How many dimensions does electricity have? My six year old daughter asked me this morning, 'how many dimensions does electricity have ?'
What would be the best answer bearing in mind the age !?
 A: 
How many dimensions does electricity have?

Three. The same as everything else. In physics however we often talk of 3+1 dimensions, the +1 being the time dimension. However this is a dimension in the sense of measure, not a dimension in the sense of freedom of motion. It isn't on a par with the spatial dimensions, so it's not quite right to talk of 4 dimensions. You can get this across to your daughter by asking her to jump forward or back a foot, then left or right a foot, then up or down a foot. Tell her that such motion maps out the spatial dimensions, then ask her to jump forwards or backwards a second. When she says she can't, you can say the time dimension is derived from motion, and that things have three dimensions, but they usually move too. 
Then you can tell her that electricity is made of electrons, but that if those electrons don't move, it isn't electricity. Blow in her face and tell her that wind isn't wind unless the air is moving. Then tell her that we're made of electricity because we're made of electrons. Tell her we're made of other things too, called protons and neutrons, but that you tell her about them another day. And when she asks you what an electron is, well, that's one for another day too...
The electron is usually described as a fundamental or elementary particle. When you look around for more information, it’s rather scant. You learn that it has spin ½. You learn that it has a mass of 9.109 x 10-31 kg or 511keV/c², c being the speed of light. You learn that it has a charge of −1.602 x 10−19 Coulombs or -1e, e being elementary charge. However you don’t learn much else. Instead you get mixed messages. Take a look at What is an Electron? by Frank Wilczek for a good example of that. He said “to understand the electron is to understand the world”. That’s good. But he also said “there are several inconsistent answers, each correct”. That’s not so good. Wilczek then said “the proper quantum mechanical description of electrons involves wave functions, whose oscillation patterns are standing waves”. That’s good. But he also said... 
A: Ask her how many dimensions a garden hose has.
A remarkable amount of electricity is well modeled using a garden hose as a metaphor.  You have solid analogues for current, voltage, and resistance.  There's also concepts that get close-but-not-quite for capacitors and inductors.
If you get her to explain her answer regarding the garden hose well enough, she should give you enough information to understand what she is thinking of when she is thinking of dimensions, and to phrase a more exacting answer regarding electricity itself.
A: 
Discussion started with a
  piece of paper which her teacher had described simplistically as 2D.
  I, foolishly, tried to explain that it really wasn't 2D. Being an
  inquisitive sort, she then asked how many dimensions electricity had -
  which had me stuck.

If you're going along that line, you could start with:


*

*since electricity (usually) runs through wires, it is basically 1D (only forwards/backwards, not sideways)

*however, just like the sheet of paper, the wire is really 3D and thus electricity can "run" in 3D as well

*
She also asked about lightning if anyone is in the mood



*

*Well, lightning happens to be just a perfect example of this 3D-electricity - simplified it looks like an upside-down tree after all


A: Here's an idea for what you maybe could say:
Well, there are kind of two "types" of things in the world. First, there are physical objects, like you, me, this house, and so on (here she might chime in with the toaster, or her doll, or something). These physical objects have the property of dimension, which we were talking about. A sheet of paper looks 2-d at first glance, but is actually 3-d, because it has a thickness, however tiny. You and I have three dimensions, height, length, and width (here you might illustrate with some object). There can be higher dimensions than 3, actually, but those are kind of hard to wrap our minds around.
The second "type" of thing is a force. Forces move the physical objects around, like when you push in a chair, or pull your doll out of a bin. There's a bunch of forces acting on you right now, like gravity, which pulls you down toward the center of the Earth. There's also things like electricity, which powers lightbulbs, and engines, and your iPad. These forces don't really have the property of dimension. They aren't physical things like you and I, they just push us and pull us around. So electricity isn't a thing that has the property of dimension. In other words, asking what dimension electricity has is a bit of a meaningless question. It's kind of like asking what kind of food is in the microwave when there is no food in the microwave. It hasn't been defined yet.
Hope this helps! I'll see if I can think up an explanation for lightning =) Teach your daughter the ways of the Force (ahem, physics).
EDIT:
In response to the comments, yes, you could get into electricity being carried by electrons. If you wanted to go this route, then here's what I'd say on top of the previous explanation:
Forces themselves aren't physical, but there are physical things that can "carry" forces. For example, let's say you shuffle your feet along the carpet and then touch a doorknob. You might get a shock, right? Well, you, a physical object, just "carried" a force, electricity, which allowed you to get that shock (side note: this obviously isn't completely how that works, but you can't really explain how that works without understanding the next part). Electricity is carried by electrons, which are tiny, tiny little particles that can hook up with other particles called a proton and a neutron to make an atom, sort of like you and I make up a family, or they can just float free, kind of like how you and I can be separated. It is when electrons are floating free, moving where they please, that they carry electricity.
So electrons transfer, or carry, the force of electricity between physical objects. But electrons themselves are physical, so they do have the property of dimension. They are 3-d. Don't look around you for electrons, though - they are so small that we can't even see them with the most powerful microscopes. Now, remember how I said earlier that you carry the force of electricity when you shuffle across the carpet and then touch a doorknob and get shocked? Well, that isn't quite right. Atoms, those things made up of protons, neutrons, and electrons, themselves make up every other physical object. They make up you, and me, and everything else. 
Normally, in an atom, the number of electrons and protons are the same. Electrons and protons each have a property called charge, and electrons have a -1 charge and protons have a 1 charge. When you add these two together, they cancel out to zero, right? (If she doesn't know about negative numbers, that might be a nice side lesson. I forget what I knew when I was six.) Well, this total charge of zero means the atom is something called neutral, which means it doesn't carry the force of electricity. However, the electrons in an atom can be pulled away from the atom, sometimes, or pushed into the atom. This leaves the atom with a non-zero charge, which means it can carry the electric force. When you rub your feet across the carpet, electrons are being pulled off and pushed on, giving you a charge, and when you touch the doorknob, electrons are pulled and pushed again, because in metal, it is easier to pull and push electrons, and this time, you feel the shock.
Note that this isn't a complete explanation, I'm still working on making it clearer (and closer to a higher level explanation).
A: Explain to her that electricity is just the flow of electrons through the wire. Electricity is similar to water flowing through a pipe, just way faster, and with electrons instead of water molecules. Now comes the key point: depending on the electrical setting, the electrons may flow through the volume of the wire, or through its surface (see Does electricity flow on the surface of a wire or in the interior?). In the former case, we could say that electricity is three dimensional, and of the size of the wire, but in the former, it is two-dimensional and it has the area of the wire. I hope this convinces her. Just encourage her to keep on asking questions and maybe she'll some day become a physicist :-)
