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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
  • showing Faradays uncertainty about this concept.

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
  • 1831 Faraday
  • 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.
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