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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Electromagnetism for Engineers
An easily accessible textbook to introduce the power of electromagnetism
Electromagnetism can be a difficult subject to grasp and to teach. Much of what we take for granted in modern life is enabled by electromagnetic effects, but it isn’t always easy to understand the impact of electromagnetism compared to other areas of engineering, such as mechanics, which are more tangibly observed and felt. Although electromagnetism is a crucial and important branch of physics with elegant mathematics, many students can find the study of electromagnetism inaccessible.
It is crucial for students of electrical and electronic engineering and physics to have a strong understanding of electromagnetism and how it impacts communications, power generation and transmission, semiconductor devices, motors, and more. Electromagnetism for Engineers aims to develop a student’s understanding of electromagnetism in the context of real effects and how they apply to such applications, whilst maintaining the sophistication of the mathematics that can be used to give deeper insight. It begins by describing the fundamentals of electromagnetism before a more detailed discussion of the basic concepts developed for specific application areas. It then considers the application of electromagnetism to transmission lines, antennas, and waveguides.
Electromagnetism for Engineers readers will find:
- A unique approach that illustrates the link between real-life applications and fundamental theory of electromagnetism
- Clear, concise language to help students gain a full understanding of the subject
- Carefully designed figures to illustrate points throughout the book
- Accompanying website at www.wiley.com/go/flewitt1418
Electromagnetism for Engineers has been written as a textbook for undergraduates studying electronic or electrical engineering. The manual can also be of interest to physics students and??to graduate-level students desirous of having a general book on electromagnetism. It is also a useful reference for professional engineers looking for a refresher on the fundamentals of electromagnetism.
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Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Acknowledgements
About the Author
Symbols
About the Companion Website
Part I: Fundamentals of Electricity and Magnetism
1 Charge and Electric Fields
1.1 Charge as a Fundamental Property of Matter
1.2 Electric Field and Flux
1.3 Electrical Potential
1.4 The Gauss Law of Electrostatics in Free Space
1.5 Application of the Gauss Law
1.6 Principle of Superposition
1.7 Electric Dipoles
Reference
2 Electric Fields in Materials
2.1 The Interaction of Electric Fields with Matter
2.2 Dielectrics
2.3 Capacitors and Energy Storage
2.4 Ferroelectrics
2.5 Piezoelectrics
2.6 Metals
2.7 Semiconductors
References
3 Currents and Magnetic Fields
3.1 Moving Charges and Forces
3.2 Magnetic Fields and Moving Charges
3.3 The Ampère Circuital Law in Free Space
3.4 Application of the Ampère Circuital Law
3.5 The Biot–Savart Law in Free Space and Superposition
3.6 The Gauss Law of Magnetic Fields in Free Space
3.7 The Faraday Law of Electromagnetic Induction in Free Space
3.8 Inductors and Energy Storage
Reference
4 Magnetic Fields in Materials
4.1 The Interaction of Magnetic Fields with Matter
4.2 The Magnetization of Ideal Linear Magnetic Materials
4.3 Diamagnetic Materials
4.4 Paramagnetic Materials
4.5 Ferromagnetic Materials
4.6 Energy Stored in Magnetic Materials
4.7 The Magnetic Circuit
References
5 The Maxwell Equations of Elecromagnetism
5.1 Introduction to the Maxwell Equations
5.2 The Gauss Law of Electric Fields
5.3 The Gauss Law of Magnetic Fields
5.4 The Faraday Law of Magnetic Fields
5.5 The Ampère–Maxwell Law
5.6 Work Done to Produce an Electromagnetic Field
5.7 Summary of the Maxwell Equations
Reference
Part II: Applications of Electromagnetism
6 Transmission Lines
6.1 Introduction
6.2 Ideal Transmission Lines and the Telegrapher's Equations
6.3 Waves on Ideal Transmission Lines
6.4 Waves on Lossy Transmission Lines
6.5 Characteristic Impedance
6.6 Loads on Transmission Lines
6.7 Connections Between Transmission Lines
6.8 Input Impedance of an Ideal Terminated Line
6.9 Ringing
Reference
7 Electromagnetic Waves in Dielectric Media
7.1 Introduction
7.2 The Maxwell Equations in Dielectrics
7.3 The Electromagnetic Wave Equation
7.4 Refractive Index and Dispersion
7.5 Properties of Monochromatic Plane Electromagnetic Waves
7.6 Intrinsic Impedance
7.7 Power and the Poynting Vector
Reference
8 Antennas
8.1 Introduction
8.2 Retarded Potentials
8.3 Short Dipole Antenna
8.4 Antenna Figures of Merit
8.5 Practical Antenna Systems
References
9 Electromagnetic Waves at Dielectric Interfaces
9.1 Introduction
9.2 Boundary Conditions
9.3 Angles of Reflection and Refraction
9.4 The Fresnel Equations
9.5 Polarization by Reflection
9.6 Anti‐reflection Coatings
Reference
10 Electromagnetic Waves in Conducting Media
10.1 The Maxwell Equations in Conducting Media
10.2 The Electromagnetic Wave Equation in Conducting Media
10.3 The Skin Effect
10.4 Intrinsic Impedance of Conducting Media
10.5 Electromagnetic Waves at Conducting Interfaces
11 Waveguides
11.1 Introduction
11.2 Rectangular Waveguides: Geometry and Fields
11.3 Rectangular Waveguides: TE Modes
11.4 Rectangular Waveguides: TM Modes
11.5 Rectangular Waveguides: Propagation of Modes
11.6 Rectangular Waveguides: Power, Impedance and Attenuation
11.7 Optical Fibres
12 Three‐Phase Electrical Power
12.1 Introduction
12.2 Synchronous Machines
12.3 Transformers
12.4 Asynchronous Machines
Epilogue
Index
End User License Agreement
List of Tables
Chapter 5
Table 5.1 Summary of the Maxwell equations.
