whopkins@csd.uwm.edu
Feb11-05, 02:51 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hippie Ramone wrote:\n> I could be totally mistaken but I was under the impression that\nMaxwell\'s\n> equations done by himself was a set of 16 coupled PDE\'s. It was left\nto\n> Heaviside to put them into the vector form we all know and love or\nloathe\n> today. :)\n\nThe equations in the treatise are:\n\nB = curl A\nE = G x B - dA/dt - grad(phi)\n4 pi C = curl H\nJ = sigma E\nC = dD/dt + J\ndiv D = rho (and a second equation for surface charge density)\nF = C x B\nB = mu H\nD = {K/(4 pi)} E\nB = 4 pi (H + I)\n\nwith A, B, C, D, E, F, G, H, I, J respectively the vector quantities\nrepresenting the magnetic potential, the B field, the "total charge",\nthe D field, the E field, the force density, the velocity, the H field,\nthe magnetication and the current density.\n\nIn modern notation, Maxwell\'s (4 pi) is equivalent to mu_0 -- indeed,\nthe setting for mu_0 (4 pi x 10^{-7} in mks units) comes straight out\nof the section on units in his treatise. Maxwell\'s H is the modern\nmu_0 H; his E is the modern (v x B + E); his mu is the modern\n(mu/mu_0); and his K is the modern (4 pi epsilon). So, in modern\nguise, his equations read:\n\nB = curl A; E = -dA/dt - grad phi\nJ = sigma (E + v x B)\ndiv D = 0; curl H - dD/dt = J\nD = epsilon (E + v x B); B = mu H = mu_0 (H + I)\n\nwith a force law given by:\nF = (J + dD/dt) x B.\n\nThe equation div B = 0 was stated as a consequence and was not included\nin the original set (which was lettered (A), (B), (C), etc.) Likewise,\nthe Kirchoff law (div C = 0) was stated as a corollary; and the\nequation involving a magnetic potential when C = 0 was stated: H =\n-grad (Psi).\n\nThe earlier 1864 treatement separately posed the charge conservation\nlaw\nd(rho)/dt + div J = 0\nas an equation. So, some time between then and the treatise, Maxwell\nrealised that it was derivable -- but only if one uses the total\ncurrent C = J + dD/dt in the equation (curl H = C) instead of the\n\'convective\' current J. (C is also needed to make the derivation for\nthe laws of capacitance go through consistently).\n\nIn actual fact, the treatise did not use vectors so much as\ndifferential forms (even the latter equation for H was explicitly\nwritten as such: H.dr = -d(Psi)). The quantities defined were the\n2-forms:\nB.dS; D.dS; J.dS; C.dS\n(where dS = (dy dz, dz dx, dx dy) in Cartesian coordinate); the 3-form\nrho dV\n(where dV = dx dy dz in Cartesian coordinate); the 1-forms:\nE.dr; H.dr; I.dr; A.dr\n(where dr = (dx,dy,dz in Cartesian coordinate) and a (1 x 3)-form:\nF.(Dr [x] dV)\nwith the 1-form (Dr) pertaining to paths across which an element of\neletrical matter is moved (particularly a circuit element); and the\n3-form (dV) pertaining to the volume element of the source of the\nfield.\n\nThe argument used to get from the product (B.dS) (C.dS) to (F. Dr [x]\ndV) was carefully delineated somewhere in the vicinity of section 600.\nThis line is CRITICAL, since that\'s where the classical divergence of\nthe field enters into play -- if one assumes that the charges,\nmicroscopically, are a distribution of point-like sources. This\nassumption was not made in the treatise, so the "classical" divergence\nis *absent*. It is, in fact, the product of a latter-day interjection\n-- Lorentz -- into Maxwell and, as such (since it leads to the\ninconsistency), it represents the unwitting insertion of an\ninconsistency, as well.\n\nSo, a careful examination of both the quantum or classical divergence\nin the force law inevitably focuses on the key argument presented in\nthis section to get from the product (B.dS) (C.dS) to (F.Dr [x] dV).\nWhatever resolution of the singularity problem -- whether in the\nclassical theory or the quantum theory -- boils down to handling this\nproperly and coming up with a consistent microscopic representation of\nthis ... as opposed to the present-day one, which is inconsistent.