A Treatise on the Magnetic Vector Potential
Kristján Óttar Klausen
Faculty Faculty of of Physical Physical Sciences Sciences University University of of Iceland Iceland 2018 2018
A TREATISE ON THE MAGNETIC VECTOR POTENTIAL
Kristján Óttar Klausen
60 ECTS thesis submitted in partial fulfillment of a Magister Scientiarum degree in Physics Education
Supervisor Viðar Guðmundsson Faculty Representative Ari Ólafsson M.Sc. committee Halldór Pálsson
Faculty of Physical Sciences School of Engineering and Natural Sciences University of Iceland Reykjavik, January 2018
A Treatise on the Magnetic Vector Potential 60 ECTS thesis submitted in partial fulfillment of a M.Sc. degree in Physics Education c 2018 Kristján Óttar Klausen Copyright All rights reserved Contact:
[email protected] Faculty of Physical Sciences School of Engineering and Natural Sciences University of Iceland Dunhagi 5 107 Reykjavik Iceland Telephone: 525 4700
Bibliographic information: Kristján Óttar Klausen. (2018). A Treatise on the Magnetic Vector Potential. M.Sc. thesis. Faculty of Physical Sciences, University of Iceland. Reykjavík: Háskólaprent.
Printing: Háskólaprent, Fálkagata 2, 107 Reykjavík Reykjavik, Iceland, January 2018
To the essence of both student and teacher; the spark of curiosity igniting the wonderful guiding force of inspiration.
Abstract Electromagnetism is a branch of physics underlying many modern developments, in both science and everyday life. The unification of the electric and magnetic fields is fundamental in our understanding of light as electromagnetic waves. Another background field lies at the heart of the unification, the magnetic vector potential field. The idea emerged in the early days of research in electromagnetism, was dismissed for more than half a century but taken up again in the formulation of quantum electrodynamics. The magnetic vector potential is a pivotal concept with ties to many aspects of physics and mathematics. Here we review the development of the concept, inquire into its nature along with calculating and plotting the field for two fundamental current distributions. We further visualize the behavior of the field using the analogy with fluid mechanics. The aim is to give the reader an intuitive understanding of the magnetic vector potential.
Útdráttur Rafsegulfræði er undirgrein eðlisfræði sem nútíma framþróun, bæði í vísindum og daglegu lífi, grundvallast á. Skilningur okkar mannanna á eðli ljóss felst í samþættingu rafsviðs og segulsviðs í rafsegulbylgjur. Annað svið liggur að baki samþættingunni sem nefnist vigurmætti segulsviðsins. Hugmyndina um tilvist þess má rekja til upphafsdaga rannsókna í rafsegulfræði. Hún var lögð á hilluna í hálfa öld en tekin upp að nýju í þróun skammtarafsegulfræði. Vigurmætti segulsviðsins snertir á fjölmörgum þáttum eðlis- og stærðfræði. Hér förum við yfir þróun og myndbirtingu hugtaksins ásamt því að velta fyrir okkur eðli fyrirbærisins. Mættið er reiknað út og teiknað upp fyrir tvær algengustu straumdreifingarnar. Hegðun sviðsins er loks könnuð nánar með hliðsjón af samlíkingu við vökvaaflfræði. Markmiðið er að veita lesandanum djúpsettan skilning á vigurmætti segulsviðsins.
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Contents List of Figures
ix
Legend
xiii
Acknowledgments
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1 Introduction 2 Conceptual emergence 2.1 Electromagnetic induction . . . . . . 2.1.1 The electrotonic state . . . . 2.2 Electrokinetic momentum . . . . . . 2.2.1 Vector potential . . . . . . . . 2.2.2 Molecular vortices . . . . . . . 2.3 Elimination for practical purposes . . 2.4 Gauge-invariance . . . . . . . . . . . 2.5 Comeback in quantized formulation . 2.5.1 The Aharonov-Bohm effect . . 2.6 In the formulation of modern physics
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3 Mathematical appearance 3.1 Electromagnetism . . . . . . . . . . . . . . . . 3.2 Potential formulation . . . . . . . . . . . . . . 3.3 Spacetime four-vector notation . . . . . . . . . 3.4 Variational approach . . . . . . . . . . . . . . 3.5 Gauge invariance in quantum electrodynamics 3.5.1 Local gauge covariance . . . . . . . . .
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4 Three dimensional solutions 4.1 Line current . . . . . . . . . . . . . . . . . . . 4.2 Method of solution for the current loop . . . . 4.2.1 The Dirac Delta distribution . . . . . . 4.2.2 Green’s function . . . . . . . . . . . . 4.2.3 Eigenfunction Expansion . . . . . . . . 4.2.4 Bilinear expansion of Green’s function 4.3 Current loop . . . . . . . . . . . . . . . . . . .
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Contents 5 Discussion 5.1 Application in telecommunications . . . . . . . . . . . . . . . . . . 5.2 Hydrodynamics analogy . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Vorticity, Acceleration and Induction . . . . . . . . . . . . . 5.2.2 The Navier-Stokes equation . . . . . . . . . . . . . . . . . . 5.2.3 The Continuity equation and Incompressible flow . . . . . . 5.2.4 Irrotational flow and the velocity potential . . . . . . . . . . 5.2.5 Table of comparison . . . . . . . . . . . . . . . . . . . . . . 5.3 Population inversion of two dimensional vortices in a finite domain . 5.3.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Negative absolute temperature . . . . . . . . . . . . . . . . . 5.3.3 Onsager’s system of parallel vortices in a plane . . . . . . . . 5.3.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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47 47 50 50 53 55 56 57 57 58 59 60 62 64
6 Conclusions
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Bibliography
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List of Figures 1.1
Amber and lodestone with magnetized sand. . . . . . . . . . . . . . .
1
2.1
A compass needle will be deflected by a lightning strike in its vicinity.
3
2.2
Iron filaments revealing the magnetic field around a bar magnet. . . .
5
2.3
A current carrying wire wound into a helical form (b) will induce a magnetic field similar to a bar magnet (a). The arrows point to the arbitrarily defined direction of the magnetic field. . . . . . . . . . . .
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Faraday’s law : Moving a magnet perpendicular to the plane of a circular conductor induces an electric current within the conducting loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A coil (2) is connected to a battery (3) and has a current flowing. Moving it perpendicular (up/down) relative to the winding of the larger coil (1) will induce a current in the latter which is detected by the galvanometer (4). . . . . . . . . . . . . . . . . . . . . . . . . . .
6
A humming top (Geupel, 2011). Pressing the handle downwards makes the humming top spin. The mechanism is semi-analogous to that of electromagnetic induction, with the handle referring to the magnet and the rotation of the tin body to the induced rotational current. Credit for the analogy goes the author’s father. . . . . . . . . . . . .
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A neuron seen through an electron microscope: The spread of electrical activity through living tissue or cells is referred to as an electrotonic current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.8
Michael Faraday in his later years and young James Clerk Maxwell. .
