WEBVTT - generated by VCS
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Have a wonderful good day, ladies and gentlemen.
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I would like to warmly welcome you to another
contribution from the series
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electromagnetic theorie.
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Today we will deal with the axiomatic foundations
of classical field theory,
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i.e. the axiomatic foundations of the Maxwell
equation.
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My name is Hans Georg Krauthäuser.
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I am the holder of the professorship for electromagnetic
theory and
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Compatibility at the TU Dresden.
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Well, let's get started right away.
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As already said, it is about the axiomatic
foundations of the classical
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Electrodynamics. Axiomatic foundations?
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Of what actually?
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Well, axiomatic fundamentals of Maxwell's
equation.
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As you know, these are the 4 equations on
which the electromagnetic
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Field theory based. Let's take that apart
a little.
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We have 2 homogeneous partial differential
equations and 2 inhomogeneous
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partial differential equations and if we have
to look the other way
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we have 2 equations that have a vectorial
character - that is, which actually
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are each 3 equations.
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And we have 2 scalar equations.
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We also note that if only the vector equations
are actually
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are time-dependent equations, i.e. only these
two equations are also
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actually dynamic equations.
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The other two do not contain time, that is,
they are purely static
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Equations. Historically, as you will know,
of course it is
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these equations were obtained independently
from observations and we
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then also have historical names: Coulomb-Gauss
law here, Faraday's law
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Induction law up here, or Ampèr's expanded
law of flux
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that we have here - expanded
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because there is still the post
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it says from Maxwell.
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And of course the relationship that maintains
the magnetic flux to the
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Expresses or
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the freedom from source of the magnetic field.
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Maxwell, as I just said, has the current of
displacement in the whole
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introduced and the equation actually merged.
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I have also given you the corresponding original
source again.
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You will not find today's formulation in this
source, but rather
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it was later developed and published by Heaviside
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on the basis of vector analysis.
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But the fundamental question that we want
to deal with here today is this
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Ask where from, that is, what is actually
the deeper foundation of this
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Equation. Is there an axiomatic basis and
if so, what is it?
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What I am going to present to you today has
been published.
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You will find a nice work by Gronwald, Hehl
and Nitsch “Axiomatics of classical
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electrodynamics and its relation to gauge field
theory“ in den „Physics Notes“.
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The source is given here, so you can actually
go back and relax
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read up. We start with the charge density,
resp.
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then also with the cargo.
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The charge appears in the Coulomb-Gauss law,
which we have just seen in
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Shape of the volume charge density roh_V and
of course I can then use a
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Volume integration of the volume charge density
on the charge in a
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Volume element come. That means, and it will
be important now, that raw_V is
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the integrand of a volume integral.
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Why is that important?
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Well, let's jump into mathematics, let's jump
to Poincaré's lemma.
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You can find that in a variety of forms, but
we'll get to that here in a nutshell
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and only represent it again for the three-dimensional
case.
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The Poincaré lemma tells us under which conditions
a quantity as a derivative of a
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other size can be displayed.
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So in principle it can be derived from a potential
.
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This is of course not clear, it also leads
to the
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Concept of calibration, more on that later.
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Here we are now first of all doing pure mathematics.
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In the special case of three-dimensional space,
only the need and interest here
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there are three important formulations or
statements of the Poincaré lemma.
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First of all, it is that as long as we focus
on one simple
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contiguous area and there we have an eddy-free
vector field,
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that this is then the gradient of a potential
field.
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That is, there is a field F such that this
invertebrate vector field is alpha, then
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can even be represented as an alpha equal to
the gradient of this potential field.
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Second, if we are in a convex area and have
one there
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source-free vector field, i.e. the divergence
disappears, then this can
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Field can be represented as the rotation of
a vector field.
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Third, if we have a scalar field density ,
then we can call it divergence
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of a vector field.
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And whenever this scalar field density gamma
is a volume integrand -
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that is, if it appears below a volume integral
- we can
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actually represent as the divergence of a
vector field Betta.
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And this third case is the one we are using
now .
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And that means we jump back to the charge
density and just do it
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further: Roh_V is the integrand of a volume
integral here and so we can use the
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apply the third statement of the Poincaré
lemma and then get it immediately
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the Coulomb-Gauss law.
