00:01
So we're given an electric field that is in the x and y direction.
00:06
So it looks something like this.
00:07
So it's like i hat plus j hat times some amplitude times the sign of k z minus omega t plus our phase angle of pi over six.
00:24
So this is propagating in the z direction.
00:28
You can kind of tell from like what wave number component is non -zero or like you know c depends on the z anyway so the magnetic field from this which we're asked to produce there's a little bit of the trick you can use so you can use you can start off with one of maxwell's equations looks like this the curl of e is the negative time derivative of b and for a plain wave note that the curl of a plane wave so let's say e is a plain wave, which this would count as a plain wave.
01:01
When we use the form of e equals some amplitude times e to the i, kr minus omega -t, something like this, this just produces a factor of i -k dot, or sorry, k -vector, cross -product with e.
01:20
You'll see where i'm going in a second.
01:22
This allows us to avoid taking a couple of curls and things like that.
01:27
And likewise, for a time derivative, so like for a time derivative of b, if we have a b of the form, and this, if you think about it, this is just the imaginary part of this, right, in some direction.
01:37
So this still, you know, would work.
01:41
This time derivative is going to be like negative i omega times b, right? assuming we have a b field of the form b, i, you know, kr minus omega t, right, which, once again, we can kind of safely assume that the reason we do this is because now this equation right here transforms into k cross e equals i omega times b because we want the b field.
02:15
And so the b field then if we divide out this, sorry, this should be an i here.
02:21
If we want the b field, what we can do is just cancel out some of these terms.
02:26
So we divide by omega on both sides...