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2.9
A magnetic field vector, B, (S-N) arises from a rotational magnetic vector potential, A. Refined original image from Maxwell (1861). . . 12
2.4
2.5
2.6
2.7
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LIST OF FIGURES 2.10 The Faraday effect: A phase shift (β) of the plane of vibration of the electric field component (E) in an electromagnetic wave, passing through a constant magnetic field (B) parallel to the direction of motion. From Silva et al. (2012). . . . . . . . . . . . . . . . . . . . . 14 2.11 Mechanism of the tea leaf paradox (Einstein, 1926). . . . . . . . . . . 14 2.12 A changing current flowing in the path A-B will induce an opposite current in the path p-q, due to changing rotational velocities of the vortices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.13 (left) Magnetic field around a current carrying conductor. (right) Plane view of the magnetic vector potential for a current element. . 16 2.14 Sketch of the magnetic vector potential due to a linear current. . . . . 16 2.15 Experimental setup for demonstrating phase interference due to a time-independent vector potential field. From Aharonov & Bohm (1959). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.16 Circular magnetic vector potential field A around a coil with current density J giving rise to a magnetic field B within it. From Feynman et al. (1963). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.17 Phase difference of wavefronts (black) in water (white) moving left to right passing a clockwise rotating vortex. From Berry et al. (1980). . 21
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3.1
The electromagnetic spectrum. From Philip Ronan (2013). . . . . . . 23
4.1
Line current in the z-direction. . . . . . . . . . . . . . . . . . . . . . . 35
4.2
Magnetic vector potential magnitude for a line current. Equipotential lines projected on to the xy-plane highlight the magnetic field lines as in figure 2.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3
Loop current density J of radius a in the xy-plane. . . . . . . . . . . 40
4.4
Magnetic vector potential magnitude in the xy-plane, current top-view. 44
4.5
Magnetic vector potential magnitude in the xz-plane, current side-view. 44
LIST OF FIGURES 4.6
Value of the magnetic vector potential field strength in the xz-plane. Side-view of the current loop with equipotential lines. . . . . . . . . . 45
5.1
Diagrams of a Josephson junction receiver (left) and dual toroidal coil radiator (right) from H. E. Puthoff (1998). . . . . . . . . . . . . . . . 47
5.2
Sketch of the vector potential field (A) of a toroidal current (J), arrows indicate direction of circulation. The magnetic field (B) points into the page inside the right cross section of the toroid and out of the page on the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3
Block diagram of the system from Nikolova & Zimmerman (2012). . . 49
5.4
Rotation in the field of c with a spatially varying magnitude. The force term − 2q ∇ × c would point into the plane of the paper, parallel to either the velocity of the magnetic vector potential, depending on initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5
Energy-Entropy graph of a limited system (left) corresponding EnergyTemperature graph (right). From Francis (2013). . . . . . . . . . . . 60
5.6
Two examples of long-lived large scale vortices. Left: Storm maturing off the east coast of the US. From NOAA (2018), lifetime in days. Right: Giant red spot on Jupiter as seen by Voyager (NASA/JPL, 2011), lifetime in centuries. . . . . . . . . . . . . . . . . . . . . . . . . 61
5.7
Clustering of vortices after the point of population inversion (dashed vertical line) with an emerging dipolar structure. From Yatsuyanagi (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.8
Population inversion of nuclear spins in a external magnetic field (B) leading to negative absolute spin temperatures. From Vrtnik (2005). . 63
Where references are not cited for images, they are in the public domain or made by the author.
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Nomenclature Vector quantities are symbolized either with arrows over-lined or with boldfaced font depending on context and/or convention therein. Dimensional analysis here assumes all phenomena can be written out in terms of the fundamental quantities of mass (M), length (L), time (T) and charge (Q). Symbol
Concept
SI-Units
Dimension
~ A
Magnetic vector potential
N·s/C or V·s/m
ML/TQ
~ B
Magnetic field
T=N·s/C·m
M/TQ
~ E
Electric field
N/C or V/m
ML/T2 Q
q
Charge
C
Q
p~
Momentum
N·s
ML/T
J~
Current density
A/m2
Q/TL2
φ
Electrostatic (scalar) potential
V=J/C
ML2 /T2 Q
E
Electromotive force
V
ML2 /T2 Q
F~
Force
N
ML/T2
f
Frequency
Hz
1/T
~v
Velocity
m/s
L/T
ω ~
Vorticity
1/s
1/T
∇
Differential operator
1/m
1/L
d`
Line element
m
L
dS
Surface element
m2
L2
dV
Volume element
m3
L3
ε0
Electric vacuum permittivity
C/V·m
T2 Q2 /ML3
µ0
Magnetic vacuum permeability
T·m/C·s
ML/Q2
c
Speed of light
m/s
L/T
~
Planck constant
kg · m2 /s
ML2 /T
χ
Gauge/phase function
J·s/C
ML2 /TQ
ψ
Wave function
m−3/2
1/L 2
3
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Acknowledgments The writing of this thesis was made possible by professor Viðar Guðmundsson who gladly took on the responsibility of mentoring and supporting me in multiple ways. My deepest gratitude and respect to you Viðar for always being open to discussion of both elementary and controversial aspects of physics, humble and firmly grounded in the laws of nature and mathematics. A warm thank you to those who proofread the thesis: Professor Ari Ólafsson, Professor Halldór Pálsson, my father Ingólfur Klausen, Professor Jón Tómas Guðmundsson, my friend Sigtryggur Hauksson and Dr. Þorsteinn Kristinn Óskarsson. Alongside my B.Sc. studies in geophysics I worked night-shifts at Iceland’s Meteorology office during a couple of summers. There I got a bit of time for independent research which I am grateful for as well as the many night long conversations I got to have with some enthusiastic meteorologists skilled in vector analysis. I am aware of the privilege of being able to access centuries of work in physics through the internet. However with every year the web gains increasing amounts of low value information or flat out dis-info, proving the value of classical textbooks with well known facts. Nonetheless one cannot help feeling that with the bird’s eye view over science made possible by the world wide web, something which was hidden in plain sight will come to light. That was what pushed me forth in the exploration of the age old concept of the magnetic vector potential. I would like to thank one of my high school physics teachers and dear friend Einar Már Júlíusson who encouraged me to pursue physics and set a true example of what good teaching is by never showing the slightest disbelief in students. His view was that every question deserved a thorough and honest response. He would often devote his coffee and lunch brakes to our exploration of fundamental questions in electromagnetism, relativity and cosmology - never settling for less than wonder. Thank you mom and dad for housing me during my time of studies and bearing with my many flaws and faults. Thank you heavenly father for patience, Iceland’s national university library and for all the people behind LaTeX typesetting with its wonderful packages and syntax. Finally, I would like to thank every future reader for his time and interest. I hope the reading will prove illuminating and enjoyable.
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1 Introduction The phenomena of electricity and magnetism must have been know to man from very early on since both manifest directly in nature. While electricity is most obvious in the form of lighting, history suggest the first experiments of the Greeks with electricity used amber, fossilized tree resin which can accumulate static charge when rubbed with wool or fur. In fact the Greek word for amber is ¯elektron from which the word electricity is derived (Brockman, 1926).
Figure 1.1: Amber and lodestone with magnetized sand. Magnetism can also be found in the mineral kingdom. Magnetite minerals in basaltic rock can be magnetized when struck by lightning. Fragments from the rock will thus become natural magnets referred to as loadstones. Needles of loadstone were used for navigation as the first compasses. The Greek word for loadstone is magnítis líthos meaning stone from the Tekin region in modern Turkey named Magnesia in ancient times, from which the word magnet is derived (Wasilewski & Kletetschka, 1999). Sparse research was done on the phenomena of electricity and magnetism up until the 17th century when William Gilbert argued that the earth itself is a magnet. We will not elaborate on this early research but begin our story at the time when the connection between electricity and magnetism became understood, in the 19th century, thus beginning the science of electromagnetism. As we shall see, at the core of the unification lie subtler fields of energy and momentum giving rise to all electromagnetic interactions.
1
In the following chapter we explore the emergence of this connection in near chronological order with the development of physics. From there we move on to the mathematical descriptions of the story in chapter three. In chapter four we calculate and plot the magnetic vector potential field for two common current distributions. Chapter five covers patented applications of the field along with exploring the analogy with fluid dynamics. Conclusions are presented in chapter six. The question in mind is “What is the nature of the field uniting electricity and magnetism?”
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2 Conceptual emergence The first hint of a link between electricity and magnetism might have been the observation that a compass needle is deflected by a lightning strike close by. This effect was experimentally observed by the Danish physicist Oersted during a demonstration in his own lecture April 21st the year 1820. When an electric current runs in a wire a compass needle in the vicinity will move. This observation is said to have been by accident, like often is the case in scientific endeavor (American Physical Society, 2008).
Figure 2.1: A compass needle will be deflected by a lightning strike in its vicinity. Oersted’s discovery ignited a bright spark of interest throughout the scientific community at the time, inspiring many to pursue research into the effect. The French mathematician and physicist Ampère found that a force can also exist between two current carrying conductors and presented a mathematical formulation of the law. Ampère remained a key contributor to electromagnetism in its early days, greatly influencing later research. Another Frenchman, the polymath Francois Arago observed that a current carrying wire wound into helical form would function similar to a bar magnet, thus discovering the electromagnet (Assis & Chaib, 2015).