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Since we do not need to take on anything,
pure mathematics that we apply here
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have and the definition that the charge is
the volume integral of a
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Charge density, a volume charge density and
we already have the divergence
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equal raw_V. Charges are also typically used
with a certain amount
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Moving speed in matter and when we can already
be at the charge
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we also look directly at this movement and
above it the current density J
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insert. It is then simply true that the current
density is nothing else than that
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Charge density multiplied by the material
speed - with the
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Charge carrier density speed .
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When we have the carrier density, we come
to simple ones
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Way to electricity, of course.
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The current is then simply defined as the flow
of these charge carriers - the
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Current density J - through a surface, so
here we have a certain
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Have surface integral.
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That was not an action yet what we have done
now.
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We saw that we can get the one equation straight
away from mathematics
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and can be derived from the definition of
the charge.
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Now we actually come to something axiomatic,
now we come to the
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Ladungserhaltung. Wir postulieren also Ladungserhaltung.
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Wenn sich also Ladung in einem Volumen ändert,
dann geschieht das nur dadurch,
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dass sie durch die Oberfläche eines Volumens
hindurchströmt.
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Die Ladungsdichteverteilung kann sich natürlich
innerhalb eines Volumens
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verschieben das würde aber nicht die Gesamtladung
im Volumen ändern.
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Die Gesamtladung im Volumen ändert sich nur,
wenn tatsächlich ein Strom durch die
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Oberfläche hindurchtritt.
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Das können wir jetzt natürlich auch wieder
mathematisch formulieren.
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Wir betrachten die materielle Ableitung - substanzielle
Ableitung sagt man auch
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often - is also closely related to the total
differential of the charge in this
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Volume. It's written down here, it should
actually go away.
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That is the axiomatic postulate and then we
only need to calculate what gives
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here just this derivation in two parts.
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On the one hand, the change in the charge
distribution in the volume and then -
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very important - just the change that is caused
by the fact that charges
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step through the surface at a certain speed
.
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Ultimately, that is nothing more than the
current density - here again
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written down. So here we have two amounts
dQ after dt, that would be the
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first post plus just that
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Stream of charge through the
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Surface of this volume.
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If we continue here, we will see that we have
a surface integral here
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to have. According to Gauss's theorem, we
can of course use this integral over the
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Identify the surface or write it differently
than a volume integral,
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now no longer about the size itself, but about
the divergence of the
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appropriate size.
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If we then have two volume integrals, we can
pull that under one integral
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and get this expression here.
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From here to here, roh_V is replaced by J,
so that the current density is also here
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to bring into play.
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It is still true that, according to our axiomatic
postulate, this total
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Expression should be 0 for any volume.
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And so that this can be correct for any volume
, nothing else remains
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left than that actually this integrand disappears.
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In that way, we pulled out an equation here
that we did too
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will be named soon.
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But we're still going on.
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If we now see again that the rho_V is already
expressed as divergence D.
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have, on the slide in front of it, then of
course we can replace this here, then have
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two thermal baths where the divergence appears,
so that we simply pull the divergence in front
of it
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can and get this simple form.
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As already said, this applies to any volume
and can accordingly
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only be correct - that can only be equal to
0 - if actually the integrand
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is always 0 for any volume.
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We'll hold on to that and I promised you that
we would use this equation too
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Give it a name, most will already know it,
that is the well-known one
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Continuity equation, which we quite obviously
immediately mathematically from the
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Get the postulate of conservation of charge.
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Now let's look again at that phrase we just
won
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and which is equal to 0.
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Und wir nehmen wieder das Poincaré Lemma –
nun in der zweiten Form - und stellen fest
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(oder können dann einfach ablesen), wenn diese
Divergenz verschwindet, dann können
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wir das was hier als Vektorfeld drin steht
auch ausdrücken als die Rotation
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eines weiteren Vektorfeldes.
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Was uns auf diesen Ausdruck führt: Damit hätten
wir also dD
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nach dt + J gleich Rotation A.
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Und wenn wir das umstellen, dann sehen wir
sofort: das ist eine weitere
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Maxwell-Gleichung. Zwischenfazit: Aus der Definition
der Ladungsdichte und Poincaré 3
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folgt sofort die Gleichung in der Form des
Coulomb-Gauß-Gesetzes.
