00:01
So let's start with maxwell's equation.
00:02
So we have the divergence.
00:05
The electric field, this is gauss's law, is the charge -sensity divided by epsilon and not.
00:11
And then we have the magnetic field, of course, is always divergences.
00:17
And then we have the curl of the electric field is equal to negative the bdt.
00:29
And then we have the curl of the magnetic field is equal to our mu not times our current density so mu not and then it is i always get the sign for this one on one over c squared plus one over c squared i guess i'm kind of cheating by saying one over c squared because that's a part they want us to answer later on so we'll write mu not times epsilon not times d t okay and we want to show that these somehow lead to a wave equation so if we take take this equation right here and let's take the divergence of it so the divergence of the curl of e and we'll just focus on the left -hand side so we're given the vector identity to use here that the divergence or sorry not divergence let's take a curl this is probably easier way to do it so we're we have the curl of a curl from the vector identity presented here is the dive or the gradient of the diver divergence of e minus the laplocene of e.
01:39
Okay.
01:40
And so the divergence of e we have is the gradient or the volume charge density divided by epsilon or not.
01:48
So this is equal to this times row over epsilon not minus this squared.
01:58
And then on the right hand side, remember we said this is equal to negative dbdd.
02:02
So this is equal to the negative time derivative, and we can commute the curl through the time derivative, because they're both derivatives.
02:13
So we can switch the order if we want.
02:15
So we have this, and then we have on the right -hand side, we have this equation, basically.
02:20
So if we plug in the equation for the curl of this, we have mu not j, and then plus, or i guess minus, the time.
02:35
Time derivative, or let me put the constants out front, minus mu -not, epsilon, not.
02:40
It's the time derivative, or the second time derivative, basically, of the electric field.
02:46
Let me rewrite this, so it looks a little better.
02:50
So, what we have is the time derivative of j with respect to t, plus, or let's see, how should we write this? plus the gradient of our charge density over epsilon not, mu not, is equal to, and then on the right -hand side, we're going to have the laplocyan of e minus the second -time derivative, or sorry, might -we -silver constant, so keep forgetting those mu -not, epsilon -not, second -time derivative of e with respect to t.
03:37
So this part of the equation is your current density part.
03:45
It's when you have sources.
03:46
If we're doing this in a vacuum where there are no charges or anything as we're told to do in the problem, then we assume that this is equal to zero because in a vacuum you have no charges.
03:54
So you have no current density...