00:01
Here i'm going to be looking at the dipole potential.
00:04
So for a point dipole.
00:08
A reminder of what p is, p is the vector dipole moment.
00:14
And what it does is it points from a positive charge, sorry, from a negative charge inside of the dipole itself to the positive charge.
00:30
So it points straight along there.
00:33
And it chose the direction that an electric field would line up the dipole if the dipole is placed in an external field.
00:45
What the r hat represents is it is a unit vector showing the orientation of p with respect to the observation point.
01:11
Of p with respect to observation point.
01:26
And i'll do some examples before we get into the heart.
01:33
And of course, r is the distance between this point, dipole, and the observation point.
01:50
So let's take a look at an example.
01:52
Maybe i'll draw just a single dipole moment.
01:56
And this is known as coordinate -free representation.
02:00
So it should not matter where x and our y is, but let's just show a p vector, maybe a little bit exaggerated, but we'll pretend that the dipole point dipole is right at the tail of the vector.
02:17
That's always an origin.
02:20
And let's pick a few observation points.
02:23
Let's pick observation point a, observation point b, and observation point c.
02:33
So those are just observers.
02:37
And let's think how we would find the potential at each of these points.
02:41
So what you would want to do is to join the tail of the p vector to your point.
02:49
And the length of that, of course, is r.
02:57
Okay, i'm not drawing very straight, but you get the idea that there are three different distances.
03:03
I can move c so it looks a little bit on top of that connector point.
03:12
And the r hat just points along each of these.
03:18
So what you're doing with p .r is you're taking a projection of p onto that little r -hat factor.
03:28
And p .r -hat is basically the size of that dipole moment, which, by the way, is just the separation.
03:42
Well, we don't even have to worry about it.
03:44
If it's a point dipole, we don't need to worry about how it's geometrically being produced.
03:51
But the dot product is p times the cosine of the angle in between those two vectors.
03:59
So if we take a look at all three of the angles, so here's theta a, theta b, and theta c.
04:14
We can actually tell the sign of the potential.
04:18
So theta a is a little bit over 90 degrees.
04:22
So that means that your cosine of something over 90 degrees is going to be a little bit negative.
04:30
So potential at point a is negative because theta a is bigger than pi over two, but less than pi.
04:46
Theta b is almost exactly pi.
04:48
That was just kind of an accident.
04:51
So vb is going to be negative because theta b is approximately pi.
05:05
And vc will be positive because theta c is less than pi over 2 and cosine of something between 0 and pi over 2 is just going to be positive.
05:22
That's a first quadrant type of idea.
05:26
So we can see that this is valuable to have the dipole potential in a coordinate free form.
05:38
Where this comes into application is studying the signals created by the heart as it polarizes and depolarizes the heart muscle.
05:51
So you can actually watch the heart muscle.
05:54
Contracting using electrical signals.
05:57
And of course, that's the idea in an ekg.
06:02
So here, we want to figure out some things about the potentials that we're studying...