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Man kann das in der Form darstellt: Es gib
ein Vektorpotential, ein Vektorfeld D, so
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dass Divergenz D gleich rho_V ist.
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Mit dem Axiom der Ladungserhaltung und Poincaré
2 und der obigen Beziehung - also
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Divergenz D gleich rho_V - , bekommen wir die
nächste Maxwellgleichung Rotation H
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minus dD nach dt gleich J.
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Wir geben den Größen jetzt sofort Namen.
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Wir haben die elektrische Anregung D - historisch
heißt das die elektrische
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Verschiebung. Wir werden in den meisten Fällen
auch bei dielektrischer Verschiebung
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später bleiben aber die moderne Ausdrucksweise
wäre elektrische Anregung.
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Wir haben die magnetische Anregung - historisch
gesehen das Magnetfeld H.
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Und was man auch feststellen muss: wir haben
bisher überhaupt nicht auf Kräfte
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zurück gegriffen. Das ist etwas, wenn Sie
mal an anderen Stellen schauen nach
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axiomatischen Einführungen, dann wird üblicherweise
sehr früh auf Kräfte
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zurückgegriffen. Ich wollte hier, das finde
ich schön an dem Ansatz von Gronwald,
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Hehl und Nitsch, dass sie eben relativ weit
kommen, ohne Kräfte zu benötigen.
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Die Ladungserhaltung, dass muss man wissen
und ist eigentlich nicht Thema hier und ich
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will es aber der Vollständigkeit halber nicht
unerwähnt lassen.
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Die Ladungserhaltung gilt auch mikroskopisch
und damit gelten tatsächlich
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auch die inhomogene Maxwell-Gleichung die wir
ja gerade schon abgeleitet haben, eben
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auch mikrophysikalisch.
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Das heißt elektrische und magnetische Anregungen
bzw.
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mit den historischen Formulierungen dielektrische
Verschiebung und das
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Magnetfeld sind mikrophysikalische Größen.
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Wichtig ist auch die Erkenntnis, dass die Ladung
auch relativistisch
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eine invariante Größe ist.
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Das heißt, dass die inhomogene Maxwell-Gleichung
auch relativistisch
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invariant formuliert werden können.
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Auch das ist eine wichtige Erkenntnis.
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Jetzt kommen wir tatsächlich, ich habe ja
gerade gesagt, dass wir bis jetzt nicht auf
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Kräfte zurückgegriffen, jetzt werden wir
das machen und zwar auch in einer
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00:16:53.000 --> 00:16:55.000
axiomatischen Art und Weise.
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Das heißt, wir werden als zweites Axiom annehmen,
dass die Lorenzkraft genau die
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00:17:06.020 --> 00:17:10.020
Form hat, wie wir sie tatsächlich auch experimentell
kennen.
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You know this expression F equals Q times E
plus U crosses B, if u is the speed
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of the corresponding charge, F is the corresponding
force acting on that charge
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would work if an E field and a B field are
present at the same time .
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This is really excellent experimentally confirmed
and if we do not continue it
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00:17:36.020 --> 00:17:39.020
can justify, let's turn it into an axiom.
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The electric field strength is introduced quite
incidentally.
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We now use the principle of relativity in further
consideration.
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The principle of relativity says that physical
laws are independent of
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00:17:55.020 --> 00:18:00.080
let the inertial systems write in the same
way.
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00:18:00.080 --> 00:18:03.080
So we now consider two inertial systems.
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00:18:03.080 --> 00:18:09.080
One is the laboratory system, I won't make
a mark on it.
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00:18:09.080 --> 00:18:17.080
That moves to the rest system of the charge
- I'll put a line on that
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do - at a speed that we call u.
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And the assumption is that we simply don't
have any
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Have E field. So E line equals 0 in the rest
field of the charge.
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00:18:38.080 --> 00:18:43.080
Of course, in the rest area of the load,
I have no speed either, that is then
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00:18:43.080 --> 00:18:44.080
the rest system of the charge.
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00:18:44.080 --> 00:18:49.080
And of course it is easy to see when E equals
0 and u must equal 0
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00:18:49.080 --> 00:18:51.080
Of course, the entire expression must also
be 0.
