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## Conceptual Origin of Maxwell Equations and Field Theory

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### ... conducting circuits, ... his way in trying to penetrate electromagnetism. ... seemed to have 2 basic geometric intuitions: magnetic lines of ... – PowerPoint PPT presentation

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Title: Conceptual Origin of Maxwell Equations and Field Theory

1
Conceptual Origin of Maxwell Equations and Field
Theory
2
• It is usually said that Coulomb, Gauss, Ampere
and Faraday discovered 4 laws experimentally, and
Maxwell wrote them into equations by adding the
displacement current.

3
• That is not entirely wrong, but obscures a most
important and fundamental fact in the history of
physics

4
• How Field Theory was Created

5
• A big step forward was the invention in 1800 by
Volta (1745-1827) of the Voltaic Pile, the first
electric battery, a simple device of zinc and
copper plates dipped in seawater brine.

6
• In 1820 Oersted (1777-1851) discovered that an
electric current would always cause magnetic
needles in its neighbor- hood to move.

7
• Ampere (1775-1836) was learned in mathematics. He
worked out in 1827 the exact magnetic forces in
the neighborhood of a current, as action at a
distance.

8
• Faraday (1791-1867) was also greatly excited by
Oersteds discovery. But he lacked Ampères
mathematical training.
• In a letter Faraday wrote to Ampère we read

9
• I am unfortunate in a want to mathematical
knowledge and the power of entering with facility
any abstract reasoning. I am obliged to feel my
way by facts placed closely together.
• (Sept. 3, 1822)

10
• Without mathematical training, and rejecting
Amperes action at a distance, Faraday used his
geometric intuition to feel his way in
understanding his experiments.

11
• In 1931 he began to compile his ltExperimental
Researchesgt, recording eventually 23 years of
research (1831-1854). It is noteworthy that there
was not a single formula in this whole monumental
compilation.

12
(No Transcript)
13
• Faraday discovered electric induction in 1831!

14
Fig. 2. A diagram from Faraday's Diary (October
17, 1831) (see Ref. 79). It shows a solenoid with
coil attached to a galvanometer. Moving a bar
magnet in and out of the solenoid generates
electricity.
15
• a state of tension, or a state of vibration, or
perhaps some other state analogous to the
electric current, to which the magnetic forces
are so intimately related.
• ltERgt vol. III, p.443

16
• Later on, the concept was variously called
• peculiar state
• state of tension
• peculiar condition
• etc

17
• (Sec. 66) All metals take on the peculiar state
• (Sec. 68) The state appears to be instantly
assumed
• (Sec. 71) State of tension

18
• Faraday seemed to be impressed and perplexed by 2
facts
• that the magnet must be moved to produce
induction.
• that induction often produce effects
perpendicular to the cause.

19
• Faraday was feeling his way in trying to
penetrate electromagnetism.
• Today, reading his ltExperimental Researchesgt,
we have to feel our way in trying to
penetrate his geometric intuition.

20
• Faraday seemed to have 2 basic geometric
intuitions
• magnetic lines of force, and
• electrotonic state
• The first was easily experimentally seen through
sprinkling iron filings in the field. It is now
called H, the magnetic field.

21
• The latter, the electro-tonic state, remained
Faradays illusive geometrical intuition when he
ceased his compilation of ltERgt in 1854. He was 63
years old.

22
• That same year, Maxwell graduated from
Cambridge University. He was 23 years old.
• In his own words, he
• wish to attack Electricity.

23
James Clerk Maxwell (1831-1879)
24
• Amazingly 2 years later Maxwell published the
first of his 3 great papers which founded

25
• Electromagnetic Theory as a Field Theory.

26
• Maxwell had learned from reading Thomsons
mathematical papers the usefulness of
• Studying carefully Faradays voluminous ltERgt he
final realized that
• Electrotonic Intensity A

27
• He realized that what Faraday had described in
so many words was the equation
• Taking the curl of both sides, we get

28
• This last equation is Faradays law in
differential form. Faraday himself had stated it
in words, which tranlates into

29
• Comment 1 Maxwell used Stokes Theorem,
which had not yet appeared in the literature. In
the 1854 Smiths Prize Exam, which Maxwell took
as a student, to prove Stokes theorem was
question 8. So Maxwell knew the theorem.

