WEBVTT - generated by VCS
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Have a wonderful day, ladies and gentlemen,
I greet you warmly
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to another video in the "Theoretical Electrical
Engineering" series. My name
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is Hans Georg Krauthäuser, I am the holder
of the professorship for "TElectromagnetic
Theory
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and Compatibility" at the TU Dresden. We are
now in our 8th
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Post in the Electrostatics block and today
it will also be the last block in the Electrical
Engineering section
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ie we will conclude the consideration of electrostatics
today. what we still lack
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now that we've looked at a whole series of
methodical things, what we're still missing,
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is actually that we take a closer look at the
area of the matter. So how is
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the interaction of the electric field now actually
with matter. With that we want now
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also begin, that brings us into a contemplation,
or into a world, between a microscopic one
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and a macroscopic view. To do this, one must
first determine
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that the Maxwell equations for the vacuum,
as we have considered them so far, are strict
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taken do not need to be changed at all. That
means we could continue to use them as is
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we have used them before. That's because matter
is essentially nothing different
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is than what we have considered so far, namely
nothing plus then charged or uncharged
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particles. We have here in matter - we do n't
want to go below the atomic level - ie
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Ultimately, we then have protons, neutrons
and electrons as elementary particles, i.e.
charged ones
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and uncharged particles. To make it clear how
thin matter actually is here,
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ie how much matter is actually in matter ,
let's do a small example. We look
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Let's look at the densest element we know of
on the periodic table. The densest element
is
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Osmium, atomic number 76, the density of osmium
is 22.6 g per cubic centimeter, atomic mass
190.23 u.
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From this information it is easy to calculate
that one mole of osmium corresponds to 8.4
cubic centimetres.
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If we now consider: What kind of particles
do we have inside? Electrons, protons, neutrons
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and we assume that they are spherical - which
is not entirely unproblematic, but let's do
it
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easy - and then assume radii for that sphere
and take values there as they are currently
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used in physics. That is, for an electron we
assume that the radius
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is about 10^ -19m. Protons and neutrons are
much larger, about a femtometer,
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so 10^-15m. Now you know from the atomic number
how many electrons, how many protons - and
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if I take the atomic mass with me - how many
neutrons you have. With the radii we can too
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calculate the corresponding volumes. We also
know how many atoms we have in one mole of
osmium
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have and we can then add up how much matter
- i.e. electrons,
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Protons, neutrons - we actually have in osmium
. And then you realize that you're in a
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Mol osmium, i.e. in 8.4 cubic centimeters osmium
contains only 0.5 times 10^ -12 cubic centimeters
of matter,
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if you really assume that they are spherical
and only count these balls. Or in other words,
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if I scale this up now so you can visualize
the numbers a bit better,
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we need 16.8 cubic hectometers of osmium -
so a cubic hectometer is a cube of
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100m by 100m by 100m and almost 17 of them
- if we have them, then we have them
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we have a cubic centimeter of matter in it.
So a piece the size of a sugar cube.
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Here, I think, it becomes clear that matter
is not at all as we normally imagine it to
be,
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but it is relatively close to the vacuum -
as crazy as that sounds. We could
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actually calculate with the Maxwell equations
of the vacuum, but then we would have to go
over
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superpose all charges that are there. We would
have to determine the locations of the charges
and
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also always take into account the changes in
location - the charges will move. the
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Information about this is not available at
all, i.e. I don't know exactly,
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where each individual particle is at the moment
and what its speed is at the moment
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is. One can of course consider that one would
use this to calculate an E-field, which is
enormous
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would be detailed because there is information
about each instantaneous position of each charge
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would put in. We must always see that most
charges will compensate each other,
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because they have different signs in a very
small place. That is if I only have one
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If I leave this place a bit, then I no longer
see any monopoly term at all from this charge.
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Going a little further away, I don't see a
dipole term from this ensemble either
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more. In other words, if we would do it like
this, then we would really make life for ourselves
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heavy and make it incredibly complicated and
what would come out of it we couldn't do that
either
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readily observe. Therefore one goes over to
a macroscopic view,
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where we try to replace the microscopic sizes
with macroscopic ones.
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This is usually done by averaging there in
a suitable way. If we look at this now
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look a little more closely, then we can say
that we have atoms, ions, molecules,
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these are larger accumulations of charge that
are in a certain place and then much further
away
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Then the next accumulation of charges is gone.
