summary and problems. The book has 167 diagrams,
65 worked examples, and 127 problems.
The material in this book is based on lectures given in the first year of a degree course
in engineering. Some lecturers, however, may feel that despite its length it does not proceed
very far, and if they wish to use it as a course textbook or as the basis of their own lectures,
parts of the book may have to be omitted. No lecturer ever finds any textbook perfectly
to his liking; there are bound to be some parts which he feels are better with another
xiii
XIV PREFACE
approach. This book is offered in the belief that many lecturers will find it provides useful
supporting material for their lectures. In particular, the problems solved in the text and
the worked examples will be found to be useful for students who need material additional
to their lectures. For those lecturers who wish to teach the concepts o ffield theory without
the more detailed application given in parts of the book, the following sections can be
omitted: Chapter 1,Introduction; Chapter 3, Flux function; Chapter 8, Potential function;
Chapter 9, Other fields; Sections 10.7-10.11, Solution to a fluid flow problem; Sections 11.7
to 11.9, Permanent magnets; and Chapters 13 and 15, Field-plotting techniques. The
student can be recommended to read these parts for himself without making it a mandatory
requirement of the course. However, readers beware; the book has not been written assum-
ing that some material can be omitted without loss.
As the interdisciplinary nature of engineering work grows, it is hoped that the advantages
of a unified approach to engineering education will be appreciated and that this book will
be found to be suitable for undergraduate courses in engineering field theory. It is
hoped to publish a companion volume consisting of worked examples in engineering
field theory.
It is difficult to thank the many people who have contributed to the development of the
field theory course at Leicester, and I hope that lack of acknowledgement will not be taken
to imply lack of gratitude. However, I should like to mention Professor E. N. Pickering,
who taught the course when it was first given at Leicester; Dr. D. J. Cockrell, who has
critically advised me in my discussion of fluid dynamics; Dr. A. R. S. Ponter, who has
advised me in my discussion of seepage; and Dr. A. C. Tory for his advice on numerical
methods of solution.
I should also like to thank all my other colleagues in the Department who have contributed
to my better understanding of the subject. Many of the problems given in the book have
been used for many years in the form of duplicated examples papers by our students. My
thanks are due to my various colleagues who have taugh tthis course for their contributions
over some years to these problems. My thanks are also due to Professor G. D. S. MacLellan,
Head of the Department of Engineering at the University of Leicester, for making the
facilities of the Department available for the preparation of preliminary copies of the
draft of the book which have been used by th feirst-year students in the Department.
Leicester
A. J. BADEN FULLER
CHAPTER 1
Introduction
1.1. INVERSE SQUARE LAW
In 1665 Newton proved the law of gravitation that is known by his name:
m m
force = G 1 2
& '
where m and m are two point masses and d is the distance between them. This equation is
t 2
restated in Chapter 5 as eqn. (5.1). He further went on to prove mathematically that the
same law is true even if the two masses are spheres of uniform density and d is the distance
between their centres.
About a hundred years later, Priestley deduced a similar law for the force between two
point charges of static electricity, and in 1785 Coulomb gave an experimental verification
of the law for both electricity and magnetism. It is now known as Coulomb's law:
force = -Ml,
d2
where q and q are two point electric charges and d is the distance between them. If the
t 2
force is measured in dynes and the distance in centimetres, the charge is in electrostatic
units (e.s.u.), and no dimensional constant is required in the equation. This equation is
res tatedin Chapter 2, including a dimensional constant, as eqn. (2.1). The same equation
for Coulomb's law may be used to describe the force between two magnetic poles, wher eq^
and q are now two point magnetic charges or point magnetic pole strengths and d is the
z
distance between them. Again, if the force is measured in dynes and the distance in centi-
metres, the charge is in electromagnetic units (e.m.u.) of magnetic charge or pole strength,
and no dimensional constant is required in the equation. This equation is restated in Chap-
ter 6, including a dimensional constant, as eqn. (6.1). Because there are at least three indep-
endent physical phenomena which exhibit the inverse square law of force acting at a distance,
it has been found useful to develop a unified theory which is applicable to all such pheno-
mena. The theory enables one set of mathematical concepts to be used in a large number of
different applications. It is a mathematical hypothesis designed to predict an effect without
explaining the mechanism of the effect.