\n\nIndeed, the problem of consistency in the classical theory is a main\npoint of focus in recent years (besides being listed in one of the\nAmerican Mathematical Society journal issues back in the 1980\'s as an\nopen problem, in the problem-challenge section); and is even listed,\nfor instance, on Baez\'s home page as one of the major open issues to be\nresolved in Physics.\n\nAnother point worth mentioning. Maxwell did a very rudimentary\nanalysis of the transformation of the various quantities under a change\nto a moving and rotating frame of reference. It\'s interesting to see\nhow it looks, in modern form, as a point of contrast to the modern\ntreatment from Relativity theory. It\'s clear from reading this and\nother parts of the treatise that Einstein\'s entire point of departure\nin the 1905 paper was a direct reply to this section (article 600 of\nthe treatise, I believe) and the treatise as a whole, and was not in\nreference to anything that was published in the intervening time. He\nwas simply filling in the gaps in Maxwell\'s somewhat botched-up and\nincomplete argument.\n\nUnder a change of frame, the velocity (as per Maxwell) goes as v -> v -\nDv, where Dv = u + Omega x r, for a rotating system (taken at an\ninstant where the origins and axes of the 2 frames coincide, so that\nboth r\'s are the same). The potential A, under a time derivative, goes\nas dA/dt -> dA/dt + (Dv.grad)A + Omega x A; the potential goes as phi\n-> phi - Dv.A; his E field is invariant, as is his B field (no explicit\naccount is given for H or D; nor even J). The net effect is that the\nexpression for E remain invariant. The identity Omega x A = grad Dv.A\n(grad applied only to the Dv) is used.\n\nAs you can see, everything was too muddled to clearly see the\nconsequence of light speed being invariant. That, in fact, does NOT\nfollows from the macroscopic equations:\ndiv D = rho; curl H - dD/dt = J\ndiv B = 0; curl E + dB/dt = 0\nat all, since these are diffeomorphism invariant and apply equally well\nto any 4-dimensional spacetime (Minkowski, Galilean, even Euclidean).\nIt only comes out of the microscopic relations D<->E, B<->H\nD = epsilon_0 E; B = mu_0 H\nwhich were notably absent from his treatise. Instead, he had (in\neffect):\nD = epsilon_0 (E + v x B);\nB = mu_0 (H - v x D).\nThe -v x D term, in fact, was mentioned early on in the (1950 edition\nof) the treatise (somewhere around article 100 or so), in a footnote\nreferencing a cite from an issue of the Philosophical Magazine from the\nlate 1880\'s. These relations are NOT Poincare\' invariant. They are\nGalilean invariant, provided one adopts the above transformation law\nfor v. This v, however, will then not measure the velocity of one\ncircuit element with respect to another, but rather the relative\nvelocity between the inertial frame where the equations are taken and\nthe ("ether") frame where they are of the form\nD = epsilon_0 E; B = mu_0 H.\n\nSo, part of the point of Einstein\'s paper (particular the statement in\nthe abstract making implicit reference to this treatement in the\nMaxwell treatise) was to point out that the latter relations should\nhold in ALL frames of reference, so that no "ether" frame or "v" is\nneeded.\n\nMaxwell stated no clear committment on whether any kind of medium was\npresent underlying the light propagation phenomenon, though he pointed\nout his prejudice on the issue in several places. In fact, he went out\nof his was in the treatise to keep the issue open.\n\nThe key section (somewhere around 600), in fact, explicitly sets out\nthe hypothesis of particle-wave duality for light! He states that\nthere are two pictures of light: that the energy is transmitted across\nthe vacuum in photons ("light-corpuscles") and that it is propagated\nacross space in a wave-like fashion. He was at a loss to explain what\nthe waves were propagating IN, and tried to outline the properties that\nmust be held of the propagation.