Electric fields in matter

Prerequisites

What do we mean by "fields in matter"?

In our previous discussions, we have defined the electric field and the electric potential felt by a test charge by looking at all the other charges and adding up their effects on the test charge. This requires identifying all the charged particles.  But it's really looking at "particles and charges in a vacuum" because if there is any other matter nearby (by matter we mean any kind of atoms or molecules) it contains both positive charges and negative charges (electrons) which will affect the electric field and electric potential felt by a test charge.  Most biology occurs inside complex matter, often in a fluid, so it is very important to learn how surrounding matter and its positive and negative charges affect the electric fields and potentials felt by a test charge. 

Lets start with one important caveat about fields and potentials in matter:  The electric field and potential gets really complex when we really zoom in to the atomic scale!  Since forces and fields increase with decreasing distance between charges, very close to any charge in the material the field will get huge and complex.  This difficulty to define a property isn't new with our treatment of electric forces and energies. We have a similar experience with quantities like temperature, pH, and chemical concentration. On the molecular level (at the nanometer scale) these quantities fluctuate wildly depending on whether a molecule happens into the tiny volume we are considering. But if we are not talking about individual molecules, but about larger structures such as e.g. membranes at the hundreds of nm scale or the pH of mitochondria, the volume we might consider will have thousands of atoms in it. At that scale, we can easily define a temperature or a concentration. And on that scale we will also agree to define an electric field and an electric potential. But what we mean by them is a kind of smoothed average.  A true electric field or potential on the molecular scale is as difficult to describe as a temperature on that scale.

Using our toy model

In some of our previous readings ( A simple electric model: a sheet of charge, the capacitor) we have made a "toy model" of a system of many electric charges spread out over a surface. Considering the effect of each individual charge would have been a horrendous mess. We would have had to add up a huge number of individual vectors, each with their own magnitude and direction, and we'd really have no way of talking about the result.

Instead, we considered a simple model where our charges were considered to be not individual particles but a smooth distribution that was spread uniformly over an infinite flat sheet. We could (fairly) easily show that the field near to such a sheet was constant and perpendicular to that sheet, and, with a little calculus, we could calculate the field strength which turned out only to depend on the charge density on the sheet. Furthermore, we figured out just when such a toy model would be reasonable: as long as there wasn't an edge too close nearby and as long as we weren't so close to the sheet that we would see the effect of individual discrete charges.

This model is ideal for helping us figure out the overall effect of an electric field on matter, seeing the average effects, and defining parameters to describe it. Let's start with the simplest case: what happens if we put a conductor (say a block of metal) into an electric field.

The fields in a conductor

In matter in general, charges may or may not move around freely. (See Polarization for a discussion.) If there are charges in matter that can move reasonably freely through the entire body of the material it is called a conductor. Two examples are: (1) a metal, where the movable charges are electrons that are shared among a dense packing of ions  (2) an ionic fluid, where the movable charges tend to be ions, for example Na+ and Cl- as a salt solution. In a "neutral" metal or fluid, positive and negative charges balance.

Consider putting a block of conducting matter between the plates of a capacitor consisting of two infinite plates of equal and opposite surface charge density (toy model alert!) as shown in the figures below. At the left, we show the capacitor before the block has been slipped in and at the right what it looks like an instant after.               

When we put the conductor in between the plates, the electric field from the two plates will be present everywhere inside the conductor. In particular, it will be present at the positions of the movable charges within the conductor. Presumably, before it was placed in between the plates, the forces on each of the movable charges in the conductor were balanced. Now, with the addition of the fields from the capacitor plates, the force is no longer balanced. The electrons (assuming a metal block) will move opposite the electric field, attracted towards the positive capacitor plate, repelled from the negative capacitor plate.

The result will be a sheet of electrons will begin to build up on the side of the conductor nearest the positive plate, leaving a sheet of unbalanced ions on the side of the conductor nearest the negative plate: something like shown in the figure at the right.

