Good morning. Let's start now. It's time. And because I didn't have all solutions to

the task, I will explain how to solve it in the end of the lecture. Hopefully some people

are late. And hopefully I will get the solutions later. So we started the subject that is called

quantization of the electromagnetic field. But in fact, I just derived based on the Maxwell's

equations, I derived the equation for the single field component. I will remind you

that we obtained the field as a function of coordinate and time in the form of a Fourier

series over harmonics k. And here stood k's harmonics as a function of time only. And

the spatial dependence was written like this. I, k, r are the exponent. And it means that

the all spatial dependence is here and the time dependence is here. The same equation

we got for the magnetic field, b, k. I will not write it explicitly. Thank you. But the

important thing is that we got the oscillator equation for this e, k of t. I will meet vectors

here. We will not be using vectors. Further on, for the time dependence, for the amplitude

of the electric field, we got the equation e, k of t, two dots, meaning the time derivative,

double time derivative plus omega k squared e, k of t is 0. And the solution, of course,

we wrote that e, k of t is 1 half. And here, two terms there, negative frequency and the

positive frequency, e, k naught, e minus i omega k t, and plus e, k naught complex conjugated,

e i omega k t. I forgot to mention probably that because of this representation, this

representation, because the field is a real value, then its Fourier component satisfies

a certain rule, and namely, this e, k of t, if we take it complex conjugated, it will

be equal to e minus k of t. Because of the field being real, because of this being the

Fourier expansion, serious expansion. Then from this dependence of the field, we can

find the dependence on time of the magnetic field. And for this, we need to use some of

the Maxwell's equations that we wrote before, namely, I'll just write it here. So the Maxwell's

equation that we needed was nabla times b minus 1 over c squared e dot is 0. So this

equation we wrote in the previous lecture. And to find the magnetic field, I apply this

equation to the harmonic, to each harmonic of the magnetic field. So we have nabla times

b k will be 1 over c squared e k dot. And then we know that the nabla operator, if the

field has such dependence on space, is just plus i k vector times the b k component. So

far I write vectors because I need to explain how this vector product looks, which is 1

over c squared i k vector e k derivative. But then if we imagine the situation, so this

is, for instance, the k vector. And this, for instance, is the b magnetic field, b k

component. Then this vector product creates a vector directed, so we have to rotate from

k to b. And so it creates a vector in this direction. So this is k times b k. And then

what I do next, I will multiply it. Sorry, this is wrong. Why did I write it here? It's

just a derivative. And then I multiply it again by k vector. And so what happens if

I multiply the vector product of k and b by k again? So so far we got a vector sticking

out of the blackboard. And then this vector I multiply by k again. And this gives me again

the vector b k, but with a minus sign. So I had a vector sticking out. And then I multiply

it by k vector. I have a vector. So this k times k times b k, this is the vector, is

just minus k squared because I have this number. And then I have a vector parallel to b k.

So I will have a b k direction over b k. So I have the same value, just multiplied by

k squared and with a minus sign because of these directions of rotation. It means that

here I will obtain i, the minus, and then k squared. And then b k is 1 over c squared.

And here because I multiplied by k, I have to write it here. Here. And from this, Alice

Clarke or Dunn-Yt? Sorry?

b k in the expression should be normal.

No, no, no, no. Yeah, yeah, yeah, yeah, yeah, yeah, right, right, right here. Yeah, yeah,

yeah, yeah, yeah. So from this equation, I can derive b k. So b k. And now I will omit

the indices. Sorry, no, let me write the vectors. So b k vector is now i goes to the numerator,

c squared denominator. And then there is the wave vector k, k squared. And then e k vector