00:01
Here we're going to find the electric field of a simple quadrupole.
00:06
So ordinarily quadrupole would mean four charges, but the more important thing is that the total amount of charge in the configuration is zero, and we do not have a dipole anymore.
00:25
And furthermore, it is a symmetric arrangement with some, reflection symmetry.
00:32
So with that, let's go ahead and follow the procedure that is typically used.
00:38
So we are going to start with the electric field of a point charge, and we'll write it in vector form with instead of an r hat.
00:52
For the direction, we'll use just a position vector.
00:55
And then we're going to label our charge as one, two, and three.
01:01
And our electric field total will be the sum of those three electric fields, which i'll just kind of write as a summation, shorthand notation.
01:17
And furthermore, the last step is to use the binomial approximation to see what happens to the behavior at large distances when the separation on the charges is much less than, sorry, yeah, the separation on the charges is much less than the distance from the charges.
01:45
So here we're going to focus, to make life simple, we'll focus our attention at a point, observation point, on the x -axis, which is a distance x away from the origin.
01:59
And that's kind of why we wrote the binomial approximation the way we did.
02:07
But furthermore, we can set up our picture.
02:12
We can kind of show what the electric fields do.
02:17
So e1 points down and to the right.
02:24
E2 is symmetric about that.
02:32
And e3, sorry, let me get the numbers right.
02:37
We called e3 down below.
02:41
And e2 from the negative charge is going to be much bigger than that and point towards the origin.
02:49
So what we can see basically is that the electric field total, so we can kind of simplify things, e total will be directed to the left, or in other words to the origin along the x -axis.
03:19
And i'll keep the argument solely on the positive side of x, but indeed if the observation point goes on the other side, the flip arrangement will occur.
03:33
That is, e total will point to the origin, to the right, along the x axis.
03:41
But we can worry about that later.
03:42
The magnitude should take care of that.
03:47
So what i'd like to do is set up the 11.
03:49
Field for each charge, and we will eventually see that the y components will cancel.
04:00
So for charge 1, the thing about the little r vector is it is the observation vector minus the source vector or positioned, if we want to think about that.
04:16
So for particle 1, the observation, of course, is at x.
04:24
I could show that on my picture.
04:27
Well, yeah, we called it x.
04:29
That's good enough.
04:34
But for particle 1, the little r vector is x, and the source point is at positive a.
04:44
So subtracting the source point, we have minus a.
04:52
And we can write e1, and i'll just concentrate on the x component, is equal to k times positive q, um, x over the square root of x squared plus a squared with a cube.
05:20
So the magnitude of r in this case is just the square root of x squared plus a squared.
05:32
Using the pythagorean theorem.
05:34
And i will point out, i like to use k, but some books really prefer to keep using 1 over 4 pi, epsilon, z as the fundamental electrical constant.
05:51
So i will just remind people that that's true.
05:55
For particle number two, the r is very simple.
05:59
It's just x and zero.
06:01
The y component, and we have e2x is equal to kq with a minus 2.
06:15
Let's go ahead and write that in the right way, minus 2 kq, and then x over x cubed.
06:28
And i'll note that on the positive y -axis, of course, this is, sorry, on the positive x -axis, this will point to, to the left.
06:41
So we'll go ahead and simplify it.
06:44
But for the opposite direction, we would flip signs for all our x components.
06:53
We won't worry about that.
06:58
It's good enough to show what happens on one side.
07:03
And for the last charge, the one at the bottom, we have the position vector is xa, and of course the magnitude is just the same as it was for the top charge and the x component of the electric field will look identical.
07:31
We can see that from the picture as well.
07:42
So yeah, let's add up all those.
07:45
So we're just going to simply add together all three of these quantities.
07:53
And if we can see that the two positive charges will conspire and make a sum together of two, so everybody has a two in front of it, but the positive charges create this complicated x situation, and the negative charge by itself is a fairly simple expression.
08:38
And if we're thinking about the magnitude, we do want to think about the absolute value.
08:49
That's going to be a negative quantity.
08:56
And if we're interested in the magnitude, we want to take absolute value...