Note the weak red (pink) charges forming on the left of the conductor and the weak blue (aqua) charges forming on the right of the conductor. These create two new sheets of charge, opposite to the ones of the capacitor.

These sheets will also produce an electric field in the conductor, but in the opposite direction of the original plates. This will reduce the total field inside the conductor, but the capacitor plates will still win and still move charges  until the sheets of unbalanced charge that have built up on the surface of the conductor are EQUAL in charge density to the charges on the capacitor plates as shown at the right.

Then the field inside the conductor will go to zero and the motion of charges will stop. There will be no field inside the conductor. This gives us our first foothold result:

The electric field inside the body of a static conductor (no moving charges) is zero.

We include the restriction "static" since if charges are moving through the conductor — like when an electric current is still flowing, we can have an electric field.

Since the change in potential between two points is the integral of the electric field times the distance, if a conductor has no field inside there can be no change in potential from one point of the conductor to another. This gives us our second important foothold result:

The entire body of a static conductor (no charges moving through it) is at the same potential.

We can see what happens to the capacitance of the capacitor if we put a block of conductor in it. Since the potential difference is the integral of the E field times the distance ($ΔV = E \times d$ if the field is constant as in our toy model), if a part of the distance now has 0 electric field, that no longer contributes to the potential difference. If our conductor has a thickness $d_c$, then there will only be E field for a distance $d - d_c$ so the capacitance will now become larger — we can store more charge separation at a lower voltage cost:

$$C= \frac{k_CA}{d-d_c}$$

The fields in an insulator: the dielectric constant

While for conductors charges can move like a fluid, for many materials, charges can only move a little, but not freely — essentially (another toy model alert!) we can think of the charges as being tethered to their counter-charges.  A polar molecule may be reoriented, or the charges on the molecule pulled slightly apart. The effect is to reduce the average E field in the material, since the charges can move a little to counter an electric field, but they can no longer move to counter an electric field all the way until the field goes to 0. The degree to which an electric field can be reduced depends on the details of the properties of the material.   The reduction of the field can be measured by determining how much the voltage goes down in a capacitor when you slip a block of the material in.  Similar to the discussion above the decreased electric field in the material leads to an increase in the capacitance of the capacitor that is filled with material.  

We define the factor by which the average field is reduced in a given material as the dielectric constant of that material, $κ$ (kappa).

$$E_{\mathrm{inside\;material}} = \frac{1}{\kappa} E_{\mathrm{if\;no\;material\;were\;there}}$$

Of course this refers to the average E field as measured by looking at the total change in the potential difference. The "real" E field will fluctuate wildly on the atomic scale.

For more details, see the pages the capacitor and the dielectric constant

Notational warning! The Coulomb constant, $k_C$, is sometimes written as $k_C=1/4πε_0$. Then the combination $κε_0$ is sometimes defined to be $ε=κε_0$. Coulomb's law for an electric field of an external charge inside matter can be written the same as in free space but with ε in the Coulomb constant instead of $ε_0$. This form includes the effects of polarizing the medium. Note, however, that in some chemistry and biology texts, "$ε$" is used to stand for "$κ$". This is misleading since then the symbols "$ε$" and "$ε_0$" which appear in the same context have different units. You often see things like $εε_0$ in a single equation to mean $κε_0$. It is then very difficult to keep units straight.

Value of the toy model

What do we learn from our toy model? Looking at all the conclusions highlighted in italics above, quite a bit. Most importantly we learned things that are true in many circumstances such as the fact that there is no field inside a conductor once the charges stop flowing. While the analysis applies our toy model as a way of thinking about a uniform field, the definitions we have developed for e.g. the dielectric constant will work even over short distances when the forces are NOT uniform (down to micrometers, anyway).

The principle that a conductor is at a constant potential turns out to be true even if there is a current in the conductor, as long as the conductor has resistance 0. This is a very valuable tool (heuristic) in solving electric circuit problems. (See Kirchhoff's principles and the follow-on examples.)

Joe Redish 2/22/12, Wolfgang Losert 3/4/13