Chapter 11
Table 11.1 The lowest eight cut‐off frequencies for modes in a rectangular w...
List of Illustrations
Chapter 1
Figure 1.1 Lines of electric flux around a point charge
+
q
in free space...
Figure 1.2 The electric field distribution around two point charges
+
q
(...
Figure 1.3 Lines of electric flux around a point charge
+
q
in free space...
Figure 1.4 An arbitrary surface divided up into small parallelograms. For on...
Figure 1.5 Lines of electric flux around a line of charge of
+
σ
per...
Figure 1.6 The electric fields
E
+
and
E
−
produced separately by ea...
Figure 1.7 A simplistic diagram of the effect of an externally applied elect...
Chapter 2
Figure 2.1 (a) A uniform external electric field
E
ext
is applied to a block ...
Figure 2.2 The electric field produced by a dipole of two charges in free sp...
Figure 2.3 A schematic diagram of the cross section of a MOSFET showing the ...
Figure 2.4 A schematic diagram of a parallel plate capacitor consisting of t...
Figure 2.5 The typical hysteresis curve for a ferroelectric material (solid ...
Figure 2.6 The effect of an applied external electric field
E
ext
on a block ...
Figure 2.7 Cross‐section of a coaxial cable with a potential difference
V
ap...
Figure 2.8 The electric field is distorted around a high aspect ratio struct...
Figure 2.9 Scanning electron microscopy image of an array of carbon nanotube...
Figure 2.10 Schematic diagram of a metallic wire of length
l
connected to a ...
Figure 2.11 The Kirchhoff Current Law states that the sum of all of the curr...
Figure 2.12 Schematic diagram of the metal oxide semiconductor capacitor str...
Chapter 3
Figure 3.1 The magnetic flux lines surrounding a positive point charge
q
1
mo...
Figure 3.2 An arbitrary line can be divided up into small vector line elemen...
Figure 3.3 The magnetic flux density
B
around a long, straight wire carrying...
Figure 3.4 A schematic diagram of the magnetic flux density
B
in a plane thr...
Figure 3.5 The Biot–Savart law allows the magnetic flux density
dB
to be cal...
Figure 3.6 The magnetic flux density
dB
produced by a small length
dl
of a c...
Figure 3.7 The radial magnetic flux density
B
x
produced by 1 A current loop ...
Figure 3.8 An electric field
E
is set up in a wire of length
L
moving with a...
Figure 3.9 There is no potential difference induced between the ends of a sq...
Figure 3.10 A rectangular loop of wire with side lengths
L
and
δx
movin...
Chapter 4
Figure 4.1 A magnetic material containing many small dipole moments
m
, each ...
Figure 4.2 A magnetic material with a surface magnetization current
J
M
insid...
Figure 4.3 A schematic diagram showing the magnetic moments in a ferromagnet...
Figure 4.4 (a) An example of an arrangement of magnetic domains in a ferroma...
Figure 4.5 Typical magnetization curves for (a) a soft ferromagnetic materia...
Figure 4.6 A coil of wire with
n
turns is wrapped around one part of a toroi...
Figure 4.7 A coil of wire wrapped around the centre of a magnetic material w...
Figure 4.8 The graph showing the relation between the magnetic field
H
and t...
Figure 4.9 A toroid of a hard ferromagnetic material that is permanently mag...
Figure 4.10 The solid line shows the relation between the magnetic field
H
a...
Figure 4.11 The magnetic flux lines passing round a horseshoe magnet attache...
Figure 4.12 A typical graph showing the approximate relation between the mag...
Chapter 5
Figure 5.1 A line of wire carrying a current
I
produces a magnetic flux dens...