3
^
4 ENGINEERING FIELD THEORY
1.2. FORCE AT A DISTANCE
In the physical phenomena of electricity, magnetism, and gravitation, one body exerts an
influence on another body which is some distance away from it. Both Newton's law and
Coulomb's law describe the action of a force between two bodies without there being any
physical contact between these bodies to transmit the force. In order to be able to under-
stand this phenomenon so that we can make practical use of it, it is necessary to postulate
a method by which a force can be transmitted from one body to another. It is not necessary
to postulate a physical model that is the truth or the latest version of the truth; it is quite
sufficient to suggest a system which fits all the observable facts. There are three proposi-
tions that have been suggested:
(1) We postulate that one body sets up a stress in the medium surrounding it which is
transferred through the medium to the other body. As the effect occurs even in a
vacuum or in inter-planetary space, where there is no medium to support the stress,
we also need to postulate an imaginary medium which exists everywhere called the
ether.
(2) We postulate that there is a steady emission of particles from each body which, on
striking the other body, give rise to a force on that body.
(3) We postulate that an imaginary fluid issues from a body and exerts a force on any other
body in its path.
The first two form the basis of two theories of the propagation of electromagnetic energy.
The third has no physical significance, but it forms the basis of field theory. It is not necessary
to know the mechanics of the origin of the forces in order to describe their effects. Field
theory makes use of the concept of the emission of an imaginary fluid called flux because
it is helpful in an understanding of field-effect phenomenon, because it fits the observable
facts, and because it enables one to predict what will happen in other situations.
1.3. FIELD THEORY
Field theory gives a unified theory which can be used in a number of different physical
situations. For the gravitational force we postulate that each body emits an imaginary
fluid called gravitational flux and that this fluid flowing away from the body causes a force
on any other body in the system. Similarly, an electric charge gives rise to a flow of electric
flux and a magnetic charge gives rise to a flow of magnetic flux. All these concepts are deve-
loped in later chapters. As the theory is built round the flow of an imaginary fluid, it also be-
comes applicable to some real systems of electric current flow, conductive heat transfer,
and certain systems of real fluid flow.
Engineering field theory is based on the concept of a hypothetical fluid in order to
explain, and more especially in order to be able to predict, certain physical phenomena.
The theory is based on certain experimental facts such as the inverse square law of force
between attracting particles. This inverse square law describes the forces in gravitation,
electrostatics, and magnetism. In electric current flow the basic experimental relationship
is Ohm's law; in seepage flow there is a similar experimental relationship which is called
INTRODUCTION 5
Darcy's law; and there is a similar relationship for conductive heat transfer. We could
expect that one theory will suffice to explain all these different phenomena.
The basis of field theory is the explanation of observed facts. The explanation is in such
a form that it may be easily used to predict results in design situations. The explanation
may be fictitious in real terms, but it is acceptable and useful to engineers in that it gives the
correct answers.
1.4. FLUX THEORY
The concept of flux is useful in predicting forces in fields characterized by the inverse
square law for variation of force with distance, such as the forces exerted between the planets.
It explains why, when the inverse square law between point elements of mass is applied to the
planets, the planets behave as if their total mass were concentrated in a point at their
centre. It describes the forces between point charges of static electricity. It describes the
forces between the poles of magnets. As a fluid is used to aid understanding of the forces
in gravitation, electrostatics, and magnetism, the same theory can also be used to describe
flow situations. The characteristics of electric flux from a point charge are the same as
electric current flow from the point of an earthing conductor in the ground. The electric
capacitance of a system of charged conducting bodies is directly related to the electric
conductance between the same system of conductors. Therefore, if the conductivity and
permittivity of a material are known, and that material is used as an insulator between
the plates of a capacitor, its leakage conductance can be calculated from a knowledge of
its capacitance and vice versa. The magnetic flux in a magnet is calculated by analogy with
electric current flow through the core of the magnet, so that it has become usual to talk
of magnetic circuits.
