Forcing charge carriers around corners

Starting with what we know: where the charge is in charged conductors

Simlarly charged objects repel. Put two in a vacuum and you can predict the forces (one on each charged object). There's a discussion of this in the SPT: Forces topic and more in the SPT: Force and motion topic on how those forces varies with separation (the charged objects don't need to be very close to each other). Here your needs are simple: the charged objects move apart, accelerated by the electrical forces acting on each charged object. In our simple universe, they'll move apart for ever. In a conductor, the charged objects are also free to move, but eventually get to an edge where they're not free to move, assuming the conductor is surrounded by an insulator.

In true physics style we'll start with the simplest conductor we can think of – a sphere. This is symmetric, so the charged particles will end up spread evenly over the surface as there's no reason for any other distribution of the surface charge. The sphere will be either positively or negatively charged (more on how to achieve this in the SPT: Forces topic), but all of those charged particles will be at the surface of the sphere.

Now we'd suggest moving to another simple shape – more like the wires which circuits are made of. To keep things simple it'll be a very very long wire, and you'll concentrate on parts far from the ends. In this way you'll be able to reason using cylindrical symmetry to predict the distributions of charged particles. Can you? We'd hope that you managed to see the connection and predict that the charged particles will be, once again, evenly spread across the surface.

Taking shorter cylinders makes the charged particle distributions harder to figure out, but the principles are the same: the charged particles are all at the surface. (The sharp corners destroy the symmetry, so you'd have to do some hard sums to predict the distributions of the charged particles, or have access to a very sensitive charge-meter to measure them – and you will not find one of those in schools.)

You can now draw the distributions of charged particles on the surfaces of isolated conductors with enough accuracy to develop the argument about where the concentrations of charged particles are in circuits. From these concentrations you'll be able to account for the steady flow of charged particles in electrical loops, however the wires are laid out on the bench.

Batteries are charge pumps

Electric circuits are not electrostatics, and the thing that makes the difference is the battery. Here we suggest that you ignore the complicated internals (leaving that to the chemists) and think of the battery as a charged particle pump.

So a simple, but very useful, model of a battery, not yet placed in a circuit, is an isolated pair of conductors, the terminals, one positively charged, one negatively charged. You can use the representations of isolated charged cylindrical conductors (imagining the terminals as short lengths of wire) to draw a functional representation of an isolated battery. It's just a linked pair of charged cylindrical conductors. The important feature of it being a charged particle pump is that the link between this pair of conductors replenishes this difference if some charged particles are removed from either cylinder. The battery pumps electrical charge because it pumps charged particles.

Adding extra uncharged cylinders to both terminals of the battery is identical in its effect to extruding the terminals. Any electrical charge that flows from the terminals to the newly added wire is replaced by the action of the electrical charge pump. You still don't have a circuit, as the ends of the wires are not joined, but you might like to think about what you expect to happen when they are.

Making connections to build electrical loops

Imagine joining the ends of two charged cylinders together. Now there's some attraction, as well as repulsion. It's not too hard to imagine the end state, and even the average intermediate behaviour, although the detail of the evolving situation is likely to be complex. There will be a steady neutralisation of the charge, from the centre outwards, as the concentrations of charge dilute by neutralisation.

Now imagine connecting two ends of the battery to two cylinders, which are the models for wires. Each cylinder will become electrically charged – it's just like the battery with extruded terminals. Now join the far ends of the cylinders together, to form an electrical loop. As the battery is an electrical charge pump, the inboard end of the cylinders, connected to the terminals, will remain charged, but the ends where the two cylinders are joined will be neutral, as the charged particles from the oppositely charged cylinders neutralise each other's effect. The situation in the cylinders will reach a steady state like one of the intermediate states when two isolated oppositely–charged cylinders were joined.

Now you can see that there is a gradient of surface charge around the circuit, and that any mobile charged particle in the circuit will have forces acting on it by these local concentrations of charge. Let's look at a straight section of such a cylinder and try to figure out the resultant force on the charged particle. On one side there will always be more surface charge than the other. So there will be a resultant force on the mobile charged particle as a consequence, down the concentration gradient. And that seems to be the case whichever way the wire is orientated.

Round the bends in a circuit

Now concentrate on a curved section of such a cylinder: again in order to figure out the resultant force on the charged particle. Start with the straight wire: the surface charge around any circumference is uniformly distributed: it varies only as you move along the wire. Now bend the cylinder to model a bend in the circuit: there is now a greater area on the outside, so more charge, if the surface charge density remains constant. Now the resultant force on the mobile charge, from this ring of charge, is directed towards the centre of the bend, which drives the charge around the bend.

Finally, again thinking about a constant straight cylinder, the surface charge is not the only charge exerting a force on the mobile charged particles – there is also the charge on the ions in the lattice. These necessarily act as retarding forces, since the conductor is neutral, and the ions are opposite in charge to the surface charge. So the average resultant force on the mobile charge is zero: the driving force due to the surface charge is equal and opposite to the retarding force due to the ions. The average speed of the mobile charged particles in this cylinder is constant.

Here is the whole discussion as a summary, from electrically charged conductors to complete circuits.