Figure 5.2 The need for a displacement current term in the Ampè...
Chapter 6
Figure 6.1 Examples of transmission lines with indicative line lengths and s...
Figure 6.2 Equivalent circuit of a short length
δx
of a transmission li...
Figure 6.3 Schematic diagram of a microstrip line of width
w
and unit length...
Figure 6.4 A wave of the form
ψ = ψ0 cos(ωt − βx)
...
Figure 6.5 A schematic showing the radial electric and circulating magnetic ...
Figure 6.6 A wave on a slinky spring showing regions of local compression an...
Figure 6.7 Fluorescent tubes illuminated by harvesting energy from the elect...
Figure 6.8 The wavelength of an electromagnetic wave in air as a function of...
Figure 6.9 The equivalent circuit of the ‘lossy’ transmission line with the ...
Figure 6.10 A wave of the form
ψ = ψ0 cos(ωt − βx)e−αx
...
Figure 6.11 The definition of voltage on the transmission line is independen...
Figure 6.12 An infinitely long ideal transmission line with a load impedance...
Figure 6.13 The sum for the forward and backward wave plotted for different ...
Figure 6.14 The sum for the forward and backward wave plotted for different ...
Figure 6.15 The forward (dashed), backward (dot) and total (solid) wave plot...
Figure 6.16 The total wave plotted at intervals of
0.1
T
over an entire perio...
Figure 6.17 Two transmission lines are connected together at
x = 0
...
Figure 6.18 A transmission line of finite length
l
which is terminated with ...
Figure 6.19 The apparent impedance along a 61.24 Ω characteristic...
Figure 6.20 A transmission line of finite length
l
which is terminated with ...
Figure 6.21 Response with time at the input end of a 50 Ω transmission...
Chapter 7
Figure 7.1 The two individual waves
ψ
1
and
ψ
2
from Eq. (7.19) each...
Figure 7.2 Illustration of sound waves with a wavelength
λ
propagating ...
Figure 7.3 Representations of a linearly polarized, monochromatic, plane ele...
Figure 7.4 The instantaneous power per unit area given by
|N|
for the monoch...
Chapter 8
Figure 8.1 The coordinate system under consideration consists of an antenna ...
Figure 8.2 The dipole antenna of length
l
at the origin of a cylindrical pol...
Figure 8.3 Schematic of the direction of the electric and magnetic fields an...
Figure 8.4 The equivalent circuit of a simple system using antennas to trans...
Chapter 9
Figure 9.1 An incident monochromatic plane electromagnetic wave propagating ...
Figure 9.2 A two‐dimensional representation of a block of dielectric materia...
Figure 9.3 A three‐dimensional representation of the same situation as in Fi...
Figure 9.4 An incident monochromatic plane electromagnetic wave propagating ...
Figure 9.5 The two polarizations of the monochromatic plane electromagnetic ...
Figure 9.6 The ratio of the incident to reflected electric field for the par...
Figure 9.7 Schematic representation of the two possible polarizations of the...
Figure 9.8 The application of an anti‐reflection coating of a quarter‐wavele...
Chapter 10
Figure 10.1 The amplitude of the electric field for an electromagnetic wave ...
Figure 10.2 Schematic of an electromagnetic wave on a coaxial cable showing ...
Figure 10.3 Estimation of the resistance per unit length of a copper wire of...
Figure 10.4 Illustration of an electromagnetic wave propagating through a co...
Figure 10.5 Estimation of magnitude of the intrinsic impedance of copper as ...
Chapter 11
Figure 11.1 Rectangular waveguide geometry where the direction of propagatio...
Figure 11.2 The (left) electric and (right) magnetic fields in vector form f...
Figure 11.3 Schematic diagram of an optical fibre showing the key angles for...
Chapter 12
Figure 12.1 The power dissipated in a resistive load by a single‐phase power...
Figure 12.2 The voltages on each of the three phases of a three‐phase power ...
Figure 12.3 Delta (top) and star (bottom) connected three phase networks.
Figure 12.4 Schematic of single pole pair three‐phase synchronous machine sh...
Figure 12.5 A single pole pair synchronous machine simplified in terms of no...
Figure 12.6 (a) Equivalent circuit and (b) Phasor diagram for one phase of a...
Figure 12.7 Operating chart for the same state of the three‐phase synchronou...
Figure 12.8 Schematic of a simple transformer with a primary and secondary c...
Figure 12.9 Equivalent circuit for a non‐ideal transformer.
Figure 12.10 Schematic diagram of the induction motor (asynchronous machine)...
Figure 12.11 The equivalent circuit of the induction motor. In the centre is...
Figure 12.12 The equivalent circuit of the induction motor where the terms o...
