And the previous two lectures, we have derived the wife equation for E and B, and we have introduced
the general solutions which were Wavves.
So these were the waves for E and also for B, where we have these exponential functions here with the
imaginary arguments.
And also we have learned already something about R and Omega.
But now in this lecture, I want to show you how the orientation of K is related to the orientation
of B and E, so on comment so that you don't wonder.
What I have done here is I have changed the sign of the of the face, which is something you can do.
So this is something that should not confuse you.
When you look up a textbook, for example, you may find that the face here is different.
So there will be different sign here.
So what this means is there is just when you have here a different, different sign here, it gives
you a different face in the exponential function.
So this is something that can be covered by this prefectly zero.
So it's something that is physically not relevant because for measuring the electric field, for example,
only the real part of this field is relevant, which is basically the cosine of this argument here.
So it does not matter if you.
Right, if you're a plus or negative, this argument here.
But of course, when you solve one particular problem, you have to use the same convention all of the
time.
But if you have two independent problems that you want to solve, you don't really have to care about
if you're right plus or negative.
It's it's not that important, really.
So here I have used this conversion where I have I and then K.R. minus only got previously.
I had eight times on McAtee minus K.R., but the physics behind it is exactly the same.
So now to understand how the orientation of K is related to eat, or B we must consider once again a
Maxwell equation.
This time we need the divergence of E is equal to zero.
So yeah, I will calculate the divergence of E.
That's what we get is once again, please think of this divergence maybe in terms of a one dimensional
problem where this is just the derivative with respect to the coordinate X and here we then would have
K Times X. So what we get is then another term, eight times K, so we get eight times K times E, zero
times.
This exponential function is equal to zero.
And now since this is the exponential function and here we have a zero, we can just divide by it.
So the condition that we are left with is that E zero times K is equal to zero.
And if you remember when you have two vectors that are or whose scalar product must be zero, this means
the vectors must either be zero themselves or they must be perpendicular to each other.
So here the solution would be that e0 is perpendicular to K.
And in a similar way, you can practice this yourself.
You can show that when you start from another Maxwell equation, where you start from the divergence
of B is equal to zero.
You will find that B zero is perpendicular to K, and you may notice this is not a vector, it is just
a scalar.
So the electric field is really given by the orientation.
I mean I mean the orientation of E is basically given by the orientation of E zero.
And the same is true for B, so we know that the the electric field is perpendicular to K and also the
magnetic field is perpendicular to K.
Now the only thing that we don't know is how R, E and B related to each other.
And to find out about this, we consider another mechanical equation, which is the rotation of E is
equal to minus the time derivative of B.
So once again, we calculate both of these equations here and we get something like this.
So on the left hand side, it's pretty similar to this one here.
But we get here a vector product and on the right hand side we get from the time derivative effect or
items minus Omega, the minus sign cancels with this minus sign.
So we just get Omega.
And now we can reintroduce E and B, so e e zero times just exponential function and B is B zero times
this exponential function.
So we get this relation here and also, I mean, it doesn't really matter that much.
We could also use here these zeroes is Provectus, because you as I told you, these are scalars.
So the orientation of B is determined by the orientation of B zero.
And also what I have done here is I have used the relation that Omega of K is equal to the velocity
of light.
And now what you can see here is that we know already that it is perpendicular to K, but we don't know
how it can be are related to each other, but now we can see how they are related.
So we have noticed two vectors here.
K and E and there are vector products is proportional to B, and we know that the vector product of
two vectors is always perpendicular to both vectors.
So this means K is perpendicular to E, E is perpendicular to be and B is perpendicular to K and the
same holds also for these zero and B zero constants.
So all of the three vectors, K, B and E. are perpendicular to each other.
So in an example that solves this problem is the following electromagnetic wave where you could say
that, for example, the magnetic field fluctuates and space and time in the X Y plane, the electric
field fluctuates in the Y Z plane, and then the vector is oriented perpendicular to this be zero and
this is zero.
And this is not the only solution for this problem.
So in the following lecture, we will discuss the different polarizations of light.