30
• With respect to the history of the present
theory, I may state that the recognition of
certain mathematical functions as expressing the
electrotonic state" of Faraday, and the use of
them in determining electrodynamic potentials and
electromotive forces is, as far as I am aware,
original but the distinct conception of the
possibility of the mathematical expressions
• arose in my mind from the perusal of Prof. W.
Thomson's papers

31
• 5 years later,
• 1861 paper 2, part I
• 1861 paper 2, part II
• 1862 paper 2, part III
• 1862 paper 2, part IV

32
• The displacement current first appeared in Part
III
• Prop XIV To correct Eq. (9) (of Part I) of
electric currents for the effect due to the
elasticity of the medium.

33
• How and why Maxwell had arrived at this
correction he never explained. Nor was there
any later historic research which had shed
light on this question.
• More historic research needed on this important
question.

34
• With this correction, Maxwell happily arrived at
the momentous.
• Prop XVI.

35
• we can scarcely avoid the inference that light
consists in the transverse undulations of the
same medium which is the cause of electric and
magnetic phenomena.

36
• Paper 3 was published in 1865. It had the title
A Dynamical Theory of the Electromagnetic Field.
In it we find the formula for energy density

37
• Its Section (74) we read a very clear exposition
of the basic philosophy of Field Theory

38
• In speaking of the Energy of the field, however,
I wish to be understood literally. All energy is
the same as mechanical energy, whether it exists
in the form of motion or in that of elasticity,
or in any other form. The energy in
electromagnetic phenomena is mechanical energy.
The only question is, Where does it reside? On
the old theories it resides in the electrified
bodies, conducting circuits, and magnets, in the
form of an unknown quality called potential
energy, or the power of producing certain effects
at a distance.

39
• On our theory it resides in the electromagnetic
field, in the space surrounding the electrified
and magnetic bodies, as well as in those bodies
themselves, and is in two different forms, which
may be described without hypothesis as magnetic
polarization and electric polarization, or,
according to a very probable hypothesis as the
motion and the strain of one and the same medium."

40
That was First clear formulation of the
fundamental principle of Field Theory
41
• 1800 Volta
• 1820 Oersted
• 1827 Ampere
• 1856 Maxwell 1
• 1861 Maxwell 2
• 1865 Maxwell 3

42
• Comment Throughout his life time, M. always
wrote his equations with the vector potential A
playing a key role. After his death, Heaviside
and Hertz gleefully eliminated A.
• But with QM we know now that A has physical
meaning. It cannot be eliminated (E.g. A-B
effect).

43
• Furthermore, A is not an ordinary vector, it
has gauge freedom.

44
• Did M. discuss this gauge freedom?
• Not in his papers.
• But he certainly was deeply aware of it, as is
evident from his use of Stoke's theorem and his
appreciation of F's geometric intuitions.

45
Developments after Maxwells death in 1879
46
• 1886 H. HERTZ EM. WAVES
• 1905 EINSTEIN SP. REL.
• 1947 LAMB RENORMALIZATION
• ----------------
• Great success for EM field theory!

47
• Many attempts to extend this success to nuclear
interactions, such as
• Tamm-Dancoff Theory,
• all without success.

48
• There followed many attempts to formulate
alternatives to field theory in the next 20
• some years
• Dispersion Relations
• Lee model
• Boot-Strap Models
• Axiomatic Field Theory
• Regge Poles
• Etc.

49
• ??????
• ???????
• ????

50
• Finally in the 1970s, physicists returned to
Field Theory, to
• NonAbelian Gauge Theory
• Spontaneous Symmetry Breaking

51
• These in turn led to great success, to
• The Standard Model

52
• It became clear that
• Gauge freedom ?is in fact the underlying essence
?of the structure of Maxwell equations.

53
• That
• Freedom implies Flexibility, and
• Symmetry restricts that Flexibility
• Furthermore
• For Maxwell Eq. the Symmetry is U(1)

54
• And enlarging that symmetry one obtains
• NonAbelian gauge theory

55
• Thus gradually there emerged the current dogma
• Symmetry dictates interactions,
• ALL interactions.