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00:18:51.080 --> 00:18:56.080
Das heißt im Ruhesystem der Ladung ist, wenn
ich kein E-Feld habe, die Kraft auf
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00:18:56.080 --> 00:18:58.080
diese Ladung gleich 0.
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00:18:58.080 --> 00:19:01.040
Damit wird auch diese Ladung nicht beschleunigt.
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00:19:01.040 --> 00:19:07.040
Die Beschleunigung der Ladung, das ist aber
ein physikalischer Vorgang, den müsste ich
218
00:19:07.040 --> 00:19:10.040
tatsächlich in jedem Inertialsysteme beobachten
können.
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00:19:10.040 --> 00:19:16.040
Wenn es im Ruhesystem nicht beschleunigt wird,
dann darf es auch in jedem anderen
220
00:19:16.040 --> 00:19:19.040
System nicht beschleunigt werden, weil das
sind ja Inertialsysteme.
221
00:19:19.040 --> 00:19:23.040
Damit muss auch die Kraft im Laborsystem verschwinden.
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00:19:23.040 --> 00:19:26.040
Die Kraft im Laborsystem ist diese Kraft.
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00:19:26.040 --> 00:19:32.040
Jetzt ist aber tatsächlich u eine Geschwindigkeit
ungleich 0.
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00:19:32.040 --> 00:19:39.040
And so that the entire expression is equal
to 0 , there is nothing left but that
225
00:19:39.040 --> 00:19:45.040
I now actually have an electric field strength
E - exactly in the form E
226
00:19:45.040 --> 00:19:48.040
equal minus u cross B.
227
00:19:48.040 --> 00:19:56.040
Historically, the two field strengths E and
B - B is magnetic
228
00:19:56.040 --> 00:20:01.000
Induction - are not independent of each other.
229
00:20:01.000 --> 00:20:07.000
They cannot be independent of one another
if I take the Lorenz force as a given
230
00:20:07.000 --> 00:20:13.000
and if the principle of relativity is valid:
a very important one
231
00:20:13.000 --> 00:20:20.000
Understanding. We continue with axiom 3 “Conservation
of a magnetic
232
00:20:20.000 --> 00:20:28.000
River ”. The magnetic flux results when
I pass the B field over a
233
00:20:28.000 --> 00:20:36.000
Integrate the surface, that means I look at
how much B field reaches through this
234
00:20:36.000 --> 00:20:38.000
Surface S through it.
235
00:20:38.000 --> 00:20:41.000
That is the magnetic flux through the surface.
236
00:20:41.000 --> 00:20:49.000
Analogous to the conservation of charge, we
can then also use the same arguments to create
a
237
00:20:49.000 --> 00:20:53.000
Get river conservation.
238
00:20:53.000 --> 00:20:58.000
Write down the same way and the arguments
are the same, so I'll shorten it
239
00:20:58.000 --> 00:21:01.060
from here and I get such an expression.
240
00:21:01.060 --> 00:21:09.060
And here I have now actually introduced a
magnetic flux, a
241
00:21:09.060 --> 00:21:12.060
magnetic flux current density.
242
00:21:12.060 --> 00:21:21.060
Here, too, I can actually transform mathematically
again and have this here
243
00:21:21.060 --> 00:21:25.060
Ring integral along the circumference of the
surface S.
244
00:21:25.060 --> 00:21:34.060
And about this I can also make an area integral
with the Stokes theorem .
245
00:21:34.060 --> 00:21:43.060
And if I still take advantage of how the flow
and the magnetic field are connected
246
00:21:43.060 --> 00:21:49.060
stand and insert that here, then you see that
there are two surface integrals
247
00:21:49.060 --> 00:21:52.060
have. I could now merge them again into a
surface integral.
248
00:21:52.060 --> 00:21:55.060
In total, it has to be 0.
249
00:21:55.060 --> 00:22:02.020
And here, too, there is nothing else left
than that the integrand disappears,
250
00:22:02.020 --> 00:22:05.020
because that would have to apply to any surface.
251
00:22:05.020 --> 00:22:20.020
If one forms the divergence of this expression,
then it follows because of the vector identity
252
00:22:20.020 --> 00:22:33.020
Divergence rotation of something is equal to
0 it follows that the divergence from dB to
dt
253
00:22:33.020 --> 00:22:41.020
disappears. In other words, that means the
divergence B, I can say that
254
00:22:41.020 --> 00:22:49.020
is something like a magnetic charge density,
it must then be constant,
255
00:22:49.020 --> 00:22:51.020
because the derivation disappears yes.