Figure 12.13 The torque as a function of the ratio of the rotor angular velo...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Preface
Acknowledgements
About the Author
Symbols
About the Companion Website
Begin Reading
Index
End User License Agreement
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Electromagnetism for Engineers
Andrew J. Flewitt
Department of Engineering and Sidney Sussex CollegeUniversity of CambridgeUK
This edition first published 2023© 2023 John Wiley & Sons Ltd
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, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Andrew J. Flewitt to be identified as the author of this work has been asserted in accordance with law.
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Limit of Liability/Disclaimer of WarrantyWhile the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging‐in‐Publication Data is Applied for:
Hardback ISBN: 9781119406167
Cover Design: WileyCover Image: © Alexandr Gnezdilov Light Painting/Getty Images
For Hannah, Miriam, Tom and Rose
Preface
There are key evolutions in human technology that have had an impact upon everyday life. History recognizes these in terms such as the ‘iron age’ or the ‘industrial revolution’. It could be argued that electromagnetism lies behind many of the really transformative technological changes since the early nineteenth century, when electromagnetism emerged as a science from what otherwise appeared to be a set of apparently magical phenomena. A list of key technologies that are enabled by electromagnetism should include the electric motor, the power station, radio transmission, the transistor and optical fibres. We live in an ‘electromagnetic age’.
Electromagnetism is not only an important subject from a technological perspective, but also one of the most beautiful areas of physics. The Maxwell equations are very elegant mathematical expressions of basic properties of electric and magnetic fields, and, particularly in differential (vector calculus) form, they can be simply manipulated to explain a variety of physical electromagnetic phenomena.
The challenge that faces those trying to either teach or learn about electromagnetism is that it can be a rather abstract subject – an electron is not visible! This perception can create a mental barrier to understanding. However, many of the physical phenomena that flow from the fundamental principles are very familiar, from leaving a building to get a better mobile phone signal to knowing not to put your fingers between two magnets in case they snap together.
In this book, I have tried to take advantage of the power of mathematics to enable quantitative explanations of the physical phenomena, while also holding on to a physical reality, and I have included examples of real engineering that flows from these phenomena. My hope is that you will appreciate the wonder of electromagnetism both in the physics and especially in how we have been able to engineer this electromagnetic age.
Andrew J. Flewitt
Cambridge, UKApril 2022
Acknowledgements
This book has only been possible as a result of the support that I have received from both family and colleagues throughout the process of writing.
My wife Hannah has been wonderful throughout. She not only has been so understanding of the challenges of the writing process, but also went well outside her academic comfort zone to proof‐read the whole book too! And to all my children who have been so enthusiastic of the idea of me writing a ‘real book’.
Bill Milne gave me the confidence to embark on writing this book at the beginning of the whole process and kindly read the final version as well. Peter Davidson has also encouraged me of the value in writing this book as the few years that it has taken me have gone by.
My thanks too to all at Wiley for their patience and dedication in preparing this text.
Finally, I should thank all those who lectured me when I was reading physics as an undergraduate at Birmingham University. As I sat in the lecture theatres of the building named after John Henry Poynting, they brought electromagnetism to life and inspired my enjoyment of electromagnetism as a subject, which I hope I have managed to convey in some small measure in this book.
About the Author
Professor Andrew J. Flewitt read for a BSc in physics at the University of Birmingham before moving to the Engineering Department at Cambridge University where he gained his PhD. He was appointed a Fellow at Sidney Sussex College in 1999 and to a lectureship in the Engineering Department at Cambridge University in 2002, and has been teaching electromagnetism to students throughout this time, both in undergraduate college supervisions and in departmental lectures. His research group works on developing novel thin‐film materials for microelectromechanical systems and large‐area electronic devices. He was appointed Professor of Electronic Engineering in 2015 and Head of the Electrical Engineering Division in 2018.