\n\nIn this light (pun intended), it\'s clear that Einstein\'s SECOND major\npaper (the photon hypothesis) was actually a just a continuation of the\n*direct* reply to Maxwell given in his Relativity paper; making\nexplicit reference to this article in the treatise and, in fact,\nadopting Maxwell\'s very particle-wave duality notion as his own.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hippie Ramone wrote:
> I could be totally mistaken but I was under the impression that
Maxwell's
> equations done by himself was a set of 16 coupled PDE's. It was left
to
> Heaviside to put them into the vector form we all know and love or
loathe
> today. :)
The equations in the treatise are:
B = curl A
E = G x B - dA/dt - grad(\phi)4 \pi C = curl H
J = \sigma EC = dD/dt + Jdiv D = \rho (and a second equation for surface charge density)
F = C x BB = \mu HD = {K/(4 \pi)} EB = 4 \pi (H + I)
with A, B, C, D, E, F, G, H, I, J respectively the vector quantities
representing the magnetic potential, the B field, the "total charge",
the D field, the E field, the force density, the velocity, the H field,
the magnetication and the current density.
In modern notation, Maxwell's (4 \pi) is equivalent to \mu_0 -- indeed,
the setting for \mu_0 (4 \pi x 10^{-7} in mks units) comes straight out
of the section on units in his treatise. Maxwell's H is the modern
\mu_0 H; his E is the modern (v x B + E); his \mu is the modern
(\mu/\mu_0); and his K is the modern (4 \pi \epsilon). So, in modern
guise, his equations read:
B = curl A; E = -dA/dt - grad \phiJ = \sigma (E + v x B)div D = 0; curl H - dD/dt = JD = \epsilon (E + v x B); B = \mu H = \mu_0 (H + I)
with a force law given by:
F = (J + dD/dt) x B.
The equation div B = was stated as a consequence and was not included
in the original set (which was lettered (A), (B), (C), etc.) Likewise,
the Kirchoff law (div C = 0) was stated as a corollary; and the
equation involving a magnetic potential when C = was stated: H =
-grad (\Psi).
The earlier 1864 treatement separately posed the charge conservation
law
d(\rho)/dt + div J =
as an equation. So, some time between then and the treatise, Maxwell
realised that it was derivable -- but only if one uses the total
current C = J + dD/dt in the equation (curl H = C) instead of the
'convective' current J. (C is also needed to make the derivation for
the laws of capacitance go through consistently).
In actual fact, the treatise did not use vectors so much as
differential forms (even the latter equation for H was explicitly
written as such: H.dr = -d(\Psi)). The quantities defined were the
2-forms:
B.dS; D.dS; J.dS; C.dS
(where dS = (dy dz, dz dx, dx dy) in Cartesian coordinate); the 3-form
\rho dV
(where dV = dx dy dz in Cartesian coordinate); the 1-forms:
E.dr; H.dr; I.dr; A.dr
(where dr = (dx,dy,dz in Cartesian coordinate) and a (1 x 3)-form:
F.(Dr [x] dV)
with the 1-form (Dr) pertaining to paths across which an element of
eletrical matter is moved (particularly a circuit element); and the
3-form (dV) pertaining to the volume element of the source of the
field.
The argument used to get from the product (B.dS) (C.dS) to (F. Dr [x]
dV) was carefully delineated somewhere in the vicinity of section 600.
This line is CRITICAL, since that's where the classical divergence of
the field enters into play -- if one assumes that the charges,
microscopically, are a distribution of point-like sources. This
assumption was not made in the treatise, so the "classical" divergence
is *absent*. It is, in fact, the product of a latter-day interjection
-- Lorentz -- into Maxwell and, as such (since it leads to the
inconsistency), it represents the unwitting insertion of an
inconsistency, as well.