And such a charge accumulation, such an atom,
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Ion molecule, I would like to simply call it
a particle - although of course it is not
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real particle is, I have that in quotation
marks here too. between such particles
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between such areas, there is almost nothing.
Now let's number these
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particle and look at the kth particle. In this
k-th particle let charges q_i, which
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we mark each with a k, i.e. q_i^k is the i-th
charge in the k-th particle.
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It doesn't matter whether the charges are bound
in any way
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or the free charges are, that doesn't matter
at all here. They are in the places
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r_i. R_i is now actually an r_i of t - which
will appear at different times
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are in different places. Then of course we
can calculate the charge density in the region
of the kth
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particle also in good approximation - good
approximation means that we now have neighboring
particles,
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because they are so far away - then we can
neglect the charge density in the region of
the kth
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Express particle in this form. That's just
the sum over the particles
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are located at location r_i. Therefore, the
delta function r minus r_i of t comes in here.
If I
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understand this in terms of a mean value, then
I can also get over the t-dependence
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average and then the time-dependent variable
becomes the average location, so to speak
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r_i. Then I also get an average charge density
in the area of the k-th particle,
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as it is written here. One can proceed analogously
with the dipole moment. Therefore
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we have to determine where the dipole moment
should start. There is
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it makes sense to simply take the center of
charge of this k-th particle and we would do
that
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denote by R^k. Again, this is usually a function,
so R^k is R^k of t. Then can
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we write the dipole moment here in this form
and k should also indicate here that it is
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for the kth particle. And of course this dipole
moment also becomes a function of time
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being. But just like above, we can also say
here, we average over the whole and then get
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this simpler relation for the dipole moment
of the kth particle. With the help of this
charge density
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and dipole moments we can now also use effective
charge densities, effective dipole moments
and
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define effective potential. For this we have
to consider that the distances within
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of particles are usually very, very small compared
to the distances between
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the center of gravity and the observation point.
I wrote this: r^k minus R^k
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Amount, i.e. everything within a particle,
is significantly smaller than the observation
point
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minus R^k. That means - you remember the multipole
expansion - that actually only
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Monopole and dipole term can play a role. The
further away I go, the faster I fall
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the higher terms. And so it makes sense and
is obvious the scalar potential of the kth
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Write down the particle as a multipole expansion,
but deviate immediately after the dipole term.
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That is , we can approximate the scalar product
of the kth particle by one divided by 4Pi
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Epsilon_0 and then here the monopole term q^k
divided by r minus R^k plus the dipole term
p^k times
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r minus R^k divided by r minus R^k magnitude
to the power of 3. If we now have n such particles,
then we can
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again also by superposition the effective charge
density, the dipole density and the potential,
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introduce in the form that we now assume that
each particle has properties
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of each individual particle, which are assigned
locally . That means we now act as if
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actually the k-th particle concentrates at
the location R^k and would then have a total
charge q^k.
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Then, via summation of k equals 1 to n for
n such particles, we can do something like
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calculate an effective charge density. Likewise,
if we now just take the dipole moment at the
location R^k,
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the k-th dipole moment at the location R^k,
then we can calculate such a sum with the
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Write down an effective dipole moment in the
delta function. And when I have it, then
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I can, of course , also write down the effective
scalar potential in the same way as above.
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Now we are slowly getting to the macroscopic
sizes, ie we want them
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Let's now look at macroscopic charge density,
polarization and the potential. We do it like
that:
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I have already said that we will ultimately
average the effective sizes again and that
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consider the resulting sizes as the macroscopic
sizes . That means macroscopic charge density
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is the charge density, which we have always
considered so far. That's actually nothing
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other than the mean value of the effective
charge density just defined. That means concretely
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all loads are actually taken into account here.
But most compensate each other
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precisely in their effect. That's why it's
so much easier with the macroscopic charge
density
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to work. We do the same with the polarization
, we come to the macroscopic one
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Polarization by averaging over the effective
dipole moment. We can do that
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first of all write it down like this, but here
we actually have to consider that we actually
always,
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certain models may be necessary to actually
use the macroscopic polarization in response
to
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predict internal and external fields. You won't
get along here without one
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actually goes back with a model presentation
and then just for - it has to be on the fabric
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that you have in front of you, that becomes
a polar crystal that has a preferred axis
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completely different model than for an ordinary
dielectric. With that we also get
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by simply substituting in the formula before,
a macroscopic scalar potential. This is also
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the scalar potential that we have always considered
so far, which in turn is the mean
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is the effective sacral potential and is then
written in this form,
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as it says here in the first line. In the meantime
we are a bit practiced and see immediately
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that we can also write r minus r' through r
minus r' amount to the power of 3 as Nabla,
related to the
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deleted component of 1 through r minus r',
i.e. gradient of 1 minus r minus r' related
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on painted component. We now want to use this
expression to actually describe the
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approximate dielectric shift. For this we first
write down the formula again,
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like we just had. That's just the last formula
from the previous slide again
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written down. We now simply calculate: What
is divergence E? Divergence E interested
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us of course because we know it from Maxwell's
equation. divergence E would be
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nothing but divergence of minus gradient phi
because E is equal to minus gradient phi.
