Mapping the Electric Field and Stuff

This is really a note for Future Rhett. You’re welcome, Future Rhett. If anyone else wants to read this, please have fun.

OK, here is the problem. How do you describe the electric field around some region? Maybe that region is a dipole, or parallel plates, or some other random charge distribution?

Here are some options:

  • Equipotential lines. I assume you know what these are.
  • Electric field lines.
  • Electric field vector plot.

Let’s talk about these three. I don’t think I’m going to make example plots because I’m not sure what I want to do. Yes, I will probably do something in the near future.

Electric Field Vector Plot

Imagine you have a dipole (a positive and negative charge separated by some distance). The electric field vector can be calculated at any position (x,y,z). So for every location, there’s a vector.

But how do you display this visually? Well, you could just pick some points and plot the electric field as an arrow. Actually, I’ve done this before so I have a picture.

Image result for rhett allain dipole

Another option is to just plot the E field every cm (or some other set distance). Of course, this too has problems:

  • What about 3D?
  • What if the electric field gets too big and you have giant arrows?
  • What if the arrows are too small?
  • Can you do this on paper?

Still, I think this is probably the best option. Historically, no one ever did it this way because you pretty much need a computer to draw all those tiny arrows.

Equipotential Lines

I want to draw a picture here. OK, this is just a rough sketch.

Each of these lines represents a series of points at the same electric potential (with respect to infinity). They are fairly easy to draw and they give a good representation of the field—even though they aren’t the field. It’s just like getting the idea of a the shape of a mountain by looking at a topographical map. It’s the same thing.

How would you create these with a computer? That’s really what I want—that will make it useful for some strange charge distribution that you would have to calculate the field using a numerical calculation. Here’s what I would do:

  • Decide on the voltage line values. Do I want to do every volt or every 0.1 volts?
  • Pick a point. I don’t know where you would start—maybe near one of the charges?
  • Calculate the electric potential. I assume it’s not an even value of the potential lines. If you get 5.5 volts, you want to move down to 5 volts.
  • Now move in some direction. Check the voltage again. Did it go down? Keep moving that way. If it goes up, go the other way. If it didn’t change, turn 90 degrees.
  • Once you get to 5 volts, plot a point.
  • Move again, but find another point that is at 5 volts. Plot it.
  • Keep doing this until you get some set distance away from the starting point or you get back to the starting point.

This seems unnecessarily complicated. There’s got to be a better way. Figure it out Future Rhett.

Oh! What about this method?

  • Calculate the electric field every dx, dy point (so like on a cm grid). If the potential is a whole number 5, 4, 3, 2, 1 volts – plot a point.
  • I like this method better. More brute force.

Electric Field Lines

I feel like electric field lines are dumb. Oh sure, they give a good sketch of the electric field, but what do they mean? From my intro physics course (many years ago), I remember the following:

  • Field lines are always perpendicular to equipotential lines.
  • When field lines are closer together, the value of the electric field is greater.
  • The electric field vector is tangent to the electric field lines.

That’s about it. But how do you create these with a computer?

Here’s what I want to try:

  • Start at some point near a charge.
  • Calculate the value of the electric field vector.
  • Move in the direction of the electric field vector (some distance dr)
  • Again calculate E and make another move.
  • Keep doing this until either the electric field gets too big (in case you get near another charge) or the distance from the starting point gets over some distance.

I think this would work. I want to try it. That’s for you, Future Rhett.

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