256
00:22:51.020 --> 00:22:54.020
That means it mustn't be time-dependent, that's
what's going on here
257
00:22:54.020 --> 00:22:56.020
Has been expressed.
258
00:22:56.020 --> 00:23:01.080
That is to say, the magnetic charge carrier
density, if it exists that
259
00:23:01.080 --> 00:23:10.080
is definitely invariant in time and ...
260
00:23:10.080 --> 00:23:13.080
Sorry, let's go back again ... was a little
too fast ..
261
00:23:13.080 --> 00:23:19.080
Please look at this equation again because
we will be in a moment
262
00:23:19.080 --> 00:23:24.080
come back to it again, it looks almost like
one of ours
263
00:23:24.080 --> 00:23:32.080
Namely, Maxwell's equation like the law of
induction, only that here at the
264
00:23:32.080 --> 00:23:48.080
Place of the E-field now here is the magnetic
current density.
265
00:23:48.080 --> 00:23:55.080
We continue with the maintenance of the magnetic
flux and use that again
266
00:23:55.080 --> 00:23:57.080
Relativity principle, we've already used that
before .
267
00:23:57.080 --> 00:24:02.040
So there is nothing new that would be added
now.
268
00:24:02.040 --> 00:24:15.040
Let us assume that the same applies in one
system as I did in the last
269
00:24:15.040 --> 00:24:20.040
Slide have just shown that the derivation
of the magnetic
270
00:24:20.040 --> 00:24:22.040
Carrier density is equal to 0.
271
00:24:22.040 --> 00:24:30.040
Then there is a value that is constant over
time at every location .
272
00:24:30.040 --> 00:24:32.040
That is the statement.
273
00:24:32.040 --> 00:24:41.040
If we now switch to another inertial system
that becomes this
274
00:24:41.040 --> 00:24:48.040
In the inertial system moves at a speed ,
then an observer would be in this
275
00:24:48.040 --> 00:24:57.040
other inertial system see a distribution that
changes over time.
276
00:24:57.040 --> 00:25:04.000
In the simplest case, just imagine that you
have a part of the
277
00:25:04.000 --> 00:25:10.000
Area, for example, a certain value of rho_magnetisch,
in the other part of the
278
00:25:10.000 --> 00:25:12.000
Area is the 0.
279
00:25:12.000 --> 00:25:20.000
Then d rho_magnetic after dt would be equal
to 0 first of all fulfilled everywhere.
280
00:25:20.000 --> 00:25:27.000
If you now change the inertial system in such
a way that you are the observer
281
00:25:27.000 --> 00:25:32.000
move relative to it, then that is the same
as if I would my hand now
282
00:25:32.000 --> 00:25:39.000
move and then from the camera perspective
it looks like it would
283
00:25:39.000 --> 00:25:44.000
the rho_magnetic actually change over time.
284
00:25:44.000 --> 00:25:51.000
This contradiction - we have just stated that
actually not
285
00:25:51.000 --> 00:25:56.000
may happen, it has to be constant over time
- this contradiction only occurs then
286
00:25:56.000 --> 00:26:01.060
does not arise when I choose a very special
form of constancy.
287
00:26:01.060 --> 00:26:07.060
Namely, that it has the same value always
and everywhere .
288
00:26:07.060 --> 00:26:10.060
The same for all places and all times.
289
00:26:10.060 --> 00:26:19.060
If it's a complete constant, then it's relatively
logical that I should then
290
00:26:19.060 --> 00:26:21.060
also just set to 0.
291
00:26:21.060 --> 00:26:29.060
Say we come here to the fact that this magnetic
charge density always and
292
00:26:29.060 --> 00:26:31.060
is 0 everywhere.
293
00:26:31.060 --> 00:26:39.060
If it wasn't, then it would never be and then
we would have it too
294
00:26:39.060 --> 00:26:43.060
in fact, magnetic field lines are always observed
somewhere in this form
295
00:26:43.060 --> 00:26:48.060
would arise or disappear what you are not
currently doing.