Symbols
A
vector magnetic potential
A
area
dA
vector element of surface area
B
magnetic field as a vector quantity
B
magnitude of magnetic flux density
C
capacitance
c
wave velocity
D
electric flux density as a vector quantity
D
magnitude of electric flux density
d
ij
piezoelectric coefficients
E
electric field as a vector quantity
E
magnitude of electric field
e
magnitude of electronic charge
F
force as a vector quantity
F
magnitude of force
f
frequency
G
conductance
g
geometry factor
H
magnetic field as a vector quantity
H
magnitude of magnetic field
I
current
i
unit vector in the Cartesian
x
‐direction
j
unit vector in the Cartesian
y
‐direction
J
current density
I
l
line current
I
ph
phase current
j
k
unit vector in the Cartesian
z
‐direction
k
B
Boltzmann constant
L
angular momentum
L
inductance
M
magnetization
m
magnetic dipole moment
m
e
free mass of an electron
N
Poynting vector
P
polarization
P
real power
p
dipole moment
Q
charge
Q
reactive power
q
charge
ρ
resistivity
reluctance
R
resistance
r
position vector
dr
vector element of length
S
apparent power
T
temperature
T
time period
U
energy
V
potential difference
V
l
line voltage
V
ph
phase voltage
v
volume
Z
impedance
Z
g
waveguide impedance
Z
0
characteristic impedance
α
attenuation coefficient
β
propagation constant
β
g
propagation constant in a waveguide
γ
propagation constant
δ
skin depth
ε
0
permittivity of free space
ε
r
relative permittivity
η
intrinsic impedance
λ
wavelength
λ
g
wavelength in a waveguide
μ
0
permeability of free space
μ
r
relative permeability
ρ
volume charge density
ρ
L
reflection coefficient
ρ
T
transmission coefficient
σ
surface charge density
σ
P
surface polarization charge
Φ
total flux through a loop
cos
φ
power factor
χ
B
magnetic susceptibility
χ
E
electric susceptibility
ω
angular frequency
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/flewitt/electromagnetism
From the website you can find the following online materials:
Questions and Answers
Videos
Part IFundamentals of Electricity and Magnetism
1Charge and Electric Fields
1.1 Charge as a Fundamental Property of Matter
All matter is comprised of fundamental sub‐atomic particles which themselves have basic properties, such as mass, charge and spin. The sub‐atomic particle which electrical engineers are probably most concerned with is the electron, which was discovered by the Nobel Prize winning physicist J.J. Thomson in Cambridge in 1897 (Thomson 1897). Among its fundamental properties, electrons have a mass of 9.109×10–31 kg and a charge of –1.602×10–19 C, the magnitude of which we call e. Of these two properties, we are probably more familiar with the concept of mass as the world around us is dominated by gravitational forces at the macroscopic scale. We understand that two particles which have non‐zero mass both experience a force between them. We rationalize this by saying that a particle with mass produces a gravitational ‘field’ which extends spatially away from the particle. Another particle with a non‐zero mass inside this field then experiences a force.
As engineers, we do not tend to worry about the exact mechanism by which this force is being exerted over some distance, leaving that important consideration to physicists. Instead, we simply apply the equations for force between masses, such as that exerted by the earth on all structures in civil engineering. Therefore, we should be content to accept the less familiar concept of charge on the same basis: a particle with a non‐zero charge produces an electric field which extends spatially away from the particle, and another particle with a non‐zero charge inside this field then experiences a force. We use the symbol Q or q to denote charge, and the SI unit of charge is the coulomb (C).
1.2 Electric Field and Flux
Fields are widely used in physics to describe regions of space in which an object with a particular property experiences a force. Therefore, an electric field is a region of space in which an object with charge q experiences a force F. As force is a vector quantity, having both a magnitude and direction, electric field must also be a vector quantity E, so that
From this equation, it can be seen that the unit of electric field is N C–1, although, as we will see in Section 1.3, the more common unit is V m–1.
In everyday life we experience objects with both positive and negative charge, because while electrons have a charge of −e, protons, which are one of the sub‐atomic constituents of the nucleus of atoms, have a positive charge of e. Therefore, the force acting on a positive charge at a particular point in an electric field will act in the opposite direction to the force on a negative charge at the same point. This is the reason why like charges repel each other whereas opposite charges attract. This is in contrast to mass, which is positive for all matter, and therefore the force between masses is always attractive.
To assist us in visualizing fields, we use the concept of flux. We imagine that the electric field is composed of lines of flux whose direction at a given point in space is the direction of the electric field at that point and whose number density per unit area relates to the magnitude (or intensity) of the electric field.
We know that charge produces electric fields, and therefore lines of electric flux begin on positive charges and end on negative charges. As the total sum of all charge in the universe is zero, it must be the case that every line of flux that begins on a positive charge must have a balancing negative charge somewhere to end on. We can now visualize the electric field around a small point charge +q in free space (a vacuum) in Figure 1.1. If we assume that the balancing charge of −q is uniformly distributed an infinite distance away, then lines of electric flux will radiate uniformly away from the point charge. This will cause the field to decrease with increasing radial distance r from the point charge as 1/r2, just as we find for gravitational fields around mass as well. This is the basis behind the Coulomb law for the magnitude of the force F that acts on a charge q2 in an electric field of magnitude E1 produced by another point charge q1:
where r is the distance between the charges.
In Eq. (1.2) we have had to introduce a new quantity ε0, which is the permittivity of free space. It is a fundamental constant with the value 8.854×10–12 F m–1, and it is required to yield a result for the force between two charges that is correct in SI units.
Figure 1.1 Lines of electric flux around a point charge +q in free space assuming that the balancing charge is uniformly distributed an infinite distance away. The direction and area density of the flux lines are a measure of the electric field E at any point.