So, a careful examination of both the quantum or classical divergence
in the force law inevitably focuses on the key argument presented in
this section to get from the product (B.dS) (C.dS) to (F.Dr [x] dV).
Whatever resolution of the singularity problem -- whether in the
classical theory or the quantum theory -- boils down to handling this
properly and coming up with a consistent microscopic representation of
this ... as opposed to the present-day one, which is inconsistent.
Indeed, the problem of consistency in the classical theory is a main
point of focus in recent years (besides being listed in one of the
American Mathematical Society journal issues back in the 1980's as an
open problem, in the problem-challenge section); and is even listed,
for instance, on Baez's home page as one of the major open issues to be
resolved in Physics.
Another point worth mentioning. Maxwell did a very rudimentary
analysis of the transformation of the various quantities under a change
to a moving and rotating frame of reference. It's interesting to see
how it looks, in modern form, as a point of contrast to the modern
treatment from Relativity theory. It's clear from reading this and
other parts of the treatise that Einstein's entire point of departure
in the 1905 paper was a direct reply to this section (article 600 of
the treatise, I believe) and the treatise as a whole, and was not in
reference to anything that was published in the intervening time. He
was simply filling in the gaps in Maxwell's somewhat botched-up and
incomplete argument.
Under a change of frame, the velocity (as per Maxwell) goes as v -> v -
Dv, where Dv = u + \Omega x r, for a rotating system (taken at an
instant where the origins and axes of the 2 frames coincide, so that
both r's are the same). The potential A, under a time derivative, goes
as dA/dt -> dA/dt + (Dv.grad)A + \Omega x A; the potential goes as \phi-> \phi - Dv.A; his E field is invariant, as is his B field (no explicit
account is given for H or D; nor even J). The net effect is that the
expression for E remain invariant. The identity \Omega x A = grad Dv.A
(grad applied only to the Dv) is used.
As you can see, everything was too muddled to clearly see the
consequence of light speed being invariant. That, in fact, does NOT
follows from the macroscopic equations:
div D = \rho; curl H - dD/dt = Jdiv B = 0; curl E + dB/dt =
at all, since these are diffeomorphism invariant and apply equally well
to any 4-dimensional spacetime (Minkowski, Galilean, even Euclidean).
It only comes out of the microscopic relations D<->E, B<->HD = \epsilon_0 E; B = \mu_0 H
which were notably absent from his treatise. Instead, he had (in
effect):
D = \epsilon_0 (E + v x B);B = \mu_0 (H - v x D).
The -v x D term, in fact, was mentioned early on in the (1950 edition
of) the treatise (somewhere around article 100 or so), in a footnote
referencing a cite from an issue of the Philosophical Magazine from the
late 1880's. These relations are NOT Poincare' invariant. They are
Galilean invariant, provided one adopts the above transformation law
for v. This v, however, will then not measure the velocity of one
circuit element with respect to another, but rather the relative
velocity between the inertial frame where the equations are taken and
the ("ether") frame where they are of the form
D = \epsilon_0 E; B = \mu_0 H.
So, part of the point of Einstein's paper (particular the statement in
the abstract making implicit reference to this treatement in the
Maxwell treatise) was to point out that the latter relations should
hold in ALL frames of reference, so that no "ether" frame or "v" is
needed.
Maxwell stated no clear committment on whether any kind of medium was
present underlying the light propagation phenomenon, though he pointed
out his prejudice on the issue in several places. In fact, he went out
of his was in the treatise to keep the issue open.
The key section (somewhere around 600), in fact, explicitly sets out
the hypothesis of particle-wave duality for light! He states that
there are two pictures of light: that the energy is transmitted across
the vacuum in photons ("light-corpuscles") and that it is propagated
across space in a wave-like fashion. He was at a loss to explain what
the waves were propagating IN, and tried to outline the properties that
must be held of the propagation.
