00:01
Here in this problem we are given the electric field component of an electromagnetic wave that travels in the negative x direction.
00:13
So we have this y and z components of this electric field and the amplitude is oscillating in the form of a cosine function with respect to time and the position.
00:30
And we are asked to sketch, draw the electric field components y and z, and the resultant electric field at certain time points, which is just t equal to zero and t equal to pi times two omega.
00:50
And later we are going to determine the polarization of this electromagnetic wave.
00:58
Okay, let's get started with the first part.
01:01
We are going to sketch the aetric field components at time equal to 0.
01:12
So the atyric field vector becomes something like this.
01:17
We have e0 cosine kx plus pi over 2 y plus e0 cosine kx, z, z hat.
01:38
Now this term, this cosine of some angle plus pi or two is just equal to sine of the same quantity.
01:53
So we have, factoring out this amplitude a zero, we have minus sine kxy plus cosine kxx.
02:15
So this is the vector that we are going to.
02:18
Draw.
02:20
Before drawing, let us observe something.
02:24
The y and z components oscillate in such a way that the total amplitude is always given by this e0, because sine squared of this argument kx plus cosine squared kx is always equal to 1.
02:41
So we expect this rotating electric field vector so that its tip, always is always on this circle of radius e zero.
02:57
So that is the first thing we should imagine about this wave.
03:02
Now let us draw the figure.
03:07
I have my z axis over here, y over here, and x right here.
03:18
I'm going to plot the z compound first.
03:22
So we have this cosine function.
03:30
So it starts at 1 and goes like this.
03:39
And this is just an envelope for the amplitude of this z component of the electric field.
03:48
So the z component goes like this.
03:51
Okay, so we have this magnitude e0 times 1, so e0.
03:58
It decreases up until to zero and starts increasing in the opposite way and decreases again and again it starts increasing in the positive direction so this is the z component of our electric field along this x -axis now let's do the y component we have this flipped sign function and by flipped i'm it carries a minus sign so it starts from zero and it will become maximum namely minus one when the cosine first becomes zero and it will continue oscillating like that so we have this sign and it is maximum order then it will be zero and the cosine is maximum and it will be maximum again when the cosine is zero and so on okay so again we have this envelope so let us sketch the y components so we have something like that and it is a bit difficult to draw it by hand but you can live with that so that is okay now these are the y components and these are the z components.
05:37
And the total, the resultant electric field, will be like that.
05:43
Okay, let's do it point by point...