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The most extensive research and experimentation on electromagnetism to this day was however done by an Englishman named Michael Faraday. Born in London the year 1791 into a relatively poor family, Faraday received only basic education and left school to begin working as an errand boy for a bookshop at the age of 13. Impressing his employer with hard work and focus, Faraday was promoted to an apprentice bookbinder. Being in the bookshop gave him access to many books, including books on science which intrigued him the most. Fascination with the entries on electricity and chemistry in the The Encyclopedia Britannica prompted him to start his own experiments in the shop’s backroom to validate the scientific concepts he read about (Thomas, 1991). A successful teacher of music, William Dance, noticed Faraday’s scientific interest and gave him an entry ticket to attend a lecture by the then world-renowned scientist Sir Humphry Davy at The Royal Institution. Faraday took down detailed notes, illustrating and adding explanations. He then bound the 300 pages of notes into a book and sent to Davy, without any expectations. The same year his seven year apprenticeship ended and Faraday began working with another bookbinder. Around the same time Davy had an accident which damaged his sight and ability to write. Remembering the accurate notes from Faraday, Davy hired him as an assistant and together they toured Europe the next two years. Upon their return in 1814 they carried on Davy’s work at The Royal Institution. Faraday was then 23 years old and now had access to one of the worlds top theoretical and experimental science facilities (Faraday, 1991). His association with The Royal Institution lasted 54 years with Faraday becoming a lecturer and a professor of chemistry. Faraday’s work on electromagnetism began in 1820 when he heard from colleagues at the Royal Institution about Oersted’s discovery of electric current influencing magnetic needles. His interest in electromagnetism was further ignited with Arago’s discovery of the electromagnet. To visualize the effect a magnet has on its surrounding space, Faraday sprinkled iron filaments on and around it thus revealing a structure as seen in Figure 2.2. From this experiment he coined the term field referring to the invisible web of force surrounding a magnet, the magnetic field. The reader should note the definition of the magnetic field lines in terms of an equilibrium state of iron filaments around a bar magnet for later reference, where we shall see that the magnetic field lines highlight axes of rotation of electromagnetic momentum.
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Figure 2.2: Iron filaments revealing the magnetic field around a bar magnet. The electromagnet, as seen in Figure 2.3, established that magnetism can originate from electricity. From there the next step in unification would be to investigate the reverse statement and one of great value: if electricity could be derived from magnetism. It was Faraday who successfully showed that was indeed possible. Even though Faraday’s work on electromagnetism was put on hold between the years 1825-1830 while working for Davy, Faraday persisted and succeeded in making an electric current flow from magnetic influence in the year 1830 (Thomas, 1991).
Figure 2.3: A current carrying wire wound into a helical form (b) will induce a magnetic field similar to a bar magnet (a). The arrows point to the arbitrarily defined direction of the magnetic field.
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2.1 Electromagnetic induction
Figure 2.4: Faraday’s law: Moving a magnet perpendicular to the plane of a circular conductor induces an electric current within the conducting loop. By moving a magnet in the vicinity of a loop of conducting wire, a current will flow within the wire. Figure 2.4 illustrates what is today known as Faraday’s law of induction. When asked about the significance of his discovery, Faraday replied along the lines “What good is a new born baby? ”. Faraday’s law of electromagnetic induction is now implemented in the majority of modern day electrical devices and can be said to have sparked the theory of relativity. The need for the wire to be in a loop comes from the necessity of closing the circuit. There is also a relation between loop windings and the magnetic field as seen in the electromagnet. The bar magnet can be replaced with an electromagnet, making inductance occur between two coils, see Figure 2.5.
Figure 2.5: A coil (2) is connected to a battery (3) and has a current flowing. Moving it perpendicular (up/down) relative to the winding of the larger coil (1) will induce a current in the latter which is detected by the galvanometer (4).
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Two aspects of the phenomenon stand out: 1. No current will flow unless there is a relative movement of the magnet and the loop, so that the loop experiences a change in the magnetic field. 2. The strongest current is induced with perpendicular movement of the magnet to the plane of the wire loop. Because of the first point, Faraday noted that a wire seems to be in a state of tension when in the vicinity of a magnet. When the magnet moves, the current flows as a result of the changing tension. This observation in conjunction with the second point led Faraday to believe there was another more subtle state or condition of matter underlying the magnetic field he termed the electrotonic state. He envisioned this field to be parallel to the wires, thus explaining the perpendicular interaction between the movement of the magnet and the current induced in the wire. The mechanism of electromagnetic induction can be likened to that of a humming top, see Figure 2.6. Pressing the handle downwards makes the humming top spin due to screw grooves in the handle. The analogy is that a rotational movement (the current) results from a force applied perpendicular to the plane of rotation. The electrotonic state in this context would correspond to the screw grooves, giving rise to the spinning when set in motion.
Figure 2.6: A humming top (Geupel, 2011). Pressing the handle downwards makes the humming top spin. The mechanism is semi-analogous to that of electromagnetic induction, with the handle referring to the magnet and the rotation of the tin body to the induced rotational current. Credit for the analogy goes the author’s father.
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2.1.1 The electrotonic state Through various experimental setups Faraday made many observations about this “peculiar state of matter” (Faraday, 1839). In his own words, the electrotonic state: • Can be taken on by all metals. • Shows no known electrical effects whilst it continues. • Does not imply force of neither attraction or repulsion. • Appears to be instantly assumed. • Appears to be a state of tension. • May be considered as equivalent to a current of electricity. Despite many attempts, Faraday did not manage to measure any direct force reaction (attraction/repulsion) from the electrotonic state alone. He concluded that his observations favored the electrotonic state to be a property of the particles of matter themselves under induction. If so, the state might be present in liquids and even non-conductors (Faraday, 1839). Other scientist as well as Faraday himself were quick to recognize the similarity of the phenomena of induction and the behavior of nerves, which are in a state of tension or readiness similar to the electrotonic state. The word quickly became a standard in the biological vocabulary of the 19th century and has persisted to this day.
Figure 2.7: A neuron seen through an electron microscope: The spread of electrical activity through living tissue or cells is referred to as an electrotonic current.
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Faraday (1839) abandoned the concept of the electrotonic state after a year of experimentation, finding the magnetic field lines sufficient to explain the phenomena of induction. He however felt the need for it again three years later, writing: [...] still there appears to be a link in the chain of effects, a wheel in the physical mechanism of the action, as yet unrecognized. If we endeavor to consider electricity and magnetism as the result of two forces of a physical agent, or a peculiar condition of matter, exerted in determinate directions perpendicular to each other, then, it appears to me, that we must consider these two states or forces as convertible into each other [...] by a process or change of condition at present unknown to us.
The struggle with the concept persisted, never being directly able to measure the electrotonic state but nonetheless seeing the need for it, especially in regards to the necessary movement for induction, since “mere motion would not generate a relation, which had not a foundation in the existence of some previous state ”. Toward the end of his investigations Faraday (1844) remarked that he firmly believed in the physical existence of the magnetic field lines and the underlying electrotonic state, which could be seen as a magnetic analog to static electricity. In concluding the review of Faraday’s legacy and foundation for electromagnetism, the story continues with his successor, James Clark Maxwell, who was handed Faraday’s experimental notes in 1855 at the age of 24 from the author himself at the end of his career, then 64 years of age (Israelsen, 2014). Recently graduated from Cambridge University’s Trinity College, Maxwell was trained in mathematics unlike Faraday. The two developed a friendship and kept in good contact, discussing for example how to express physical and mathematical truths in ordinary language (Bence, 1870).
Figure 2.8: Michael Faraday in his later years and young James Clerk Maxwell.