In this light (pun intended), it's clear that Einstein's SECOND major
paper (the photon hypothesis) was actually a just a continuation of the
*direct* reply to Maxwell given in his Relativity paper; making
explicit reference to this article in the treatise and, in fact,
adopting Maxwell's very particle-wave duality notion as his own.
> I could be totally mistaken but I was under the impression that
Maxwell's
> equations done by himself was a set of 16 coupled PDE's. It was left
to
> Heaviside to put them into the vector form we all know and love or
loathe
> today. :)
The equations in the treatise are:
B = curl A
E = G x B - dA/dt - grad(\phi)4 \pi C = curl H
J = \sigma EC = dD/dt + Jdiv D = \rho (and a second equation for surface charge density)
F = C x BB = \mu HD = {K/(4 \pi)} EB = 4 \pi (H + I)
with A, B, C, D, E, F, G, H, I, J respectively the vector quantities
representing the magnetic potential, the B field, the "total charge",
the D field, the E field, the force density, the velocity, the H field,
the magnetication and the current density.
In modern notation, Maxwell's (4 \pi) is equivalent to \mu_0 -- indeed,
the setting for \mu_0 (4 \pi x 10^{-7} in mks units) comes straight out
of the section on units in his treatise. Maxwell's H is the modern
\mu_0 H; his E is the modern (v x B + E); his \mu is the modern
(\mu/\mu_0); and his K is the modern (4 \pi \epsilon). So, in modern
guise, his equations read:
B = curl A; E = -dA/dt - grad \phiJ = \sigma (E + v x B)div D = 0; curl H - dD/dt = JD = \epsilon (E + v x B); B = \mu H = \mu_0 (H + I)
with a force law given by:
F = (J + dD/dt) x B.
The equation div B = was stated as a consequence and was not included
in the original set (which was lettered (A), (B), (C), etc.) Likewise,
the Kirchoff law (div C = 0) was stated as a corollary; and the
equation involving a magnetic potential when C = was stated: H =
-grad (\Psi).
The earlier 1864 treatement separately posed the charge conservation
law
d(\rho)/dt + div J =
as an equation. So, some time between then and the treatise, Maxwell
realised that it was derivable -- but only if one uses the total
current C = J + dD/dt in the equation (curl H = C) instead of the
'convective' current J. (C is also needed to make the derivation for
the laws of capacitance go through consistently).
In actual fact, the treatise did not use vectors so much as
differential forms (even the latter equation for H was explicitly
written as such: H.dr = -d(\Psi)). The quantities defined were the
2-forms:
B.dS; D.dS; J.dS; C.dS
(where dS = (dy dz, dz dx, dx dy) in Cartesian coordinate); the 3-form
\rho dV
(where dV = dx dy dz in Cartesian coordinate); the 1-forms:
E.dr; H.dr; I.dr; A.dr
(where dr = (dx,dy,dz in Cartesian coordinate) and a (1 x 3)-form:
F.(Dr [x] dV)
with the 1-form (Dr) pertaining to paths across which an element of
eletrical matter is moved (particularly a circuit element); and the
3-form (dV) pertaining to the volume element of the source of the
field.
The argument used to get from the product (B.dS) (C.dS) to (F. Dr [x]
dV) was carefully delineated somewhere in the vicinity of section 600.
This line is CRITICAL, since that's where the classical divergence of
the field enters into play -- if one assumes that the charges,
microscopically, are a distribution of point-like sources. This
assumption was not made in the treatise, so the "classical" divergence
is *absent*. It is, in fact, the product of a latter-day interjection
-- Lorentz -- into Maxwell and, as such (since it leads to the
inconsistency), it represents the unwitting insertion of an
inconsistency, as well.
So, a careful examination of both the quantum or classical divergence
in the force law inevitably focuses on the key argument presented in
this section to get from the product (B.dS) (C.dS) to (F.Dr [x] dV).
