Electric currents

Prerequisites

In our studies of electric forces we have developed ideas about the forces between charged particles (Coulomb's law), the energies of charged particles (Electric potential energy), and have learned about how charges can move in matter (Polarization). In particular, in conductors (metals, ionic fluids) charge can move reasonably freely. These movements are responsible for all the electrical devices in our current (pardon the pun) large arsenal and for many of the critical processes of life. In the next few readings we will work out the concepts and laws that control the motion of charge in matter.

We will first work these out in the context of the motion of charges in metals and later turn to ionic fluids (mostly in examples and problems). One reason for this is that the basic concepts can be isolated more cleanly in the metallic context and our definitions and principles more clearly established; another is that we can do hands-on demonstrations with batteries, bulbs, and capacitors that make our results physically real with direct sensory experience.

The basic example: A battery and a bulb

If we connect the two ends of a battery to a small bulb, the bulb will light. What's going on? We know a few things that are relevant. We know that a battery is a chemical store of energy and that it's labeled by a voltage. We also know that conductors are kinds of matter in which some of the electrical charges of which the matter is made can move. But we also know that the positive and negative charges in matter attract each other strongly. Separating them, as we do when we move charge from one plate of a capacitor to another, takes work. (See The energy stored in a capacitor.) What could be happening?

A tentative model of what might be happening

We know enough about metals to know that it is composed of (positively charged) ions and (negatively charged) electrons in a very close balance. And we know that the electrons can move through the metal reasonably freely. We can build a picture of what's happening by imagining putting an electric field on a conductor. This will push the electrons in one direction and the ions in another, but since the mass of the ions are so much larger than the mass of the electrons (127,000 times as great for Copper), the electrons will move and the ions will not. If we make a loop, the electrons can push round in a circle and continue to move without building up an unbalanced charge anywhere. The picture of a piece of it might be imagined to look something like this:

We've drawn the electrons as small red spheres, bouncing around with some randomness as they are pushed to and fro by the random thermal motion of the ions, but if there is a constant E field, there will be a constant force pushing the electrons through and resulting in a drift.

Note that in our picture, the charges are always balanced in any small slice of our piece of metal, despite the fact that the electrons are moving. Any moving electrons push other electrons out of their way in front of them and so on around the loop. So the loop of electrons is moving and there is no charge build up,

If the electric force were the only force acting on the electrons, we would expect them to speed up (at a rate $a = F^{net}/m$ — see Reading the content in Newton's 2nd law). Since the bulb doesn't get any brighter, it seems like there is some kind of steady state. Maybe the electrons move at a constant velocity. If that's the case, there must be a balancing resistive force since objects moving at constant velocity experience balanced forces. (See Resistive forces.)

What we will find is that the experimental observations imply that there is a resistive force opposing the flow that is proportional to the velocity, just like the familiar viscosity in a fluid like air or water. This will lead us to a gradient-driven flow equation: Ohm's law — a result analogous to the Hagen-Poiseuille equation for fluid in a pipe.

What's really happening

In order to think about what's going on in electrical circuits, we will find it useful to think of a variety of analogies in which a driving force is opposed by a resistance. Each of these models will stress one or more aspects of the electric situation and will help us build up a good mental picture of electric current that we will be able to use both in thinking about electric devices and in currents in biological systems.

But we are occasionally asked, "what's really happening when a wire carries a current?" This turns out to be extremely complicated and involves quantum mechanics. Even electrical engineers use a mixed set of analogies involving diffusion and the shifts in quantum energy levels called band structures. But if you know a little chemistry, you can get an idea of how it works.

Conductivity in a metal is a generalization of covalent bonding in a molecule. In a covalently bound molecule, one or more electrons is "shared" by more than one atom.

All quantum objects are "delocalized" to a certain extent — spread out in space. The amount they do this goes inversely with their mass. Electrons are therefore spread out the much more than atomic nuclei. In a single atom, electrons are spread out over the atom and when covalently bound in a molecule,  the shared electron is spread out so that its probability cloud covers both atoms in the molecule. The molecule is bound because the potential energy of the shared electrons are more negative than it is when they are separated.

In a single crystal of a metal, it is like many, many atoms are covalently bound. One or two electrons from each atom are shared over many, many atoms. The electrons become highly delocalized and their delocalization results in there being a large number of closely spaced energy levels available to the electrons called an electron band. The way the electrons distribute themselves in these levels are what controls the conductivity properties of the metal.

We are not going to try to think about conductivity in this way here. Rather, we will build up a series of analogies that allow us to reason effectively about electric currents, both qualitatively and quantitatively. But first, we will quantify what we mean by electric current.

Joe Redish 2/25/12

Article 633