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2.2 Electrokinetic momentum Starting from the data of Faraday’s observations Maxwell penetrated the logical structure in the natural laws of electricity, magnetism and the electrotonic state. He equated the electrotonic state to a mathematical quantity which “may even be called the fundamental quantity in the theory of electromagnetism” (Maxwell, 1873). ~ in multiple ways. It constitutes the momentum Maxwell approached this quantity, A, in a circuit, termed the electrokinetic momentum ˆ ~ · d`, p= A (2.1) C
where p is the electrokinetic momentum and d` an element of the circuit C. Maxwell found that the inductive- or electromotive force, E, arising from electromagnetic induction was the rate of decrease of the electrokinetic momentum of the circuit dp E =− . (2.2) dt From the above equations it is evident that the electromotive force stems from the ~ or change in the quantity A ˛ d ~ · d` E =− A (2.3) dt which is a mathematical formulation of Faraday’s law in terms of the electrotonic ~ The loop integral coincides with the conductor being a closed loop. The state A. electomotive-force is the force per charge in the conductor, in modern terms a rotational electric field ˛ ~ · d`. E= E (2.4) Together Equations (2.3) and (2.4) give ~ ~ = − dA E dt
(2.5)
meaning that an electric field will be induced with a time varying electrotonic state, either in magnitude and/or direction. The relationship to momentum can also be seen with a classical analog since the electric field is equivalent to force per charge ~ ~ =F E q
(2.6)
and force is the time derivative of momentum d~p F~ = − dt
10
(2.7)
substituting (2.6) and (2.7) together into (2.5) reveals that ~ = p~ A q
(2.8)
thus the electrotonic state can be understood as momentum per charge. By differentiating (2.1) with respect to time and taking into consideration the motion of the circuit itself, Maxwell obtained a general expression for the induction of an ~ and electromotive force, with ~v denoting the particle velocity in the magnetic field B φ denoting the electrostatic potential: ~ ~ = ~v × B ~ − dA − ∇φ. E dt
(2.9)
Maxwell (1873) recognized a fundamental difference in the nature of the vectors ~ and B ~ for the magnetic vector potential and magnetic field respectively. By A dimensional analysis with the fundamental units of length (L), mass (M), time (T) and charge (Q) in SI, starting out with the dimensions of energy, U , 2 LM (2.10) [U ] = T2 Maxwell found the dimension of the electrokinetic momentum of a circuit element to be 2 ` LM [~pek ] = p~ · = . (2.11) q QT ~ and B ~ can then be written out as The dimensions of A LM p~ek [A] = = , qT L M p~ek [B] = = . qT L2
(2.12)
(2.13)
The electrotonic state has dimensions of electrokinetic momentum per length but the magnetic field has dimensions of electrokinetic momentum per area, thus belonging to the category of fluxes. In the next section we explore the relationship between the electrotonic state and the magnetic field uncovered by Maxwell. Let us keep in mind that the dimension of the differential operator ∇ is inverse length or 1/L. From now on in this thesis, the electrotonic state will be referred to as the magnetic vector potential or, for short, the vector potential.
11
2.2.1 Vector potential Maxwell showed that the magnetic induction in Faraday’s law, see Figure 2.4, depended on the conducting wire loop and not on the surface enclosed by it. Therefore it should be possible to determine the induction by a quantity residing within the ~ related to the magnetic field, wire itself. That could be done by finding a vector, A, ~ in such a way that the line integral of A ~ would be equal to the surface integral of B, ~ B, or in modern notation ˛ ¨ A · d` = B · dS (2.14) where d` and dS denote the line- and surface elements respectively. Invoking Green’s theorem (see the Appendix) one obtains ˛ ¨ A · d` = ∇ × A · dS (2.15) therefore revealing the relation B=∇×A
(2.16)
meaning that A is the vector potential for the magnetic field B. This mathematical relationship is often denoted B = rot(A) or B = curl(A) since the geometrical interpretation of the equation is that the vector B is equal to a rotational flow of the field A. This geometrical interpretation can be seen in Figure 2.9 below.
Figure 2.9: A magnetic field vector, B, (S-N) arises from a rotational magnetic vector potential, A. Refined original image from Maxwell (1861). Furthermore, Maxwell noted that one can add the gradient of a scalar function χ to the vector potential without changing the value of the magnetic field B, since ∇ × (A + ∇χ) = ∇ × A + ∇ × ∇χ = ∇ × A
(2.17)
due to the fact that the curl of a gradient of any scalar function f is zero, ∇ × ∇f = 0.
12
(2.18)
Maxwell (1873) stated that for a given distribution of electric currents there is ~ where A ~ is everywhere finite and one, and only one, distribution of the values of A continuous, satisfying the equation ~ = −4πµJ, ~ ∇2 A
(2.19)
the operator ∇2 being the Laplacian operator, interpreted as concentration by Maxwell, µ being the magnetic permeability and J~ the total electric current density. One supplementary condition was necessary though, known as the solenoidal condition in vector analysis, ~=0 ∇·A (2.20) ~ is solenoidal and thus divergence free. The value of the stating that the field A magnetic vector potential would then be ˚ ~=µ A V
J~ dV. r
(2.21)
This expression was put forth by Maxwell in quaternion form without direct derivation in his treatise (1873). We will analyze it further in Chapter 4, solving for the vector potential of two fundamental current distributions and plotting the results. One of Maxwell’s great insights was equating the magnetic vector potential with Faraday’s electrotonic state, describing the result of Equation (2.14): The entire electrotonic intensity round the boundary of an element of surface measures the quantity of magnetic induction which passes through that surface, or, in other words, the number of lines of magnetic force which pass through that surface (Maxwell, 1873).
Another key insight was equating and comparing the following ratio to the speed of light in vacuum, 1 c= √ (2.22) ε0 µ 0 where ε0 is the electric permittivity and µ0 the magnetic permeability of vacuum. From there on Maxwell argued that light can be understood as an electro-magnetic wave with “the vacuum and the electromagnetic medium being one the same” (Maxwell, 1873). The electromagnetic medium being the substance giving rise to the various phenomena of electricity and magnetism through deformation of it, stress, tension, pressure and density gradients. This viewpoint is further supported by the hydrodynamics analogy analyzed in Chapter 5.2.
13
2.2.2 Molecular vortices
Figure 2.10: The Faraday effect: A phase shift (β) of the plane of vibration of the electric field component (E) in an electromagnetic wave, passing through a constant magnetic field (B) parallel to the direction of motion. From Silva et al. (2012). In order to explain the Faraday effect (see Figure 2.10) and Fardays’s law, Maxwell turned to a mechanical model comprising of vortices in the medium, inspired by the magnetic vector potential. The main hypothesis of the model was that the rotation would be of very small elements of the medium. Observing the magnetic field lines drawn out by iron filaments around a magnet, Figure 2.2, Maxwell (1861) noted that the lines “indicate the direction of minimum pressure at every point of the medium”, the difference in pressure being caused by molecular vortices with their axis parallel to the magnetic field lines, as depicted by Figure 2.9. The iron filaments in Figure 2.2 can thus be seen as being at rest within the vortex center, analogous to the stillness in the eye of a storm. To get a feel for this, the reader can refer to the common experience of stirring a cup of clear tea whereby particles in the fluid such as tea leaves will collect in the center of the vortex formed by stirring. Known as the tea leaf paradox, explained by Einstein (1926) being due to velocity differences at the top and bottom of the cup. The analogy is not exact since the cup forms boundary conditions different than in the case of the magnetic field lines. Nonetheless in both instances the particles collect at the center of a vortex.
Figure 2.11: Mechanism of the tea leaf paradox (Einstein, 1926).
14
Figure 2.12 shows the original model of Maxwell (1861) for explaining mutual induction. Hexagonal cells signify vortices, the hexagonal shape being solely to simplify graphic representation. Between the vortices are hypothesized particles, representing electricity, capable of spin as well. Plus and minus symbols in the vortex centers signify both handedness of rotation and the direction of the magnetic vector normal to the plane of observation, clockwise (-) being into the plane and counter-clockwise (+) pointing outwards, in accordance with the arbitrary directional convention of the curl in Equation (2.16).
Figure 2.12: A changing current flowing in the path A-B will induce an opposite current in the path p-q, due to changing rotational velocities of the vortices. When current flows from A to B, rotational momentum will be formed on either side of the current with opposite orientation of spin, see also Figure 2.13. With the row of vortices g-h being set in motion, the row k-l row above will start rotating, inducing a current in the opposite direction to the original one in the path q-p. By applying the model to static electricity Maxwell was led to his famous correction to Ampére’s law by adding the displacement current. The mechanical model also seems to have been a crucial step for Maxwell to the realization that light is a transverse wave of magnetic and electric oscillations (Yang, 2014). The model highlights the fact that Maxwell’s line of thought in approaching electromagnetism was greatly influenced by the concept of the magnetic vector potential, Faraday’s electrotonic state, understood as vacuum vortices with rotational momentum, having the magnetic field lines as their axes of rotation.