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Divergence gradient is Laplace. So that's minus
Laplace's phi of r. I have here at now
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Laplace operator again explicitly stated that
the Laplace operator regarding the not
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deleted component, i.e. with respect to r,
because we need the distinction right away.
Good,
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so let's write that down. Divergence E - we
simply insert into the formula,
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that we had - it's still up here. We can push
through the Laplace operator,
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it acts on r, so we can easily pull it under
the integral.
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We can also sweep past the Rho_V of r' so that
it just pops up here in the first buzzer
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1 acts through the amount of r minus r'. In
the second summand we can also pull it through
to the
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P from r' past the Nabla', so it also only
acts on 1 through r minus r' amount.
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And Laplace of 1 divided by r minus r' absolute
value, but we already know that, that's nothing
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other than minus 4 pi times the delta function.
And the delta function under the integral comes
to us
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of course very convenient, the minus 4Pi of
course just outside and then we have here,
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in the first summand, an integral over Rho_V
of r' times delta r minus r'. Well that's easy
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Rho_v of r and here with the second addend
, the delta function also helps us, of course,
but
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First of all, let's write that down explicitly
. And then see that here Nabla re
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is formed from the deleted component of delta
r minus r' , but the Nabla operator is related
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applied to the coated component and that applied
to delta relative to the uncoated component
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from r minus r' differ only in the inner derivation,
which I have here. This means,
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if I omit the dash and get it replaced by the
nabla operator regarding r
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I just use a negative sign because then I have
to compensate for the inner derivative. Yes
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and of course that has the nice advantage -
if we write it down like this - that we then
have this
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have nabla under the integral with respect
to r, which we can easily pull forward - that
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we can pull out of the integral - so that we
finally get the minus. That is
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this minus what is written here and then simply
divergence and here is P itself, because yes
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from Integral, when I've pulled out the Nabla
, then that's just the Integral
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of P of r' times Delta r minus r' and that
is P of r. So it just says minus divergence
here
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P. And yes, of course, that now calls for the
two thermals with the divergence
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summarizes and this then gives the macroscopic
equivalent of the Coulomb-Gauss law in matter,
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where you now introduce the dielectric displacement
D , i.e. we put it on one side,
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then here is divergnez Epsilon_0 times E plus
P. The Epsilon_0 is of course also
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been brought over. That's divergence D, so
say D I now define as Epsilon_0 times E
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plus P. And on the other side just stays Rho_V
of r. Lead on the way
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do we now have the dielectric shift or do we
now have the dielectric shift
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introduced. We have already become acquainted
with the axiomatic consideration, from a quite
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different perspective here actually again from
the transition from the microscopic considerations
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for macroscopic observation. And with that,
of course, we can now use Maxwell's equations
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and also simply have to write down the continuity
conditions again.
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We now have a basic equation in electrostatics
in the final form here
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changed, a changed Coulomb-Gauss law, the rest
remains as it was before.
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And for the sake of completeness, we write
down the continuity conditions again. if
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If you don't remember where that came from,
there 's a video on the behavior at interfaces,
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in general, not only in electrostatics, but
in general.
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But just write it down here again. So we have
two continuity conditions, right
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discontinuity conditions, on the one hand we
state that the normal component of the dielectric
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displacement is discontinuous. And the discontinuity
is just a surface charge density. To the
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Convention again: the normal vector n is of
course perpendicular to the trend surface and
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it is directed so that the normal vector of
the volume is 1, or of the surfaces is 1 ie
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it points from 1 to 2. And the second condition,
which is now really a continuity condition,
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which yields that the tangential component,
more precisely each tangential component of
the E-field,
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is continuous during the transition from medium
1 to medium 2 . Any tangent components because
it is clear
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if they have a normal component, then they
can have any number of tangent components
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have to. The statement is that each of these
tangential components of the E-field is continuous
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in the transition between media. Now we already
know quite a lot about these media,
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said this matter, now we should classify it
a bit more systematically. Although it is
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said from the outset, we can only scratch the
surface here, actually it will be
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extremely exciting at this point and you should
now do your own lecture on material
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about materials, we get into solid state physics
very quickly and that just blows up here
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the framework, but is very interesting. Types
of dielectrics: First of all, we have dielectrics
in the
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actual sense, or I could also say ordinary
dielectrics, these are those that
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have no internal dipole moments with no external
field . Alos if there is no external field,
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then they also have no internal dipole moment.
