Electrodynamics/Tutorials/3/4/4
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.
Electric Field of a Dipole
Let's start with a charge geometry with dipole moment p.
THe potential is r hat dot p, which is simply p cos theta, the angle between p and r that goes to the point P we are interested in.
(formula)
The field is the negative of the gradient of the potential:
(formulas for Er, Etheta, Ephi)
Thus, E is:
(full formula of E)
Note that the field falls off at 1/r^3, compared to 1/r^2 for a point charge.
You might expect, and you'd be correct, that quadrupoles fall off at 1/r^4, octopoles, 1/r^5, etc...
This isn't surprising: the electric field is always 1/rth the potential -- the gradient's d/dr adds in another 1/r.
When theta is 0 or pi, then the electric field points in the same direction as r. When theta is pi/2, then the field points in the theta direction -- downwards.
It looks something like this.
(Draw)
For comparison, here's the field of a "physical" dipole:
(draw)
The difference is rather obvious: the bit near the point charges simply compress and disappear in a "pure" dipole.
Conclusion of Chapter 3
That's it for Chapter 3. In Chapter 4 we're going to see how insulators, also known as dielectrics, behave in electric fields. Hint: They're tiny dipoles!
Thanks for your time.