15
~ B
~ ∇×A
J~
J~
Figure 2.13: (left) Magnetic field around a current carrying conductor. (right) Plane view of the magnetic vector potential for a current element. The mechanical model gives an explanation of the rotational magnetic field formed around a current carrying conductor, see left image in Figure 2.13, which Faraday had observed with iron filings and was a well established phenomena at the time. The flow of the current J~ sets in motion vortices with opposite spin direction, see right image in Figure 2.13 which give rise to the magnetic field. Like the magnetic field, the geometrical shape of the magnetic vector potential has rotational symmetry around the conductor. This implies a spinning torus shape called a toroidal or ring vortex, see Figure 2.14, known from flow in liquids (Lamb, 1895), the smoke ring being the most familiar example. A natural question to ask in this context is what substance is set in motion by the flow of the current? For the majority of scientist in the 19th century it was the aether, an all pervading fluid like substance, the vibrations of which were the cause of electromagnetic vibrations. It was dismissed in the favor of an empty vacuum in the beginning of the 20th century. Still, the vacuum has properties of magnetic permeability and electric permittivity, both a magnet and a charge keep their qualities in empty space. In the final words of his treatise on electricity and magnetism vol. 2, Maxwell (1873) suggests future research should be aimed at the properties of the electromagnetic medium itself. In concluding the review of Maxwell’s work on the magnetic vector potential, one must bear in mind that at the time of his writings only primitive atomic models had been put forth, the electron was yet to be discovered and the spin property of particles was to be considered more than half a century later.
Figure 2.14: Sketch of the magnetic vector potential due to a linear current.
16
2.3 Elimination for practical purposes Even though Maxwell had successfully unified the magnetic and electric field, ~ and E, ~ via the magnetic vector potential A, ~ no practical benefit was evident B from the unification. In fact, Maxwell himself later put forth the electromagnetic theory of light without the potentials. The physicists Heaviside og Hertz, went a step further and felt the potentials did not even have to be mentioned due to their elusive nature and lack of direct measurability and eliminated them completely from electromagnetic theory in the beginning of the 20th century (Wu & Yang, 2006). This downplay of the importance of potentials can still be seen in some modern textbooks on electromagnetism. It’s worth noting that the reformulation of the equation of electrodynamics goes hand in hand with a crisis in mathematics at the same time. Mathematical description of dynamics in three dimensions was in development with two promising systems, Hamilton’s quaternion analysis and Grassman’s system of multivectors. Modern vector analysis can be seen as a simplified and reduced mixture of the two systems. For example, the convention of writing ~i, ~j, ~k as the vector basis in cartesian coordinates comes from the quaternionic notation, where i, j and k are all complex numbers. This has to be kept in mind when reading Maxwell’s papers. Also, the divergence and curl with their differential notation of ∇· and ∇× have their roots in quaternion analysis (Crowe, 1967). William Clifford united the two systems at the end of the 19th century and put forth a complete vector algebra for all dimensions known as Clifford algebra. Due to his early death the system was quickly forgotten but has resurfaced in later years and proven to be a most excellent mathematical language for electromagnetism along with other fields of physics, the reader can refer to the Appendix for a short introduction to Clifford’s geometric algebra.
2.4 Gauge-invariance One of the main reasons for the viewpoint that the potentials have no physical significance is the freedom in their definition. An addition of a scalar or gradient field to the electrostatic or magnetic vector potential respectively, will result in the same electromagnetic field. Changing the potentials in this way is referred to as a gauge transformation and implies a certain symmetry of the theory called gauge symmetry or gauge-invariance. The magnetic field will be unaltered by the following transformation, as before in Equation (2.17), ~→A ~0 = A ~ + ∇χ. A
(2.23)
17
In order that the electric field will be the same, the electrostatic potential φ, will have to be transformed as well with φ → φ0 = φ −
1 ∂χ c ∂t
(2.24)
where χ is a nonsingular smoothly differentiable scalar function. This hints at an ~ and φ, which will intrinsic spacial/temporal relationship between the potentials A be clarified in Chapter 3.3.
2.5 Comeback in quantized formulation In quantum mechanics, probability amplitudes replace particle trajectories in the equations of motion. Instead of working with forces, interactions are described in terms of momentum and energy. The force fields of the electric and magnetic fields were therefore not readily worked with nor quantized. However, the electromagnetic ~ and φ, could be directly incorporated into the framework of quantum potentials, A ~ as the elecmechanics in order to implement electromagnetic interactions, with A tromagnetic momentum and φ as the energy. Thus the importance of potentials was revived (Feynman et al., 1963). The Hamiltonian for purely electromagnetic interactions takes the following form (Griffiths, 1995) ˆ = 1 (ˆ H p − qA)2 + qφ 2m ∂ ˆ then becomes and Schrödinger’s equation, i~ ∂t ψ = Hψ ∂ 1 2 i~ ψ = (ˆ p − qA) + qφ ψ. ∂t 2m
(2.25)
(2.26)
The soviet physicist Vladimir Fock found that for the quantum dynamics of the electromagnetic interactions of charged particles to be invariant for the gauge transformations, the wave function ψ would have to be transformed as well with the following rotation or phase transformation, implying a relationship between the phase of the wave function and the electromagnetic potentials: iq 0 ψ → ψ = ψ · exp χ (2.27) ~ where i is the unit complex number, q is the charge, ~ the reduced Planck constant and χ is the phase/gauge function (Jackson & Okun, 2001).
18
2.5.1 The Aharonov-Bohm effect
Figure 2.15: Experimental setup for demonstrating phase interference due to a time-independent vector potential field. From Aharonov & Bohm (1959). The relationship between the electromagnetic potentials and the phase of the wave function was further hypothesized by Aharonov & Bohm (1959) where they proposed an experimental setup for interference, see Figure 2.15. A beam of electrons is deflected on either side of a metal foil, in front af a magnetic solenoid. The magnetic field is mainly concentrated in the center of the solenoid and can be made effectively zero outside of it. The interference at point F will depend on the phase difference between paths AB and AC even though the magnetic field strength along the paths is effectively zero. The magnetic vector potential in the region is however non-zero as depicted in Figure 2.16.
Figure 2.16: Circular magnetic vector potential field A around a coil with current density J giving rise to a magnetic field B within it. From Feynman et al. (1963).
19
Still referring to Figure 2.15, the phase for each beam will be ˆ q θAB = A · d` ~ AB ˆ q θAC = A · d` ~ AC since by the gradient theorem (see the Appendix) ˆ b ˆ b A · d`. ∇χ · d` = χ(b) − χ(a) = a
(2.28) (2.29)
(2.30)
a
Integrating along the closed path ACFB one can write the relative phase difference as a closed path integral ˛ q ∆θ = A · d`. (2.31) ~ Even though the magnetic field is zero along both paths there is nonetheless an electromagnetic influence since the Hamiltonian includes the magnetic vector potential, as before in Equation (2.25). The closed path integral of the magnetic vector potential is equal to the magnetic flux of the solenoid by Equation (2.16) and Green’s theorem; ˛ ¨ ¨ A · d` = ∇ × A · dS = B · dS = ΦB (2.32) with ΦB symbolizing the magnetic flux through the area enclosed by the line integral. Changing the current in the solenoid, and therefore the magnetic flux, should then shift the interference pattern. This is termed the magnetic1 Aharanov-Bohm effect. It was first experimentally observed by Chambers (1960) but the result received criticism since the magnetic field was not completely zero outside the region. Tonomura (1986) shielded the electron beams completely from the magnetic field using toroidal ferromagnets shielded with both a superconducting layer and a copper layer but still observed the effect, thus basing it on a solid experimental foundation. The shift of the interference pattern could be due to non-local interaction of the magnetic field, however, that conflicts with the relativity principle. In order that the interaction be a local one, the physicality of the vector potential becomes a requirement (Aharonov & Bohm, 1961). Despite this argument and validation from experiment, interpretation of the effect remains controversial to this day with many claiming that it is a purely classical phenomena or that the potentials can be eliminated all together. Interestingly Berry et al. (1980) proposed an analogous system with wave fronts passing a vortex in water, reminiscent of Maxwell’s model, thus demonstrating the phase change in a classical setting, see Figure 2.17. 1
Aharonov & Bohm (1959) also proposed an similar experimental setup showing the interference effect of the electric scalar potential in a region with an electric field free region. The relative phase change is then given by a time integral of the potential differences.
