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# Surface charges and the Poynting vector

In this section we try and answer the question: 'How does energy really flow in electric circuits?'.  The answer is fairly standard university-level physics but is rarely mentioned in text books.  Richard Feynman described the theory as 'mad' (though he agreed it must be true).

## Energy is transferred through empty space outside the wires

The essential point is that energy is transferred through empty space around (and NOT in) the wires of an electric circuit via an electromagnetic field called the Poynting field, named after the 19th century English physicist John Poynting.  The direction of the Poynting field depends on electrons distributed over the surfaces of the wires in the circuit.

I have to admit it's quite a way outside my comfort zone but here is the gist.

## Distribution of charge for an isolated conductor

Imagine you have an isolated length of wire.  If you push some extra electrons onto the wire, how do they distribute themselves?  The answer is that they end up on the surface.

Electrons are different from atoms of gas in an empty container.  Atoms only exert forces on their immediate , and only when they collide with them.  The arrangement that the energy of the system is to have the atoms spread evenly throughout the container.  In other words each atom tends to get as far away as possible from its .

Electrons, on the other hand, are charged and so exert forces on ALL the other electrons.  It turns out that the energy-minimising arrangement is for the electrons to be spread around the surface of the conductor.  In other words each electron is as far away as possible from every other electron not just its .

The negative terminal of a battery tends to push extra electrons onto any wires it's connected to.  These extra electrons all end up on the surface of the wire.

In a simple DC circuit these surface charges stay static once the circuit is set-up and is running normally.  It's the electrons actually inside the wires that move.

## Surface charges help the current turn corners

But the surface charges aren't arranged evenly around the circuit.  They're bunched up in some places and spread out in others.

For example when the wire bends round a corner then the electrons have to move in a curve.  This means there have to be more surface electrons on the outside of the corner to provide the centripetel force.  If the arrangement of surface charges isn't quite right then electrons from inside the wire overshoot the curve, joining the surface charges that are already there and increasing the force on the next electron to attempt the turn.  This process happens extremely quickly and continues until the charge distribution is just right.

## The electric and magnetic fields

There is also an electric field running down the middle of the wire, which extends to just beyond its surface.  This electric field pushes the charges along against the resistance and adds to the electric field caused by the surface charges.  The resultant electric field changes its direction around the circuit as the wires form a loop back to the battery.

The moving charges inside the wire cause a magnetic field around it in the normal way.

Energy flows in a direction perpendicular to both the electric field and magnetic field at every point in space.  It's a right-hand rule.  It turns out that energy flows out of the battery in both directions just outside the wires and then into the bulb.  It's important to remember that the current doesn't flow in both directions, just the energy.

So that's the Poynting field in a nutshell.  There are lots of questions that spring to mind.  I can see that extra electrons distribute themselves on the surface but it's not clear to me that when you withdraw electrons from the positive side the net positives should be on the surface with the remaining electrons only in the centre.  I can't really convince myself of the exact mechanism for heat and light being produced in the bulb.  You probably have lots too.