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Conceptual Origin of

Maxwell Equations

and

Field Theory

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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.

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

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How Field Theory was Created

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• 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.

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

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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.

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Faraday (1791-1867) was also greatly excited by Oersted’s discovery. But he lacked Ampère’s mathematical training.

In a letter Faraday wrote to Ampère we read:

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“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)

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Without mathematical training, and rejecting Ampere’s action at a distance, Faraday used his geometric intuition to “feel his way” in understanding his experiments.

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

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Faraday discovered electric induction in 1831!

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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.

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“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.”

<ER> vol. III, p.443

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

showing Faraday’s uncertainty about this concept.

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(Sec. 66) All metals take on the peculiar state

(Sec. 68) The state appears to be instantly assumed

(Sec. 71) State of tension

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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.

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• Faraday was “feeling his way” in trying to penetrate

electromagnetism.

• Today, reading his <Experimental Researches>, we have to “feel our way” in trying to penetrate his geometric intuition.

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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.

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The latter, the electro-tonic state, remained Faraday’s illusive geometrical intuition when he ceased his compilation of <ER> in 1854. He was 63 years old.

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• That same year, Maxwell graduated from Cambridge University. He was 23 years old.

• In his own words, he

“wish to attack Electricity”.

23James Clerk Maxwell (1831-1879)

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Amazingly 2 years later

Maxwell published the

first of his 3 great

papers which founded

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Electromagnetic Theory as a Field Theory.

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• Maxwell had learned from reading Thomson’s mathematical papers the usefulness of

• Studying carefully Faraday’s voluminous <ER> he final realized that

Electrotonic Intensity = A

AH

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• He realized that what Faraday had described in so many

words was the equation:

• Taking the curl of both sides, we get

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This last equation is Faraday’s law in differential form. Faraday himself had stated it in words, which tranlates into:

dHdt

ddlE

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

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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…

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5 years later,

1861 paper 2, part I1861 paper 2, part II1862 paper 2, part III1862 paper 2, part IV

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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.”

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• 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.

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With this correction, Maxwell happily arrived at the momentous.

Prop XVI.

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“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.”

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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:

.8

1 22 HE

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

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“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.

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“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."

That was

First clear formulation ofthe fundamental principle of

Field Theory

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1800 Volta1820 Oersted1827 Ampere

1831 Faraday1856 Maxwell 11861 Maxwell 21865 Maxwell 3

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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).

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

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• 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.

Developments afterMaxwell’s death in 1879

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• 1886 H. HERTZ EM. WAVES• 1905 EINSTEIN SP. REL.• 1947 LAMB… RENORMALIZATION

----------------

Great success for EM field theory!

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• Many attempts to extend this success to nuclear interactions, such as

Tamm-Dancoff Theory,

all without success.

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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.

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一时群雄竞逐 各领风骚五六年 好不热闹

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• Finally in the 1970s, physicists returned to Field Theory, to

• NonAbelian Gauge Theory +

• Spontaneous Symmetry Breaking

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• These in turn led to great success, to

The Standard Model

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It became clear that:

Gauge freedom is in fact the underlying essence of the structure of Maxwell equations.

That

• Freedom implies Flexibility, and

• Symmetry restricts that Flexibility

Furthermore

• For Maxwell Eq. the Symmetry is U(1)

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And enlarging that symmetry one obtains:

NonAbelian gauge theory

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Thus gradually there emerged the current dogma:

Symmetry dictates interactions,

ALL interactions.