1.3 Electrical Potential
Let us imagine that we have two point charges of equal magnitude but opposite sign, +q and –q, which initially exist at the same point in free space, so that there is no electric field around the charges. If we slowly move the charge −q away from the +q charge so that the distance r between them is increasing, then an electric field distribution will be created around and between the charges as shown in Figure 1.2. As there will be a force acting on the charge that is being moved, given by Eq. (1.2), there must be some change in energy – work is being done against the force that is attracting the charges together. In this case, mechanical energy is being converted into an electrical potential energy, which could be converted back into mechanical energy again by allowing the two charges to accelerate back towards each other once more. It is a key concept that whenever a field (whether electric, magnetic or gravitational) occupies a volume of space, then some potential energy has been stored.
We can use basic mechanics to relate electric field to potential energy. If we have a charge q, in an electric field of magnitude E, which is moved by a small distance δx in the direction of the electric field, then there will be a change in potential energy of the charge given by
where F is the magnitude of the force acting on the charge due to the electric field. The change in potential energy is negative as the force acting on the charge is in the same direction as the movement. We define a new quantity, the potential difference, which is the change in energy per unit charge between two points in space. The potential difference is given the symbol V and has units of volts. Therefore, the small potential difference δV between the two points separated by the distance δx over which we have moved our charge q is
Equating δW in Eq. (1.3) and (1.4) gives
and therefore, by basic calculus, we have the result that
In other words, the electric field is the negative of the potential gradient. For readers who are familiar with vector calculus, we can rewrite this in three dimensions as
It should be noted that we can only ever talk about a potential difference between two points, for example the potential at a point a with respect to a point b which we could denote Vab. The direction is significant, as Vba = − Vab. We often use arrows to denote a potential difference where we are considering the potential at the tip of the arrow with respect to the tail. If we know the electric field distribution between the two points, then we can evaluate this. A simple integration of the two sides of Eq. (1.6) would suggest that
Figure 1.2 The electric field distribution around two point charges +q (right) and −q (left) separated by increasing distance from (a) to (c).
However, this is a slight simplification as we know that the electric field is actually a vector quantity, and it is therefore only the component of a small movement dx parallel to the direction of the electric field that will lead to a change in potential. If we use vectors to express both the electric field E and the small movement dx, then the scalar (dot) product of the two yields exactly this result. Therefore, Eq. (1.8) is more generally expressed as
In practice, Eq. (1.8) is more commonly used in simple calculations. This is achieved by ensuring that a path between two points is chosen so that E and dx are always either parallel or perpendicular to each other so that the scalar product is either a simple product or zero at any point.
This picture is very similar to the situation for gravity, where we can define a gravitational potential difference as being the difference in gravitational potential energy per unit mass between two points. In practice, we often take the surface of the earth as a reference point from which to measure gravitational potential as a function of height h above the surface of the earth which, assuming the gravitational constant g to be uniform, is simply gh. Likewise, it is helpful to take a constant reference point from which to measure electric potential difference. The earth again provides a good practical reference point, as it is so large that small changes in the charge on the earth have no significant impact upon its electrical potential energy. For other situations, such as the point charge in free space shown in Figure 1.1, the earth is simply not present, and so we have to choose an alternative reference from which to measure potential difference. In this example, an infinite distance away from the charge (r = ∞) provides a good reference point. We can therefore use Eq. (1.8) to calculate an expression for the potential difference with respect to this reference V(r) around the point charge in free space from the expression for the electric field around the charge in Eq. (1.2):
Note that we have been able to use the scalar Eq. (1.8) rather than the vector Eq. (1.9) by choosing a radial path which is parallel to the radial electric field. To help visualize potentials, we often plot lines of equipotential, rather as we plot contour lines on a map of constant height and also therefore constant gravitational potential, and this is shown for the point charge in Figure 1.3. As electric field points down a potential gradient according to Eqs. (1.6) and (1.7), lines of electric flux always cut perpendicularly through lines of equipotential.
It is clear from Eq. (1.10) that the potential difference with respect to r = ∞ at any point in free space around the point charge is uniquely defined. This is always true as we are dealing with a linear system. Therefore, it does not matter what path is actually chosen over which to evaluate the potential difference between two points using the integral in Eq. (1.9) – the result will always be the same. It is for the same reason that if you walk up a hill, the change in your gravitational potential energy will not depend on which path you take. Likewise, if we choose a closed path that ends back at its starting point, then there will be no potential difference between the start and end of the loop, which can be expressed mathematically from Eq. (1.9) as
Figure 1.3 Lines of electric flux around a point charge +q in free space with dashed lines of equipotential calculated with respect to r = ∞, with the graph of how potential varies as a function of distance r from the charge shown underneath on the same length scale.
where ∮C indicates the integral round a closed loop.
