So far, we have talked about charges, Curran's fields and forces and the previous lecture, but one
thing is missing, it's the energy because typically, for example, the classical mechanics, you can
also solve physical problems using energy equations.
For example, you can apply energy conservation laws where you transform potential energy into kinetic
energy.
So this is something we can also do for electrodynamics.
And for this, we also need to Loren's force.
So this is why I introduced it in the previous lecture.
And once again, I really hope I did not confuse you too much.
What I want to tell you is that the physicists in the 19th century could not derive this Lawrence force,
but still used it.
And this is how we want to handle the things in this course as well.
We just want to use it here.
So what we want to calculate here is energies and quantities that are related to energies, for example,
to power, which is the energy generation or consumption over time.
So the consumption rate.
So this power W is equal to E of a T, as I just mentioned.
And you can tell if you if you say the energy or the work that you need to apply is equal to the force
times distance, then this becomes minus to four times the velocity.
For example, consider such a geometry here when you want to pull some mass up a mountain.
Then there were the power that you that you ordered, the energy that you consume.
So that's the power which tells us the energy per second, for example, is equal to the force, so
is proportional to the mass times, the velocity with which you are pulling.
So if you are pulling faster than you require more power and if you're poor slower, then it's less
power.
Now, in our case, the force is the of course, the Lawrence force.
So we have the power is equal to the negative charge of the particle times, the electric field, plus
this vector product here.
And then we have another velocity term.
And now what's really interesting is we have two terms here.
First one is minus cubed times.
He taught V, but then we have another term which is proportional to the vector product of three times
B and then scalar product with B.
Now what happens here is that this or every every vector product of two vectors is always perpendicular
to both of these vectors.
So V times B is perpendicular to V, but when we then calculate the scalar product, we have to scale
our product of two perpendicular vectors, which always is zero.
So the second term does not contribute to the power and this is well known the power that there is no
power consumption from magnetic fields.
So we only need to consider the electric field contribution.
Now, another quantity that we can introduce is the power density.
This is our power minus Q e.V., but it's normalized to some volume.
So it's we can also say it's the charge over the volume.
So this is just a charge density.
And now we have for you to charge density times the velocity.
This is equal to a current density.
So our power density, which is you could say the energy generation density is equal to minus the current
density dot the electric field because it's a quantity that we will need in the following.
Now, what I will do is I will once again consider a Maxwell equation, this time we take the equation
that considers the rotation of the magnetic field is equal to this is actually a partial derivative
of the electric field with respect to time.
And then we have the currents as well.
So the current itself is equal to these two terms.
So I have just taken this turn, put it on the other side and you get something like this.
Now, what we can do is we can multiply with a scale of product from the right hand side, so we multiply
by the electric field.
So we do it also on the right hand side, so here I have written this down and a bit of a different
way.
So the first term Epsilon zero idot times E can be written down like this to please follow along and
see that this is really true.
So what we have to do is we have to calculate a time derivative of each square.
So what does that give us?
It gives us two times E but don't forget the inner derivative gives us two times E times the time derivative
of E, so it gives us in total Epsilon zero over two times, two times E times e dot.
So absolute zero times.
Eight times.
So that's really the first term.
Now, in the second term, we have this novela operator, which is basically the vector of partial derivatives,
acting on B in terms of a rotation, but also it will act on it.
So we have to apply we can apply a product rule.
So what we can do is we can write this down in terms of two terms.
So this is a bit more complicated, but you can really figure it out yourself if you write all of this
down in terms of the components and you will see that it's equivalent to one of them zero times beat
times to rotation of E, where at this time the novel operator only acts on E, and then we have heard
this term, one of them use zero and then the divergence of this vector product of B.
And so here you first have to calculate the the vector product and then calculate the divergence.
OK, so what can we do here?
What you can see here is the rotation of E!
And if you think about a Maxwell equation, the rotation of E is equal to minus B dot.
So we have here one over Muzio times B times B dot.
So this is very similar to this one here.
But we had E times e dot so we can use again such a trick, very write it down as to in terms of a derivative,
with respect to time and some factors.
And then we write here B square and not E Square.
So I done this here we have now another term, the time derivative with respect to time, as I, as
I said.
And then we have these three factors which are in this case, one of them use zero and then one half.
And if we calculate the time derivative of this P squared term here, we had two times B times B dot.
So this gives us the two councils.
With this two, we get A B and we get B dot, which is this one.
And this one is actually minus B dot.
But here's minus, here's A plus.
So it really works out.
And here we get another term which where we can just swap around the order.
So if you swap the two terms of a vector product, it gives us a negative sign.
So this makes it here.
Plus the divergence of this term here and now we can identify the shape of this equation with a continuity
equation.
So we have some term here, some scalar that is equal to the time derivative of some scalar because
E squared, B square, both scalars and another term, which is a divergence.
So this one is a vector.
So we have here this is just from the previous slide to last lines.
We have here a continuity equation for the term minus J times E, which is our power density as we have
established in the beginning of this lecture.
So we have established a continuity equation for the energy density where we have here a source or sink
term.
So this is how power can be generated.
This is proportional to the current density times eat.
Then we have the energy density itself, which is this term here and where we have to act.
That's the time derivative act on.
So this is Epsilon Zero Square and then we have another term proportional to be square and then we have
another term, which is this divergence here, which is the where does this as vector?
Here is the energy current density.
This is the pointing vector.
And you could say this tells us along which direction energy is transferred.
And we will use this pointing vector in the next main section where we will discuss electromagnetic
waves.
So I will come back to these results here.
But what we have done in this lecture is we have taken and one of the Maxwell equations and have shown
that it can be transformed to a continuity equation for the energy density.
And this allowed us to establish several terms, for example, the pointing vector or the energy density
and show how they relate to the electric and magnetic fields.
For example, you can see the energy density is proportional to the sum of each square and B square,
and the pointing vector, which gives us the direction of energy transfer, is perpendicular to the
electric field and perpendicular to the magnetic field.
Due to this vector product, you.