The concept of flux is also applicable to heat transfer by conduction. The calculation of
the heat lost through the lagging around a hot-water pipe is similar to the calculation of
the capacitance or conductance of an electric cable in the form of two concentric conductors.
The laws of flux also apply to certain systems of real fluid flow and to seepage flow through
permeable soils. The theory is used to calculate seepage effects through earth dams or
embankments and to calculate the pressure distribution for ideal fluid flow around certain
immersed bodies.
In electrostatics, the field concept of flux considers that the imaginary fluid is relentlessly
pouring out of the charged body and by virtue of its motion exerts a force on any other
charged body in its path. In one sense, flux is an imaginary fluid because it is neither a
liquid nor a gas and we cannot detect it with our normal senses. However, electric flux is
real in another sense because an electric charge will always detect it. Similarly, a magnet
will always experience a force due to a magnetic flux and any lump of matter will experience
a force due to gravitational flux.
1.5. SYSTEMS OF UNITS AND DIMENSIONS
This book uses the international system of units which are given in Table 1.1, but because
historically Coulomb's law was used as the basis of two different systems of units and
because other systems are in practical use and will continue to be for some time, it is
6 ENGINEERING FIELD THEORY
TABLE 1.1 INTERNATIONAL SYSTEM OF UNITS
Basic Units
Quantity Unit Symbol
Length metre m
Mass kilogram kg
Time second s
Temperature kelvin K
Electric current ampere A
Supplementary Units
Plane angle radian rad
Solid angle steradian sr
Derived Units
Area square metre m2
Volume cubic metre m3
Frequency hertz Hz (c/s)
Density kilogram per cubic metre kg/m3
Velocity metre per second m/s
Angular velocity radian per second rad/s
Acceleration metre per second squared m/s2
Angular acceleration radian per second squared rad/s2
Force newton N (kg m/s2)
Pressure newton per square metre N/m2
Energy, quantity of heat joule J (Nm)
Power, rate of flow of heat watt W (J/s)
Thermal conductivity watt per metre kelvin W/mK
Electric charge, electric flux coulomb C (As)
Electric potential difference, volt V (W/A)
electromotive force
Electric intensity, field strength volt per metre V/m
Electric resistance ohm Ω (V/A)
Resistivity ohm metre Ω m
Conductance Siemens S (1/Ω)
Conductivity Siemens per metre S/m
Capacitance farad F (A s/V)
Electric flux density coulomb per square metre C/m2
Magnetic flux, pole strength weber Wb (Vs)
Magnetic potential difference, ampere A
magnetomotive force
Magnetic intensity, field ampere per metre A/m
strength
Inductance henry H (Vs/AÏ
Magnetic flux density tesla T (Wb/m2)
INTRODUCTION 7
thought necessary to digress about units here. In mechanical systems it is generally necessary
to have the three dimensions of length L, mass M, and time T, and all other quantities
are derived from these: for example, acceleration is length divided by time squared [L T~2]
and force is mass times acceleration [M L T~2].
EXAMPLE 1.1. Find the dimensions of work or energy
Answer. Work is done when the point of application of a force moves in the direction of the force. Work
= force x distance = ML2T"2. Alternatively, the answer can be obtained from the formula for kinetic
energy, energy = mass X (velocity)2.
Our choice of basic units is completely arbitrary, and we could have chosen length,
mass, and energy as the basic units and derive time as a function of these. For the electrical
sciences, a further dimension is needed. In the International System of units the electric
current is the fourth basic unit, and then all electric and magnetic units can be expressed
in terms of length, mass, time, and electric current. For heat-flow calculations, a further
basic unit of .temperature is needed, and for light problems one of luminous intensity is
required. The unit of luminous intensity is not included in Table 1.1 because it is not used
in this book.
