# Why do charges redistribute themselves on parallel conducting plates? Vector field vs field line view?

Is it possible to explain the migration of charge on parallel conducting plates using field lines? I've been reading about the application of Gauss's Law to determining the electric field between two conducting plates of opposite charges, same magnitude. For this question, I'm considering plates of infinite area.

A textbook that I'm using, Halliday, Resnick, and Walker makes a special point to explain that the field between the two plates is twice the field of a lone plate, but not because of superposition of the two fields, but rather because the charges redistribute themselves on the opposite facing sides to double the surface charge density. The text asserts that the charges migrate due to mutual attraction, but doesn't develop the idea further. I was puzzled because I imagined two competing effects: the attraction between the charges on opposite plates, but also the increased repulsion between like charges that results when the charge density doubles.

What strikes me as interesting is that when I think about this system with field lines (not vectors), I don't see how the field lines from one plate reach the charge on the far side of the opposite plate to draw it in.

In contrast, looking at the system with vector fields seems to offer more insight. I drew out the two plates in close proximity to each other, and consider the electric fields BEFORE any redistribution occurs (non-equilibrium state: charges distributed equally on both sides of each plate). From what I can tell, in general, the field inside each plate is the sum of the vector field contribution from its own charges (native field) and the charges on the opposite plate (external field). Since the native field in an isolated conductor is 0, only the external field influences the charges inside, and the charges are attracted to the opposite facing sides. (I don't really understand the dynamics of how the charges migrate from one side to the other, in terms of whether there is an intermediate buildup of charge in the bulk, or does the bulk act as a wire, with no accumulation of charge, just depletion on one side and accumulation on the other.) If someone knows this, I'd love to see an explanation.

Some "hand-waving" ideas.

Starting with a historical term which went through a number of manifestations - lines of force.

These lines a force can be thought of as having a number of properties which include:

• a line of force stars on a positive charge and ends on a negative charge
• lines of force are in a state of tension
• lines of force repel one another

So what you have are "stretched rubber bands" between the charges of opposite sign on the plates and as you bring the opposite charges on the plates closer together the charges at the "back" are pulled round towards the "front" overcoming the mutual repulsion of the lines.

Although the idea of a line of force may seem rather strange/weird? it is used as a visual aid and indeed up to not so many years ago these lines were counted with the unit of magnetic flux, the gauss, being one line per square centimetre with one line (of induction) being called 1 maxwell.

Using lines of force one can "explain" many electrical and magnetic phenomena and this is often done when when students start studying Physics.

An electric field line has the property that at any point on the line a tangent to the line gives the direction of the force on a point charge placed at that point and the term is sometimes used instead of line of force.
Considering the energy stored in an electric field per unit volume, $\frac 12 \epsilon E^2$, gives an alternative prospective.