1.4 The Gauss Law of Electrostatics in Free Space
Our starting point for the discussion about charge as a fundamental property of matter in Section 1.1 was that charge produces an electric field. In the following discussion we then refined this to say that lines of electric flux begin on positive charges and end on negative charges. The Gauss law of electrostatics takes this basic statement about the origin and nature of electric fields and expresses it in a rather elegant mathematical formulation, namely,
where the left‐hand side is an integral over a closed Gaussian surface and q is the net charge enclosed by the surface. We will now consider the mathematical basis for this equation.
Let us imagine that we have an arbitrary surface in three dimensions, such as that in Figure 1.4. We could work out its surface area by laying it out onto a flat surface and measuring its area, but calculus and vector algebra allow us to take a more elegant approach. We could imagine dividing up the surface into lots of small elements of the surface, each of which approximates to a flat parallelogram, as shown in Figure 1.4. We define two vectors which point along adjacent edges of the small parallelogram; if we take the cross product of these two vectors then the result is a third vector which points perpendicularly away from the plane of the parallelogram and whose magnitude is equal to the area of the parallelogram. We call this vector dA. If we were to integrate the magnitude of all of the dA vectors over the whole surface, then we would have the whole surface area,
Figure 1.4 An arbitrary surface divided up into small parallelograms. For one parallelogram, the side vectors have been shown as small arrows and the resulting vector element of surface area dA is shown.
Let us now imagine that an electric field is passing through the surface. Taking the dot product of two vectors, such as a and b, gives the component of a in the direction of b multiplied by the magnitude of b, or, expressed mathematically,
where θ is the angle between a and b. Therefore, if we take the dot produce of E with any one of the small dA vectors, the result will be the component of E perpendicular to the surface multiplied by the area of the small surface element. Therefore, as |E| is the number of flux lines per unit area, E · dA is the number of flux lines passing perpendicularly through the small element of surface. We are taking a very similar approach to that used when calculating the force exerted by a ladder that is resting against a wall. For this mechanics problem we would resolve the force to calculate the component of force acting perpendicular to the surface of the wall. Here we are just resolving the electric field to take the component perpendicular to the surface.
We can therefore calculate the total number of electric flux lines passing perpendicularly through any surface by simply integrating E · dA over the surface,
Therefore, if the surface is closed (such as a box, sphere, or cylinder) then this integral would give the total flux passing out of the surface. We should note that the mathematical convention is that vector elements of surface area, dA, always point out of closed surfaces.
We should now be able to intuitively understand the meaning of the Gauss law of electrostatics as expressed in Eq. (1.12). We can call any closed surface that we can imagine in space a Gaussian surface. By taking the integral of E · dA over that surface, we are effectively counting flux lines. A line leaving the volume enclosed by the surface counts positively, while one entering counts negatively. As lines of electric field begin and end on charges, if there is no net charge enclosed within the Gaussian surface then the result of the integral must be zero. As many lines of flux leave through the Gaussian surface as enter. The result can only be non‐zero if a line of flux has either begun or ended on a charge that is somewhere in the volume enclosed by the Gaussian surface. This is exactly what Eq. (1.12) states: that ∮SE · dA is proportional to the net charge q enclosed by the surface. We only now need 1/ε0 as a constant of proportionality to ensure that the correct numerical values are produced.
1.5 Application of the Gauss Law
The Gauss law of electrostatics allows us to calculate the electric field produced by distributions of charge in space, and consequently the potential using Eqs. (1.6) or (1.7). The presence of the surface integral in Eq. (1.12) can make the Gauss law appear rather intimidating in the first instance. However, in practice we can often calculate the electric field produced by quite complex charge distributions through careful choice of the Gaussian surface. This may be achieved by ensuring that the electric field and vector elements of surface area are either parallel or perpendicular to each other, normally by reflecting the geometry of the charge distribution in that of the Gaussian surface.
