Electrodynamics/Tutorials/3/4/3
Video Intro
Hi, this is Jonathan Gardner.
We're covering [section reference] of Griffiths Introduction to Electrodynamics.
I'm going to move fast, but you can always rewind.
Thumbs up and share if you appreciate my effort.
As always, questions in a video response or comments.
Let's get started.
Origin
A point charge at the origin is a "pure" monopole. When you move the origin, then your vectors to your point change, and you no longer have a "pure" monopole.
This highlights how important choosing the origin can be.
Let's say we set up a point charge over here. (Draw picture).
The dipole moment is easily calculated: (calculate it) Although there's a single charge, there is a dipole moment.
There's also a quadrupole, octopole, etc... terms because this charge is not at the origin anymore.
We could calculate the potential by using <math>\vec{\mathfrak{r}}\;</math> and thus get a clear answer. But remember, the multipole expansion doesn't care about curly r, it answers the question, "What is the potential based on the geometry of the charges, and a point r away from the origin?"
Moving a charge, or if you prefer, moving the origin, affects the multipole terms. When we moved the point charge, the monopole term didn't change, but we had to add in all the other terms to get the correct answer.
Origin and Dipole Moment
What about the dipole moment and the origin?
The rule is that if the net charge is zero, then the dipole moment doesn't depend on the origin. If there is some net charge, then the origin is important.
It should be rather obvious to see why. If you have net charges of zero, then adding up all the vectors with positive charges and subtracting all the ones with negative charges will eliminate all the vectors except vectors pointing from negative charges to positive ones. Let me show you.
If we move the origin by d, then recalculate, we can rewrite the new dipole moment in terms of the old, d, and the net charge.
(Draw, show formula)
So, next time someone says, "What's the dipole moment?" Check the net charge. If 0, then you can simply look at what the dipole moment would be for ANY origin.
If it's not 0, then you have to ask, "What's your origin?"