The international system of units (abbreviated to SI units from the French) uses metres,
kilograms, seconds, and amperes as the basic units. The unit of temperature difference is the
kelvin, and scientific calculations are performed in terms of the thermodynamic or absolute
temperature in K. However, temperature is usually measured according to the Celsius scale,
°C, which has the same temperature increment as the standard unit but which has the zero
of the temperature scale at the freezing point of water. Therefore for measurement of
temperature difference,
1°C = 1 K,
but for absolute measurement of temperature,
0°C = 273-15 K or -273-15°C = 0 K.
There are other systems of units in common use such as the Imperial units—foot, pound,
second (f.p.s.), and the centimetre, gram, second (c.g.s.) units. These two sets of units have
been standards in the past and were adopted because their sizes were convenient. Altern-
atively, the Imperial units could have been standardized on the yard, ton, fortnight; and,
probably, 3000 years ago the standards were the cubit, shekel, and day. Any system of units
could have been adopted as the standard units from which all other units are derived, but
the SI units have been adopted as the system which provides the best compromise between
units which are in common use in part of the world, multiples of the internationally used
scientific units, and the commonly used electrical units.
The normally used multiple and submultiple prefixes in the SI units are given in Table 1.2.
It is recommended that the prefix is applied to a unit in the numerator and not to one of the
component dimensions of a derived unit. For example, a pressure of 106 N/m2 might be
written as 1 MN/m2 but not as 1 N/mm2. It is normal only to use the multiples in powers
8 ENGINEERING FIELD THEORY
of 103, but the use of deci-, centi-, and deka- (for 10" *, 10""2, and 10 respectively) is allowed.
If there is ever any doubt about the possible meaning of multiple or submultiple prefixes,
they ought not to be used. The quantity can be presented as a number multiplied by a power
of 10, for example, 1-38 x 106 kg.
TABLE 1.2. MULTIPLE AND SUBMULTIPLE PREFIXES
Multiple or
submultiple Prefix Symbol Pronunciation
1012 tera T tér'â
109 giga G jî'gâ
106 mega M mëg'â
103 kilo k kïi'o
io-3 milli m mïl'ï
io-6 micro μ mï'krô
IO"9 nano n nän'ö
io-12 pico P pê'cô
io-15 femto f fëm'tô
io-18 atto a at'tô
1.6. VECTOR QUANTITIES
Some quantities can be represented completely if their amplitude or strength is known,
whereas other quantities can only be adequately represented if their direction of action
as well as their amplitude is known. An example of the first quantity is time. It is only
necessary to know the length of time that has elapsed or the number of seconds that have
elapsed to give a completely unambiguous measure of time. Similarly, it is only necessary
to know the number of metres vertically above sea-level to know the height of any point
on a map, and the height is always the same at the same point irrespective of the path used
to reach that point. Such a quantity which can be completely represented by its amplitude
is called a scalar quantity. Scalar quantities are represented by ordinary type in mathematical
equations. All the equations so far quoted in this chapter have been in terms of scalar
quantities. Mass, electric charge, and magnetic pole strength are all scalar quantities.
However, when discussing these equations, the direction of the force was not given; the
equation only gave the amplitude of the force. The force between point masses or between
opposite sign point charges is one of attraction. That is, the force acts along the line joining
the point masses or joining the point charges. For its complete specification, a force needs
to be described by both its amphtude and its direction of action.
Most quantities which need to be described both by direction and amplitude are called
vector quantities. Therefore we see that force is a vector quantity. In mathematical equations
and expressions, vector quantities are represented by the use of bold type. The same symbol
is used for^both a vector quantity, i.e. F, and its amplitude, i.e. F. When only the amplitude
of a vector quantity is given in an equation, its direction must be specified in some other