Conceptual Origin of
Maxwell Equations
and
Field Theory
1
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.
2
That is not entirely wrong,
but obscures a most
important and fundamental
fact in the history of physics:
3
How Field Theory was Created
4
• A big step forward was the
invention in 1800 by Volta (17451827) of the Voltaic Pile, the first
electric battery, a simple device of
zinc and copper plates dipped in
seawater brine.
5
In 1820 Oersted (17771851) discovered that an
electric current would
always cause magnetic
needles in its neighborhood to move.
6
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.
7
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:
8
“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)
9
Without mathematical training,
and rejecting Ampere’s action
at a distance, Faraday used his
geometric intuition to “feel his
way” in understanding his
experiments.
10
• 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.
11
12
Faraday discovered
electric induction in
1831!
13
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.
14
“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
15
Later on, the concept was variously called
• peculiar state
• state of tension
• peculiar condition
• etc
showing Faraday’s uncertainty about this concept.
16
(Sec. 66) All metals take on
the peculiar state
(Sec. 68) The state appears to
be instantly assumed
(Sec. 71) State of tension
17
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.
18
•
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.
19
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.
20
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.
21
•
That same year, Maxwell
graduated from Cambridge
University. He was 23 years old.
•
In his own words, he
“wish to attack Electricity”.
22
James Clerk Maxwell (1831-1879)
23
Amazingly 2 years later
Maxwell published the
first of his 3 great
papers which founded
24
Electromagnetic Theory
as a Field Theory.
25
•
Maxwell had learned from
reading Thomson’s mathematical
papers the usefulness of
H A
•
Studying carefully Faraday’s
voluminous <ER> he final realized
that
Electrotonic Intensity = A
26
•
He realized that what Faraday
had described in so many
words was the equation:
•
Taking the curl of both sides,
we get
27
This last equation is Faraday’s
law in differential form. Faraday
himself had stated it in words,
which tranlates into:
d
E

dl


H

d



dt
28
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.
29
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…
30
5 years later,
1861
1861
1862
1862
paper 2, part I
paper 2, part II
paper 2, part III
paper 2, part IV
31
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.”
32
•
•
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.
33
With this correction,
Maxwell happily arrived
at the momentous.
Prop XVI.
34
“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.”
35
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:


1
2
2
E H .
8
36
Its Section (74) we read a very
clear exposition of the basic
philosophy of Field Theory:
37
“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.
38
“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."
39
That was
First clear formulation of
the fundamental principle of
Field Theory
40
1800
1820
1827
Volta
Oersted
Ampere
1831
1856
1861
1865
Faraday
Maxwell 1
Maxwell 2
Maxwell 3
41
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).
42
•
Furthermore, A is not an
ordinary vector, it has
gauge freedom.
43
• 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.
44
Developments after
Maxwell’s death in 1879
45
• 1886 H. HERTZ
EM. WAVES
• 1905 EINSTEIN
SP. REL.
• 1947 LAMB…
RENORMALIZATION
---------------Great success for EM field theory!
46
• Many attempts to extend this success to
nuclear interactions, such as
Tamm-Dancoff Theory,
all without success.
47
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.
48
一时群雄竞逐
各领风骚五六年
好不热闹
49
• Finally in the 1970s, physicists returned to
Field Theory, to
• NonAbelian Gauge Theory +
• Spontaneous Symmetry Breaking
50
• These in turn led to great success, to
The Standard Model
51
It became clear that:
Gauge freedom ​is in fact the
underlying essence ​of the
structure of Maxwell equations.
52
That
• Freedom implies Flexibility, and
• Symmetry restricts that Flexibility
Furthermore
• For Maxwell Eq. the Symmetry is U(1)
53
And enlarging that symmetry
one obtains:
NonAbelian gauge theory
54
Thus gradually there emerged
the current dogma:
Symmetry dictates interactions,
ALL interactions.
55