Whatever resolution of the singularity problem -- whether in the
classical theory or the quantum theory -- boils down to handling this
properly and coming up with a consistent microscopic representation of
this ... as opposed to the present-day one, which is inconsistent.
Indeed, the problem of consistency in the classical theory is a main
point of focus in recent years (besides being listed in one of the
American Mathematical Society journal issues back in the 1980's as an
open problem, in the problem-challenge section); and is even listed,
for instance, on Baez's home page as one of the major open issues to be
resolved in Physics.
Another point worth mentioning. Maxwell did a very rudimentary
analysis of the transformation of the various quantities under a change
to a moving and rotating frame of reference. It's interesting to see
how it looks, in modern form, as a point of contrast to the modern
treatment from Relativity theory. It's clear from reading this and
other parts of the treatise that Einstein's entire point of departure
in the 1905 paper was a direct reply to this section (article 600 of
the treatise, I believe) and the treatise as a whole, and was not in
reference to anything that was published in the intervening time. He
was simply filling in the gaps in Maxwell's somewhat botched-up and
incomplete argument.
Under a change of frame, the velocity (as per Maxwell) goes as v -> v -
Dv, where Dv = u + \Omega x r, for a rotating system (taken at an
instant where the origins and axes of the 2 frames coincide, so that
both r's are the same). The potential A, under a time derivative, goes
as dA/dt -> dA/dt + (Dv.grad)A + \Omega x A; the potential goes as \phi-> \phi - Dv.A; his E field is invariant, as is his B field (no explicit
account is given for H or D; nor even J). The net effect is that the
expression for E remain invariant. The identity \Omega x A = grad Dv.A
(grad applied only to the Dv) is used.
As you can see, everything was too muddled to clearly see the
consequence of light speed being invariant. That, in fact, does NOT
follows from the macroscopic equations:
div D = \rho; curl H - dD/dt = Jdiv B = 0; curl E + dB/dt =
at all, since these are diffeomorphism invariant and apply equally well
to any 4-dimensional spacetime (Minkowski, Galilean, even Euclidean).
It only comes out of the microscopic relations D<->E, B<->HD = \epsilon_0 E; B = \mu_0 H
which were notably absent from his treatise. Instead, he had (in
effect):
D = \epsilon_0 (E + v x B);B = \mu_0 (H - v x D).
The -v x D term, in fact, was mentioned early on in the (1950 edition
of) the treatise (somewhere around article 100 or so), in a footnote
referencing a cite from an issue of the Philosophical Magazine from the
late 1880's. These relations are NOT Poincare' invariant. They are
Galilean invariant, provided one adopts the above transformation law
for v. This v, however, will then not measure the velocity of one
circuit element with respect to another, but rather the relative
velocity between the inertial frame where the equations are taken and
the ("ether") frame where they are of the form
D = \epsilon_0 E; B = \mu_0 H.
So, part of the point of Einstein's paper (particular the statement in
the abstract making implicit reference to this treatement in the
Maxwell treatise) was to point out that the latter relations should
hold in ALL frames of reference, so that no "ether" frame or "v" is
needed.
Maxwell stated no clear committment on whether any kind of medium was
present underlying the light propagation phenomenon, though he pointed
out his prejudice on the issue in several places. In fact, he went out
of his was in the treatise to keep the issue open.
The key section (somewhere around 600), in fact, explicitly sets out
the hypothesis of particle-wave duality for light! He states that
there are two pictures of light: that the energy is transmitted across
the vacuum in photons ("light-corpuscles") and that it is propagated
across space in a wave-like fashion. He was at a loss to explain what
the waves were propagating IN, and tried to outline the properties that
must be held of the propagation.
In this light (pun intended), it's clear that Einstein's SECOND major
paper (the photon hypothesis) was actually a just a continuation of the
*direct* reply to Maxwell given in his Relativity paper; making
explicit reference to this article in the treatise and, in fact,
adopting Maxwell's very particle-wave duality notion as his own.