20
Figure 2.17: Phase difference of wavefronts (black) in water (white) moving left to right passing a clockwise rotating vortex. From Berry et al. (1980).
2.6 In the formulation of modern physics Invariance under a gauge transformation as in Equation (2.23) or a phase transformation such as in (2.27) sparked a new approach to theories in physics. By insisting on invariance under a specific transformation with a certain symmetry, physical interactions could be uncovered. These transformations became known as gauge transformations and the corresponding theories gauge theories. Thus began a new paradigm in theoretical physics of symmetry dictating interactions (Yang, 2014). The seminal paper of Emmy Noether (1918) on the connection between symmetries and conserved quantities had set the stage for this line of thought. For example the conservation of charge stems from global phase symmetry (Brading, 2002). From being understood as the electrotonic state to the magnetic vector potential, the understanding further evolved to the concept of a gauge field. Gauge theories incorporate the mathematics of group theory, already fully developed before its application, with gauge fields being described by the mathematical concept of a connection on a fiber bundle. The gauge theory of electromagnetic interactions rests on the symmetry of the phase transformation in (2.27) or U(1) symmetry groups. By exploring more complex symmetries, in 1954 Yang & Mills came up with a gauge theory of the strong nuclear force interaction with the SU(2) symmetry group and later others extended gauge symmetry to SU(3) symmetry, resulting in quantum chromo-dynamics. In 1998 a team from Cambridge university presented a gauge theory of gravity using Clifford’s geometric algebra (Lasenby et al., 2004). Furthermore the Standard Model, describing all known elementary particles, can be described by the internal symmetries of the group product U(1)×SU(2)×SU(3). The Higgs mechanism, based on spontaneous symmetry-breaking of the gauge field, is needed in order to incorporate the masses of particles (Griffiths, 1987). In this way, the magnetic vector potential can be said to continue to play a key role in the development of physics to this day as the generalized concept of a gauge field.
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3 Mathematical appearance We now venture to explore the various roles taken on by the magnetic vector potential in the mathematical description of light and electromagnetic interactions in general.
3.1 Electromagnetism As the name suggests, the theory of electromagnetism covers the sources, interactions and applications of electric and magnetic fields as well as electromagnetic waves. Visible light makes up only a little part of the frequency spectrum, the band most abundant in solar radiation, which our eyes are well equipped to receive and interpret.
Figure 3.1: The electromagnetic spectrum. From Philip Ronan (2013).
The classical mathematical description of electromagnetic waves consist of four well known equations known as Maxwell’s equations, which arguably are more directly from Heaviside and Hertz, as covered in Chapter 2.3. However, as is the case with most human advances, they are the result of a combined effort of many.
23
The differential formulation of the equations in vacuum in SI-units is ∂B ∇×E=− ∂t ∇ × B = µ 0 J + µ 0 ε0 ∇·E=
∂E ∂t
(3.1) (3.2)
ρ ε0
(3.3)
∇·B=0
(3.4) (3.5)
In plain words the equations state the following relations: i. A rotational electric field will be induced by a time varying magnetic field. ii. A rotational magnetic field will be induced by a current density J and/or a time varying electric field. iii. A source for the electric field is a charge density, ρ. iv. The magnetic field is without sources or in other words divergence-free. Without sources, ρ = 0 and J = 0, the equations become symmetric with respect to interchangeable roles of the magnetic and electric field. Thus it becomes evident that one field will induce the other and wave propagation is implied. The integral formulation holds more information about the relative dimensions of its components, with N signifying an arbitrary three dimensional volume with the boundary ∂N and M an arbitrary two dimensional surface with the boundary ∂M . ¨ ˛ d B · dS (3.6) E · d` = − dt M ∂M ˛
¨ B · d` = µ0
∂M
"
1 E · dS = ε0 ∂N
d J · dS + µ0 ε0 dt M
¨ E · dS
(3.7)
M
˚ ρ dV
(3.8)
N
" B · dS = 0
(3.9)
∂N
The equivalence of the differential and integral formulations becomes clear by using the boundary theorem in the appropriate dimension, see the Appendix.
24
Only when the integral manifolds N and M are fixed can the time derivative be brought under the integral. In case of a time varying manifold one has to use the so called Leibniz Integral theorem, also known as Reynold’s transport theorem in fluid dynamics. For a time varying vector field F(r, t), ˛ ¨ ¨ d ∂ F + [∇ · F] v · dS − [v × F] · d` (3.10) F · dS = dt M (t) ∂t ∂M (t) M (t) where v is the velocity of the contour ∂M . Applying this to Equation (3.6) along with ∇ · B = 0 results in ˛ ¨ ˛ ∂ 0 E · d` = − B · dS + [v × B] · d` (3.11) ∂M (t) M (t) ∂t ∂M (t) denoting the induced electric field with E0 . From Equation (3.1) a time varying magnetic field by itself will induce a rotational electric field so ¨ ˛ ¨ ∂B · dS = ∇ × E · dS = E · d` (3.12) − M (t) ∂M (t) M (t) ∂t Inserting Equation (3.12) into Equation (3.11) and writing the closed loop integral of the electric field on the left hand side of (3.11) in terms of force per charge we obtain ˛ ˛ ˛ F/q · d` = E · d` + [v × B] · d`. (3.13) ∂M (t)
∂M (t)
∂M (t)
Rearranging the charge q, we uncover the Lorentz force law F = q (E + v × B)
(3.14)
which is often said to be a necessary supplement in order to make the equations complete (Cheng, 1983). The above analysis however shows that the force law is embedded in the time varying integral and corresponding surface.
3.2 Potential formulation Starting with Maxwell’s two original equations relating the magnetic and electric field to the potentials, ∂A E = −∇φ − (3.15) ∂t B=∇×A (3.16) let us see if we can recover Equations (3.1) to (3.4) above. By taking the curl of Equation (3.15) we recover Equation (3.1) ∇×E=−
∂ ∂B ∇×A=− ∂t ∂t
(3.17)
25
since ∇×(∇f ) = 0 for any scalar field f and the space and time derivatives commute. Taking the curl of Equation (3.16) gives ∇×B=∇×∇×A
(3.18)
for an arbitrary vector or scalar field f , we have the vector identity, ∇ × (∇ × f ) = ∇(∇ · f ) − ∇2 f
(3.19)
∇ × B = −∇2 A + ∇(∇ · A).