For example, we can very easily derive the expression for the electric field around a point charge that is implicit in the Coulomb law (Eq. (1.2)). Intuitively, we would expect a point charge q to produce a spherically symmetric electric field, as shown in Figure 1.3. Therefore, we should choose a Gaussian surface that is a sphere of radius r centred on the charge, where the Gauss law will allow us to evaluate the electric field at that radius. As both the electric field E and the surface vectors dA both point radially outwards, the dot product of E and dA becomes just a simple product. Also, as can be clearly seen in Figure 1.3, the magnitude of the electric field E(r) is a constant at a given radius. Therefore, for this scenario using Eq. (1.13) we have
As the area of the sphere is 4πr2, Eq. (1.12) reduces simply to
which then rearranges to the familiar expression
We can also handle more complex charge distributions, such as an infinitely long line of charge with a charge per unit length of σ. In this case, the system has cylindrical symmetry, and so we should choose a Gaussian surface which is a cylinder axially centred on the line of charge, with some arbitrary length l and a radius r, as shown in Figure 1.5. In this case, the vector elements of surface area on the end caps of the cylinder will point in the direction of the axial line of symmetry of the system, whereas the electric field will point radially outwards. Hence, E · dA = 0, so these do not need to be considered further. Over the curved surface of the cylinder, E and the surface vectors dA both point radially outwards, and so we can use Eq. (1.16) again. The area of the curved surface of the cylinder is 2πrl, and therefore the Gauss law reduces to
where σl is the charge enclosed within the cylinder. This rearranges to
Figure 1.5 Lines of electric flux around a line of charge of +σ per unit length in free space. A cylindrical Gaussian surface of length l and radius r is shown using a dashed line. Two elements of surface area dA are shown: one on an end of the cylinder pointing axially outwards and one on the curved surface pointing radially outwards.
Therefore, although Eq. (1.12) may appear rather complex at first sight, involving surface integrals of vector quantities, in many cases the actual application of the Gauss law only requires scalar multiplication of fields and areas.
1.6 Principle of Superposition
We have seen that the Gauss law of electrostatics can be used to calculate the electric field around geometrically ‘simple’ distributions of charge. We can then use this as a basis for calculating the field around more complex distributions using the principle of superposition.
The principle of superposition in its most general form as it applies across diverse branches of physics states that if there is a linear relationship between some stimulus and a response, then the total response due to many stimuli acting simultaneously is the same as the sum of the responses that each stimulus would have produced individually. In the case here, the stimulus is the charge which produces a response in the form of an electric field, and the Gauss law in Eq. (1.12) clearly shows that these are proportional to each other (in other words, they are linearly related). Therefore, we can break down a complex charge distribution into many small elements of charge, calculate the electric field distribution due to each of the elements and then simply add all the contributions to the electric field at each point in space to yield the total field at that point.
We could have used superposition to calculate the electric field around the line of charge shown in Figure 1.5 by dividing the line up into infinitesimally small elements of charge and then integrating the resulting fields at any point to determine the total field, but the symmetry of this charge distribution makes the approach using the Gauss law directly a significantly simpler means of deriving Eq. (1.20).
1.7 Electric Dipoles
Two charges of equal magnitude but opposite sign separated by a small distance are called a dipole. The electric field distribution around the dipole of two point charges +q and –q in Figure 1.2 has been calculated using the Principle of Superposition. Each of the two charges would independently produce their own electric field distribution, one pointing radially outwards and the other radially inwards, as shown in Figure 1.6. If we imagine the plane passing through all points that are equidistant between these two charges, then the components of the electric fields due to the two charges pointing parallel to this plane will cancel out when superposed, leaving only the perpendicular components which sum, as is clearly the case in Figures 1.2 and 1.6. If the two charges are separated by a distance d then, from Eqn. (1.18), the electric field at the equidistant point along the line between the two charges due to each of the two charges independently will be
To calculate the total electric field at this point, we would need to sum Eqs. (1.21a) and (1.21b). However, the two expressions have been calculated in different frames of reference: one pointing away from the positive charge and one pointing away from the negative charge. We have to use a common frame of reference when summing the two fields. In the frame of reference used for the positive charge, the electric field due to the negative charge at the equidistant point is −E−, so the total field is
Therefore, the electric field at this equidistant point decreases as the distance between the dipole increases, and this can be seen in the density of the flux lines around this point shown in Figure 1.2, which are densest in Figure 1.2a where the dipole separation is smallest. At long distances away from the dipole (much greater than d) the two charges appear to be at the same point in space, and their electric fields will cancel out to leave no net field, and so the electric field is effectively concentrated in the region of space close to the two charges. This effect is also most pronounced in Figure 1.2a.
Figure 1.6 The electric fields E+ and E− produced separately by each of the point charges in a dipole and the resultant total electric field calculated by superposition. The field passes perpendicularly through the plane of equidistance between the two charges.
In practice, dipoles frequently appear in real situations as most objects have no net charge, and so the act of removing some charge by some distance from a particular object immediately creates a dipole between the removed charge and the original object. At the atomic scale, when an atom experiences an externally applied electric field, the electrons around the nucleus will be displaced from their equilibrium position, as shown in Figure 1.7. Just as we can think of a distributed mass acting from a ‘centre of mass’ point, so we can think of the electrons around the nucleus as acting from a ‘centre of charge’. The action of the electric field is therefore to move this effective centre of charge by a vector displacement d from the positive charge, resulting in the formation of a dipole.
Figure 1.7 A simplistic diagram of the effect of an externally applied electric field E