296
00:26:48.060 --> 00:26:54.060
That is the statement of divergence B equal
to 0, which can be read here immediately.
297
00:26:54.060 --> 00:27:00.020
And divergence equal to 0 is the closest Maxwell
equation, which we thus too
298
00:27:00.020 --> 00:27:03.020
axiomatically derived.
299
00:27:03.020 --> 00:27:09.020
But now, as I have already said, it is still
not entirely clear what
300
00:27:09.020 --> 00:27:12.020
because actually with this j ^ phi is.
301
00:27:12.020 --> 00:27:17.020
It has the same dimension as the electric
field strength and we had yes
302
00:27:17.020 --> 00:27:22.020
from the conservation of flux already this
equation dB according to German
303
00:27:22.020 --> 00:27:23.020
plus rotation J ^ phi equals 0.
304
00:27:23.020 --> 00:27:28.020
If we compare that to Maxwell's equation for
induction -
305
00:27:28.020 --> 00:27:37.020
Rotation E plus dB after dt equals 0 - then
one could assume that here
306
00:27:37.020 --> 00:27:43.020
actually the J ^ phi is nothing else than the
E.
307
00:27:43.020 --> 00:27:47.020
And yes, it is.
308
00:27:47.020 --> 00:27:54.020
And you can do that too, we'll abbreviate it
here, from the principle of relativity to
the full
309
00:27:54.020 --> 00:27:57.020
in a similar way as we have done so far .
310
00:27:57.020 --> 00:28:06.080
So we would actually have found our last Maxwell
equation.
311
00:28:06.080 --> 00:28:14.080
At this point we could break up, but let's
not do it yet,
312
00:28:14.080 --> 00:28:20.080
we just go ahead and take stock.
313
00:28:20.080 --> 00:28:27.080
We have now derived the 4 Maxwell equations
from 1.
314
00:28:27.080 --> 00:28:29.080
Mathematics, 2.
315
00:28:29.080 --> 00:28:31.080
the principle of relativity, 3.
316
00:28:31.080 --> 00:28:34.080
charge retention, 4.
317
00:28:34.080 --> 00:28:36.080
the Lorenz force and 5.
318
00:28:36.080 --> 00:28:42.080
the maintenance of the magnetic flux and real
postulates are only
319
00:28:42.080 --> 00:28:46.080
Conservation of charge, Lorenz force and conservation
of magnetic flux.
320
00:28:46.080 --> 00:28:55.080
In addition, we actually found the continuity
equation from the
321
00:28:55.080 --> 00:29:02.040
Charge retention. And we noticed that E and
B are actually not one another
322
00:29:02.040 --> 00:29:05.040
are independent phenomena.
323
00:29:05.040 --> 00:29:11.040
Now if we add up what we have, we have a total
of 4
324
00:29:11.040 --> 00:29:18.040
Field sizes with 3 components each that would
be 12 field components and we have
325
00:29:18.040 --> 00:29:27.040
2 vectorial differential equations, so a total
of 6 components and then come
326
00:29:27.040 --> 00:29:33.040
add the equations for the divergence and that's
two more times, so make
327
00:29:33.040 --> 00:29:38.040
a total of 8 partial differential equations
in component wise.
328
00:29:38.040 --> 00:29:45.040
Of these, however, only 2 dynamic Maxwell
equations are the other 2
329
00:29:45.040 --> 00:29:46.040
Equations, namely the
330
00:29:46.040 --> 00:29:48.040
Divergence equation are always
331
00:29:48.040 --> 00:29:49.040
met when they become one
332
00:29:49.040 --> 00:29:52.040
Time are fulfilled, so something like ancillary
conditions.
333
00:29:52.040 --> 00:29:58.040
So we only have 6 dynamic equations for 12
components.
334
00:29:58.040 --> 00:30:05.000
What is still missing here are actually the
so-called material equations.
335
00:30:05.000 --> 00:30:09.000
That's 6 more equations,
336
00:30:09.000 --> 00:30:12.000
for example for the vacuum
337
00:30:12.000 --> 00:30:15.000
just written down.
338
00:30:15.000 --> 00:30:22.000
These are the well-known equations as you
know them.
339
00:30:22.000 --> 00:30:27.000
The 4.