(3.20)
therefore Comparing with Equation (3.2), restated here for clarity: ∇ × B = µ0 J + µ0 ε 0
∂E ∂t
we find that ∇2 A = −µ0 J
(3.21)
the same as Equation (2.19), which Maxwell himself had uncovered (Maxwell, 1873). The second term, ∂E µ 0 ε0 = ∇(∇ · A) (3.22) ∂t depends on the value of ∇ · A, or the so called gauge. The Coulomb gauge, ∇ · A = 0 is thus not in accordance with Equation (3.2). However with the Lorenz gauge, ∇·A=−
1 ∂φ c2 ∂t
(3.23)
Equation (3.22) adds up; ∂E 1 ∂φ ∂E 1 ∂ µ 0 ε0 =∇ − 2 = 2 (−∇φ) = µ0 ε0 ∂t c ∂t c ∂t ∂t
(3.24)
since c−2 = µ0 ε0 and E = −∇φ (in case of a time-invariant vector potential). In order to have Equation (3.2) consistent with the Coulomb gauge, we then have to add to the current density in equation (3.21) the term (Schwinger et al., 1998) 1 ∂ (∇φ) = JL c2 ∂t
(3.25)
which, as Jackson (1999) notes, is the irrotational or longitudinal part of the current with ∇ × JL = 0. The value of the divergence of A is therefore not arbitrary, having implications for the term ∂E/∂t, known as the displacement current, which was famously added by Maxwell (1861) to Ampère’s law, allowing him to derive the electromagnetic wave equations (Yang, 2014). Continuing with our analysis, we have yet to recover Equations (3.3) and (3.4) describing the divergence of the electric and magnetic field. The results that the
26
magnetic field is divergence free is directly implied in Equation (3.16) since the divergence of a pure rotational field is zero, ∇ · B = ∇ · (∇ × A) = 0. For the electric field, taking the divergence of (3.15) gives ∂A 2 ∇ · E = −∇ φ − ∇ · ∂t
(3.26)
(3.27)
keeping in mind that the divergence of a gradient is the same as the Laplacian operator ∇2 = ∇ · ∇. Assuming a static vector potential we then recover Equation (3.3), since ρ ∇2 φ = − . (3.28) ε0 If we were not to assume a static vector potential and using the Lorentz gauge as above, rearranging derivatives, Equation (3.27) becomes ∂ 1 ∂φ (3.29) ∇ · E = ∇ · (−∇φ) − ∂t c2 ∂t and in the light of Equation (3.3) we then have a wave equation for the electric scalar potential 1 ∂ 2φ ρ ∇2 φ − 2 2 = − . (3.30) c ∂t ε0 We can also obtain a wave equation for the magnetic vector potential in a similar manner, putting together Equations (3.20) and (3.2) gives ∂E . (3.31) ∂t Replacing E with the potentials in Equation (3.15), rearranging and again using that c−2 = µ0 ε0 for a more classical form 1 ∂ 2A 1 ∂φ 2 ∇ A− 2 2 −∇ ∇·A+ 2 = −µ0 J (3.32) c ∂t c ∂t − ∇2 A + ∇(∇ · A) = µ0 J + µ0 ε0
where the Lorentz gauge cancels the third term on the left hand side leaving 1 ∂ 2A = −µ0 J c2 ∂t2 which has the general solution (Jackson, 1999) ˚ ˆ 0 0 µ0 1 3 0 0 J(r , t ) 0 0 A(r, t) = dr dt δ t + |r − r | − t . 4π |r − r0 | c V ∇2 A −
(3.33)
(3.34)
We have thus seen how the Lorentz gauge seems to be a natural expression of the divergence of the magnetic vector potential, as in Equation (3.23) which, along with the wave equations for the potentials (3.30) and (3.32), forms a complete description of the classical electromagnetic field (Griffiths, 1999).
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3.3 Spacetime four-vector notation By working within in four dimensional spacetime, Equations (3.30) and (3.33) combine into a single equation, describing the behavior of the electromagnet field: Aα = µ0 J α
(3.35)
where the electric scalar potential φ and the magnetic vector potential A are joined together in electromagnetic four-potential, α ∈ 0, 1, 2, 3 with A0 = φ/c φ α ,A (3.36) A = c and similarly the charge and current density form the four-current J α = (cρ, J).
(3.37)
The superscript refers to the fact the the four-vectors can be seen as rank one contravariant tensors, obtained by a projection onto the manifold with the metric, as opposed to the dual covariant vectors. The time and space derivatives come together in the operator 2 known as the d’Alembertian ≡
1 ∂2 − ∇2 . c2 ∂t2
(3.38)
The four-potential, four-current and the d’Almbertian all turn out to be Lorentz invariant, meaning that they stay the same in reference frames moving relative to each other. Underlying is the relativity postulate that the speed of light is a finite constant with the same value for all reference frames (Thidé, 2004). One interesting aspect of the four-vector formulation is that the electric scalar potential becomes the time component of the four-potential whereas the magnetic vector potential is the space component. Recalling that the magnetic vector potential signifies momentum per charge in space, the electric scalar potential can then be seen as momentum in time. In fact, this might be generalized to say that energy is momentum in time. The potential energy of a static charge can then be seen to stem from its movement at the speed of light through the dimension of time. In the same way a stationary charge density can be viewed as a current through time. Another interesting feature of the spacetime formulation is that both the Lorentz gauge condition from Equation (3.23) and the continuity equation become simple statements of both quantities being divergent-less in spacetime (Thidé, 2004). Defining the covariant derivative operator 1∂ ∂α = ,∇ (3.39) c ∂t the divergence of the four-potential becomes the invariant ∂α Aα =
28
1 ∂φ + ∇ · A = 0. c2 ∂t
(3.40)
In the same way the continuity equation or the law of charge conservation ∂ρ +∇·J=0 ∂t
(3.41)
∂α J α = 0.
(3.42)
can be written as To see more clearly how Equation (3.41) describes the conservation of charge, lets rewrite Equation (3.41) with the volume integrals ˚ ˚ ∂ρ dV. (3.43) ∇ · J dV = − ∂t Invoking the divergence theorem gives " ˚ ∂ρ J · dS = − dV ∂t
(3.44)
stating that any change of charge density in time within a volume will result in a equal and opposite current density over the boundary of the volume. Since Equations (3.41) and (3.40) are identical in structure, the same goes for the magnetic vector potential, meaning that for any change in time of an electric potential within a volume will result in a flux of magnetic vector potential (electrokinetic momentum) over the boundary, " ˚ 1 ∂φ A · dS = − 2 dV. (3.45) c ∂t Equation (3.45) holds only in the Lorentz gauge. Given that the scalar potential represents energy per charge and the magnetic vector potential the momentum per charge, as supported by dimensional analysis, Equation (3.45) would state that a change of energy within a volume will result in a flux of momentum through the boundary of the volume. An arbitrary choice of the value divergence of the magnetic vector potential or gauge, would likewise mean an arbitrary choice of the sources of charge momentum. Therefore we stress the argument that the gauge conditions have physical implications and that the Lorentz gauge is a natural one.
3.4 Variational approach Calculus of variations allows for finding equations of motion by determining the minimal extrema of the action integral ˆ t2 S=δ L dt = 0 (3.46) t1
29
where L (qi , q˙i , t) = T − U
(3.47)
is the well known Lagrangian density in terms of generalized coordianates qi , with the dotted q symbolizing the time derivative, equal to the difference between kinetic and potential energies T and U respectively. The equations of motion come from solving the Euler-Lagrange equation ∂L d ∂L − = 0. (3.48) dt ∂ q˙i ∂qi Electromagnetic interactions are dictated by the potentials with the Lagrangian LEM = qv · A − qφ,
(3.49)
the total Lagrangian for a charged particle in an electromagnetic field has an added term from the free particle (Landau & Liftshitz, 1971) 1 LTot = mv2 + qv · A − qφ. 2
(3.50)
By formulating the Lorentz force law in terms of potentials the form of the Lagrangian can be found and vice versa, as solving the Euler-Lagrange equation results in the Lorentz force law. The canonical or general momentum, p is given by p=
∂L ∂L = = mv + qA ∂ q˙i ∂v
(3.51)
in agreement with Equation (2.8). The Hamiltonian can be found with a Legendre transformation H=p·v−L (3.52) with the Lagrangian from Equation (3.50) resulting in 1 H = mv2 + qφ. 2
(3.53)
Rewriting Equation (3.51) shows that v=
1 (p − qA) m
(3.54)
plugging back into Equation (3.53) gives the Hamiltonian H=
1 (p − qA)2 + qφ 2m
same as referred to in Equation (2.25).