But then an outer field actually shifts
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Charges eg in an atom against each other and
thus generates dipole moments, thus generates
a
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Polarization. And such a polarization is also
called a deformation polarization,
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because an atom is actually deformed by the
external field. There are so-called paraelectrics,
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those are those that actually have permanent
internal dipole moments. celebrity,
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perhaps the most prominent example is water.
Water, H_20 you know. So you have two
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hydrogen atoms, one oxygen atoms. The oxygen
atom is more electronegative,
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pulls the bonding electrons a little to its
side, so that the side
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where the oxygen is is a little more negative
than the other side where the hydrogen is.,
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it's a little bit more positive. In this way
a permanent,
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internal dipole even without having an external
field. But by the thermal disorder
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these dipoles will always be oriented arbitrarily
ie in the superposition macroscopic I see
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just no polarization. But if I now create an
outer field, then that changes,
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then I get a preferred orientation of these
dipoles, which are oriented in the outer
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Field. This is temperature dependent because
temperature is thermal motion
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always the force that works against it, that
is always wanting to mess things up. So it's
coming
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to a temperature-dependent, and of course field-strength-dependent,
macroscopic one
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Polarization. This is called an orientation
polarization, because dipoles
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orient in the field. The next class are the
so-called ferroelectrics. For ferroelectrics
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there are permanent internal dipoles even without
an external field, just like before, but below
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a critical temperature. But below an internal
temperature, a critical temperature
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the internal dipoles spontaneously align with
each other. That is something,
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which is actually added here. An example is
barium titanate. Above the
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critical temperature, these substances appear
paraelectric, since they lose
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so this ferroelectric property. In the ferroelectric
field you can
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Actually reverse the direction of the spontaneous
alignment as well . It's easy to imagine
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if I now create an outer field, then I can
really force it into the other
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bring direction, but it happens, just like
you might know it from ferromagnetic,
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to hysteresis phenomena. So if I plot P over
E , I get typical hysteresis curves.
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Ferroelectric materials is an important one,
also for technical applications,
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important class of materials because ferroelectric
materials are pyroelectric and they are too
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piezoelectric. Pyroelectric means yes, if you
heat that up, you get one
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temperature-dependent strain on the sides of
these crystals. Piezoelectric means
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if you deform it, if you apply pressure , then
you get tension
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the faces of the crystals. These are both important,
also technically important, effects,
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which, as I have already said, we will not
consider further here in the context of this
event
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be able. That is, we will limit ourselves to
ordinary dielectrics and paraelectrics. And
at
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where the polarization depends on the E field,
so P is a function of E. But since we
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have no permanent polarization without an E-field,
as we might have with ferroelectrics,
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also P of E equals 0 equals 0. And with that
we can, in general, of course,
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expand the polarization in powers of E. I wrote
it down very concisely here,
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so P_i is now the ith component of the polarization
and we use that
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Summation convention, i.e. over double indices,
over 123 or xyz is summed up,
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so here we have Gamma_ij E_j is a sum, a simple
sum,
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and if I write Beta_ijk E_j E_k here, it's
a double sum over i and k. and
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the higher thermals would now typically be
omitted. Normally you just come with one
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linear and a quadratic term. These tensors
that show up here, so we
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have here a second-order tensor Gamma_ij and
a third-order tensor Beta_ijk, which are
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material-specific variables, material constants.
And the important special cases are the one
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linear dielectric, that's what we're talking
about when we're actually dealing with the
linear
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Term get along here. That is, if we don't have
a square or higher term, if that
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is a sufficiently good approximation, then
we speak of a linear dielectric. And it is
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clear that of course the linear approximation
for sufficiently small field strengths, E-field
strengths,
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actually always applies. And the bigger I make
the E, the more likely I'll be that
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quadratic therm must also be taken into account.