340
00:30:27.000 --> 00:30:33.000
axiomatic assumption now is actually that
the material equation of the vacuum
341
00:30:33.000 --> 00:30:36.000
have exactly this shape.
342
00:30:36.000 --> 00:30:43.000
It can be shown that the material equations
of the vacuum are precisely these
343
00:30:43.000 --> 00:30:48.000
Must have shape if 3 points are met.
344
00:30:48.000 --> 00:30:49.000
1.
345
00:30:49.000 --> 00:30:57.000
The material equations of the vacuum are invariant
for translations and rotation.
346
00:30:57.000 --> 00:31:03.060
As a side note: this is a property of vacuum.
347
00:31:03.060 --> 00:31:05.060
2.
348
00:31:05.060 --> 00:31:16.060
The matter equations of the vacuum are local
and linear, i.e. fields am
349
00:31:16.060 --> 00:31:20.060
same place and at the same time are linked.
350
00:31:20.060 --> 00:31:25.060
Local, linear: this is also a property of the
vacuum.
351
00:31:25.060 --> 00:31:36.060
and 3rd: There is no mixing of electrical
and magnetic effects and
352
00:31:36.060 --> 00:31:38.060
we want to postulate that for the vacuum.
353
00:31:38.060 --> 00:31:46.060
in the end it is also a property of the vacuum,
because where there is nothing - where there
is none
354
00:31:46.060 --> 00:31:49.060
Matter is - how is there supposed to be a
mixture of electrical and magnetic
355
00:31:49.060 --> 00:31:50.060
Effects come about.
356
00:31:50.060 --> 00:31:59.060
You can then show, and I'm not going to do
that in great detail, that you can do that
357
00:31:59.060 --> 00:32:05.020
you may read that if spacetime is still flat,
358
00:32:05.020 --> 00:32:13.020
the material equations must result exactly
in this known form.
359
00:32:13.020 --> 00:32:22.020
That is remarkable or these are material equations
of the vacuum
360
00:32:22.020 --> 00:32:32.020
remarkable because they not only share the
excitations and the field strength with one
another
361
00:32:32.020 --> 00:32:37.020
link - that's wonderful - but they actually
link too
362
00:32:37.020 --> 00:32:44.020
electromagnetic fields with the structure
of spacetime.
363
00:32:44.020 --> 00:32:51.020
This was not yet the case with the first axioms
we considered,
364
00:32:51.020 --> 00:32:57.020
Now only at the fourth axiom, do we actually
get a shortcut to it
365
00:32:57.020 --> 00:33:03.080
Spacetime. That means here it becomes extremely
fundamental.
366
00:33:03.080 --> 00:33:12.080
Indeed, there is strong evidence that the
propagation of the
367
00:33:12.080 --> 00:33:20.080
electromagnetic field determines the metric
structure of space-time and even
368
00:33:20.080 --> 00:33:24.080
not the other way around - as you might now
from the representation I have chosen,
369
00:33:24.080 --> 00:33:30.080
could have concluded - that is the electromagnetic
370
00:33:30.080 --> 00:33:33.080
Properties follow from space-time.
371
00:33:33.080 --> 00:33:38.080
If you are really interested in this, I have
two more sources for you
372
00:33:38.080 --> 00:33:46.080
specified. In fact, it goes well beyond that
373
00:33:46.080 --> 00:33:51.080
what we will do in the course of the further
course .
374
00:33:51.080 --> 00:34:03.040
I hope you have realized that you can actually
do relatively simple things
375
00:34:03.040 --> 00:34:11.040
Assumptions and with relatively few steps
the Maxwell equations and also
376
00:34:11.040 --> 00:34:15.040
the material equations and also the continuity
equation for
377
00:34:15.040 --> 00:34:19.040
can discharge the charge.
378
00:34:19.040 --> 00:34:28.040
That is, these are equations that are actually
not somehow perfectly in
379
00:34:28.040 --> 00:34:34.040
hang out of the air, but they have a very
good and solid rationale.
380
00:34:34.040 --> 00:34:44.040
At this point I would like to thank you for
your attention.
381
00:34:44.040 --> 00:34:51.040
You can find more information and also how
to contact me on the website
382
00:34:51.040 --> 00:34:55.040
I wrote down for you here as well.
383
00:34:55.040 --> 00:34:59.040
Thank you and see you next time.