Let us start a section by driving the wave equation, so in the previous section, we have introduced
the Maxwells equation and now we want to consider these equations in a vacuum.
So what does that mean?
Vacuum means we have no charges and no currents.
So both of these quantities will be zero.
So what you can see will happen is there is some kind of symmetry in the electric and magnetic fields
because now both the divergence of E and the divergence of B will be zero and the rotation of E will
be proportional to the time derivative of B and vice versa, because this term is zero and this term
is zero.
So we have these four Maxwell equations.
And also something I will introduce in the following is the speed of light, which is actually this
factor here, the square root and then the power of minus one.
So let's take our whiteboard here and I want to show you how we can derive the wave equation for E and
also for B based on these maximal equations.
So let us take this maximal equation here.
The rotation of E is equal to the time derivative of B and A minus sign.
So this is what is written here and here.
So of course what we can do is we can multiply a KNOBLER operator from the left using a vector product
and the equation is still true.
So we have the rotation of the rotation of E is equal to the rotation of minus the time derivative of
B, so we can now do is we can switch around these two derivatives.
Is something you can basically always do if you here you have the derivative with respect to the coordinates
and to you with respect to the time.
So these are independent properties coordinates.
So you can switch around these derivatives so we can write down that this is equal to minus the time
derivative of the rotation of B.
Now you can see here we have the rotation of B so we can take the second Maxwell equation.
The rotation of B is equal to this prefectly here, which is equal to one of the C Square.
And then we have the time derivative of E.
And since we have here already another time derivative, this is equal to the second derivative of E
with respect to the time.
OK, so here you can see I have still these three factors, and here I have introduced a velocity of
light.
So we have here an equation.
The rotation of the rotation of E is equal to minus one of a C square and then the second derivative
of E with respect to time.
Now, this is almost the Maxwell equation, sorry, almost a wave equation, but we need one more thing.
We need to transform this first term here and here.
We use something, some relation using these Nabala operators, which is true for every vector field.
So this is not a particular relation for E, but it is true for every vector.
So the rotation of the rotation can be expressed as this gradient of the divergence minus a dollar plus
operator of each.
So if you don't believe me, write down this equation in all of the coordinates, X, Y and Z, and
then you will see that you can transform it to this expression here now.
So so far, this is true for every week, as I said, and now we use a special property of the electric
field.
So here you can see we have the divergence of E and we know from the Maxwell's equations that the virgins
of E is zero.
So we are left only with this term here.
And this is something where it's really important that we consider vacuum, because if we would not
be in a vacuum, then this term you would not be equal to zero, because then we would have here a term
proportional to the charged density.
And then this whole derivation of the WAF equation would not be true.
But since we consider here a vacuum, we have now that this term is equal to that one here and also
it is equal to minus the plus operator of E!
So we can write down this term is equal to this term.
So this is our wave equation.
Now, to find a solution, it's most easy to use only a one dimensional problem, so then this class
operator here becomes just the second derivative with respect to the coordinate, let's say X.
So we have here one of a C squared, the second derivative of E with respect to time, minus the second
privative of E with respect to the coordinate, X is equal to zero and now we can guess the solution.
So the solution will be some cosine or sine function or more general and exponential function with an
imaginary argument here.
So the solution is some vector is zero times this exponential function e to the power of AI and then
here we have omega T minus the scale of product of a vector K and a vector.
So in the following we will learn what these quantities Omega and K are.
But so far they are just coefficients.
So if we calculate the second derivatives, you will see that since this is here in E function and an
exponential function, this exponential function will remain and we will just get the derivative.
So for this term here, we will get a square which is minus one, and then we will have Omega Square.
And for this term here we get a square and then we get K Square.
And as I mentioned in the following, we will relate Omega and K to see and then we will understand
what Omega and K are.
But as you can see, this is an exponential function within an imaginary argument.
So the real part, as you probably know, is then a cosine function and it is fluctuating in time and
it is fluctuating in the coordinated R, so it is a wave and this is an electromagnetic wave.
And as it turns out, this is light because we are now in a vacuum, which means we have no charges,
we have no currents.
So this is the assumption that we put into these maximal equations here.
So the only thing that we can have is light, because light is not a charge and is not a current.
It's just photons and it's just some excitation in forms in terms of an electromagnetic wave.
So this is the wave equation for light in vacuum and this is the solution for the electric field.
Next, we want to find the same equation and a similar solution for the magnetic field.
And this is actually a nice exercise for you to see if you really understood what we did on the previous
slide.
So I think it would be best if you would pause the video now and take a sheet of paper and a pen and
write down a very similar derivation of the wave equation for beep.
But of course, I will now show you also the solution.
So what you have to do is you once again have to consider here this Maxwell equation.
So we have the rotation of P is essentially you the time derivative of E with some perfecta.
So we have this one here.
And once again we can switch around these two derivatives, which I have done here.
And now we see that this is the rotation of E, which is essentially the time derivative of be so we
can get another time derivative.
And here the vector field of work, the field of B.
So this is the first term that we want to have.
And now the second term.
First of all, we use this vector identity, which gives us these two terms here.
And then we see that we have here the divergence of B, which is zero, which is true generally not
only in vacuum.
So we have that these two terms here are equal.
And this gives us the wave equation for B, which looks exactly the same as for E, and therefore our
solution will be the same.
But to please note that B zero can still be a different vector to e0.
But like the electric field, also the magnetic field is an electromagnetic wave.
It fluctuates in time and in the coordinates.
And then the following section we will determine the dispersion relation.
And I want to show you what Omega means and what K means.