And we are talking about an isotropic one
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Dielectric, if these tensors reduce to scalars,
i.e. if it actually
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there is no directional dependency. That, it
must be said, is already a tougher limitation
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and is no longer related to whether I now have
a small E field strength or not,
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but that actually typically depends on the
structure. And very, very many solids,
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that have a crystal structure are actually
anisotropic dielectrics and not isotropic
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dielectrics. For linear isotropic dielectrics
- and these are the ones that we actually use
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want to look at here, or have to, because we
don't have time for much more - you define
them
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electrical susceptibility. This is a scalar
quantity in the form that you say: well, well,
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this constant of proportionality, which we
still have between P and E, we call that
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now as Chi_e times Epsilon_0. And the chi_e
is now the electrical susceptibility.
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So we can then also write the dielectric displacement
by using the electric
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Bring susceptibility in here. That is then
equal to 1 plus Chi_e times Epsilon_0 times
E. And
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1 plus Chi_e, which is then typically referred
to as Epsilon_r. And Epsilon_r is then
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the relative dielectric constant. Now of course
it would be nice to have this
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could actually calculate the dielectric constant
from atomic considerations
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ie if there were a link between the microscopic
world and the macroscopic one
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World. And yes, in fact there is that and the
answer to the question of the connection of
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Both are just the atomic polarizability or
just the Clausius-Mosotti equation. And when
we
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so we know - from models or from calculations
- how the answer of matter to a
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electric field, then - if we have this size
- then we can use it, like we do in the same
way
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will see - make the link to macroscopic sizes
. But this way is actually relative
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time- consuming, because we actually need models
of solid-state physics for this. We need
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possibly quantum mechanical calculations. In
principle, we could also say that we can
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calculate the whole thing numerically, but
I wrote in principle behind it,
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because the number of particles that we have
to take into account is so immense that
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is such a complex many-body problem that such
calculations really only work in whole,
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very large exceptional cases are really feasible.
That may change at some point, but
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At the moment, we're not really getting anywhere
with numerical calculations. The most
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one actually comes with the development of
solid state and quantum mechanical
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models - or out after a combination of solid
state physics and quantum mechanics. And such
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Models then provide the so-called atomic -
sometimes also called molecular - polarizability
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Alpha. And the link between the atomic and
the macroscopic quantities then provides the
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Clausius-Mosotti equation, which I will only
mention here. We will not derive them.
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And the Clausius-Mosotti equation actually
links the atomic side here, the
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atomic polarizability, with the macroscopic
sizes, with the macroscopic size Epsilon_r,
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i.e. the relative dielectric constant. And
here at the front still appears in this constant
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the small n. And the small n is calculated
as Avogadro's constant times density divided
by
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molar mass, i.e. number per mole times mass
per volume divided by mass per mole,
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so that overall it is nothing more than one,
in terms of dimension, a number denser
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number per volume. Good. However, as I said,
we cannot do this here
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but I find it very reassuring that you have
this link between the
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atomic and the macroscopic world can write
here in this way.
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We still have a little bit more to do in the
area of electrostatic energy
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and also the energy density. We had already
explained this in detail in the "Introduction"
video .
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I think that was. And had considered that for
the vacuum and about a consideration
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of work - how much work do I need to test charge
from infinity to a
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to bring in certain potential - and had actually
derived this formula here - the
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I was just writing down here now and which
applies to the vacuum. And they have to
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we actually change it a bit. And without me
doing the same derivation again
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do - it doesn't actually change essentially,
so we can do it
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go through the same steps again - you come
up with a slightly different form and instead
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Epsilon_0 times E square now appears here,
so E times D. And I don't think so at all
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surprising. Epsilon_0 times E would be the
D in the vacuum, so to speak, and we replace
the D in the vacuum
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now actually through matter, so that these
are absolutely compatible formulas. And of
course
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a slightly different, but completely analogous
, result then also for the energy density
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Expression. With that we would actually have
reached the end of electrostatics. We need
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don't introduce anything again with all the
formal methods. All calculation rules
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that we have found you can continue to use
without any problems. You just have to come
with me now
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the correct Maxwell equations - you always
have to work with the correct Maxwell equations
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work - but use the appropriate thermal baths
. And then typically you will
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the whole thing can only be calculated specifically
for the special case that we are dealing with
here
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have now also considered, namely for linear
and isotropic dielectrics. Good,
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that would be it too. It only remains for me,
as always , to thank you for visiting the website
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refer for more information and then hopefully
see you next time.