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(3.55)
3.5 Gauge invariance in quantum electrodynamics Cohen-Tannoudji et al. (1989) note that inserting the gauge transformations ∂χ ∂t 0 A → A = A + ∇χ, φ → φ0 = φ −
(3.56) (3.57)
which leave the electric and magnetic fields invariant, into the Lagrangian in (3.50) gives an extra term of the arbitrary gauge function χ(r, t). Omitting sums over indices for an ensemble of particles; 1 d L = mv2 + qv · A − qφ + qχ 2 dt
(3.58)
with the added total time derivative dχ ∂χ = + v · ∇χ dt ∂t
(3.59)
and thus making no change to the equations of motion. This is essentially a translation that amounts to the unitary transformation iq T = exp χ (3.60) ~ which is the same phase factor as mentioned in Equation (2.27). The observable´s O, before the gauge transformation symbolized with O1 and after it with O2 , have the relation ˆ1 = T O ˆ2 T † O (3.61) where † denotes the conjugate transpose, whereas the state vectors relate directly with the same transformation |ψi1 = T |ψi2 . (3.62) The transformation in Equation (3.60) is then the link between different formulations of quantum electrodynamics with its roots in the gauge invariance of the potentials. Aitchison & Hey (2003) put forth an trivial yet interesting observation, writing the wave function in parts ψ = ψR + i ψI (3.63) one can take a closer look at the effect of the global phase transformation, by Euler’s formula ψ 0 = eiα ψ = (cos α + i sin α) (ψR + i ψI ) (3.64) with α being the argument of (3.60) as above. Therefore, ψR0 + i ψI0 = ψR cos α − ψI sin α + i (ψR sin α + ψI cos α)
(3.65)
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and so the transformed parts of the wave function are given by ψR0 = ψR cos α − ψI sin α ψI0 = ψR sin α + ψI cos α
(3.66) (3.67)
which can be seen as a rotation in the internal space of the wave function, in the same way that a rotation in the complex plane by an angle θ is described with z 0 → eiθ z. With the phase transformation the angle corresponds to ~q χ, a quanta of charge in a sense, however one might question how far the analogy can be stretched or given physical meaning. Observing that T T † = eiα · e−iα = cos2 (α) + sin2 (α) = 1
(3.68)
the phase transformation is said to be unitary, with the set of all such transformations forming a group of U(1) symmetry, corresponding to the rotational symmetry of a circle. Furthermore the group is abelian meaning that applying two consecutive phase transformations will give the same result, independent of the order, since eiα eiβ ψ = ei(α+β) ψ = eiβ eiα ψ.
(3.69)
The transformations eiα and eiβ are then said to commute, with the commutator relation (3.70) [eiα , eiβ ] = eiα eiβ − eiβ eiα = 0. This symmetry extends to both the Pauli equation and Dirac equation, with the former being the non-relativistic limit of the latter (Feynman, 1961). The Pauli equation adds to the Schrödinger equation a description of electromagnetic interactions for spin 12 particles, fermions in general and the photon in particular; ∂ψ 1 = [σ · (p − qA)]2 ψ + qφψ (3.71) ∂t 2m where σ = (σ1 , σ2 , σ3 ) are the Pauli matrices describing the spin. As a side-note, the algebra of the Pauli matrices is isomorphic to the Clifford algebra C`(3) (Hestenes, 2015), yet another case of mathematics being reinvented for explanation of physical behavior. Jackson (1999) notes that for a quantum-mechanical description of the photon only the magnetic vector potential has to be quantized, further supporting the viewpoint that the vector potential is the root cause of electromagnetic interactions. i~
In order that Equation (3.71) be invariant for the phase transformation iq 0 χ ψ = eiα ψ (3.72) ψ → ψ = exp ~ we need to make the following changes to the electromagnetic potentials, same as before A → A0 = A + ∇χ ∂χ φ → φ0 = φ − , ∂t
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(3.73) (3.74)
since
∂ψ 0 ∂α iα ∂ψ ∂χ iα iα ∂ψ i~ = i~ e +i e ψ = eiα i~ − iq e ψ ∂t ∂t ∂t ∂t ∂t
(3.75)
and ˆ eiα ψ = −i~∇(eiα ψ) = iq∇χ · eiα ψ − eiα i~∇ψ = (ˆ p p + iq∇χ) eiα ψ.
(3.76)
Feynman (1961) pointed out that the magnetic vector potential enters into the Hamiltonian for the Pauli equation above as a perturbation potential for transitions between states. This could be a clue to the ill-known mechanism for photon emission and absorption, with momentum exchange being governed by the vector potential in process similar to electromagnetic induction.
3.5.1 Local gauge covariance To approach gauge invariance from the opposite point of view is to start with a description for a free particle, impose gauge invariance and make necessary corrections. For simplicty we use Schrödinger’s equation throughout this section, analogous analysis can be applied on both the Pauli and Dirac equation. Schrödinger’s equation for a free particle reads ∂ψ ~ 2 i =− ∇ ψ. (3.77) ∂t 2m By a global phase transformation ψ → ψ 0 = eiα ψ with α being a constant, it is trivial to show that the equation is globally phase invariant. If however α is a function of local coordinates, the equation is not invariant for the transformation since one obtains extra terms of derivatives of the phase function α as shown before in Equations (3.75) and (3.76). We can however make the equation locally covariant by introducing the gauge covariant derivative Dµ = ∂µ + iqAµ
(3.78)
∂µ = (∂t , −∇).
(3.79)
with the defined four-gradient
Keeping in mind that ∂0 = ∂t and A0 = φ, with the substitution ∂µ → Dµ , Equation (3.77) then becomes ∂ 1 i − iqφ ψ = (∇ − iqA)2 ψ. (3.80) ∂t 2m
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Here we have Schrödinger’s equation for a particle in an electromagnetic field, with the same Hamiltonian as in Equation (3.55). Rewriting with the momentum operator p = −i~∇
(3.81)
resulting in i~
∂ψ ˆ = Hψ ∂t
(3.82)
where
ˆ = 1 (p − qA)2 + qφ. H 2m Imposing the local phase transformation iq 0 ψ → ψ = exp χ(x, t) ψ ~
(3.83)
(3.84)
the symmtery holds only if the fields A and φ transform in the familiar manner φ → φ0 = φ −
∂χ ∂t
A → A0 = A + ∇χ. By starting out with an equation for a free particle and imposing U(1) phase/gauge symmetry we have uncovered the fundamental interaction of electromagnetism. The boson following the symmetry is the photon described by the magnetic vector potential A, with charge being the single corresponding quantum number. As Aitchison & Hey (2003) note, the covariance is manifested in the observationally equal effect of changing the phase locally and the influence of the field A in which the particle moves. The particle is no longer free however, since we have introduced the interaction with the potentials. This relation between the phase and matter fields is known in the literature as minimal coupling. From this line of thought one could try imposing more complex symmetries and see if that leads to other fundamental interactions, which indeed they to. This can be said to be the paradigm of 20th century physics; that symmetries of a theory dictate the interactions. In the same way that U(1) symmetry can be understood as the rotational symmetry of a circle, the SU(2) symmetry can be understood as the rotational symmetry on a sphere. More complex symmetries such as SU(2) turn out to be non-commutative and are said to be non-abelian. Non-commutative operations can be visualized by rotating a book on a table. Rotation of a quarter of a circle on to the top edge of the book and again sideways will result in a different final state then if the rotations were done in the reverse order. Diving into the realm of non-abelian gauge symmetries is a matter for another thesis and would be sidetracking from our story about the magnetic vector potential. Nonetheless it is worth while to stop and ponder the richness of the concept, first intuitively sensed by Faraday when experimenting with magnets.
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4 Three dimensional solutions In this chapter we solve the Laplace equation for the magnetic vector potential from two fundamental current distributions. The solution for the line current is relatively straight forward. For the current loop we use the method of eigenfunction expansion.
4.1 Line current To obtain the value of the magnetic vector potential for a given current distribution we solve Equation (3.21) which has the integral expression, as in Equation (2.21), ˚ J(r0 ) µ0 dr0 . (4.1) A(r) = 0| 4π |r − r V Here we let the current flow in the z-direction of a Cartesian coordinate system (x, y, z), see Figure 4.1, with unit vectors (ˆ ex , ˆ ey , ˆ ez ) respectively.
z I · eˆz |r − r0 | x
0 y
r
Figure 4.1: Line current in the z-direction. We align the coordinate system such that the current lies along the z-axis, then the current density can be described with J=Iˆ ez ,
(4.2)
ignoring the thickness of the wire.
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Only the z-component of the integral in Equation (4.1) contributes and since the integrand is even and both the unit vector and current are constant, ˆ µ0 ∞ Iˆ ez A(r) = dz √ (4.3) 2 4π −∞ r + z2 ˆ ∞ dz Iµ0 √ ˆ ez = (4.4) 2 2π r + z2 0 ˆ L dz Iµ0 √ ˆ ez (4.5) = lim L→∞ 2π r2 + z 2 0 ! √ Iµ0 r 2 + L2 + L = lim ˆ ez ln . (4.6) L→∞ 2π r We see that as L → ∞, the value of the magnetic vector potential becomes infinite as well. However, at a distance r, orthogonal to a wire of finite length L we have the expression ! √ Iµ0 r 2 + L2 + L ˆ ez ln , (4.7) A(r) = 2π r and in the vicinity of the wire, where r
