Class Content

Prerequisites:

• A simple electric model: a sheet of charge 
• The capacitor
• Polarization 

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 the effects. This is essentially identifying all the charged particles. It's really looking at "particles in a vacuum". But most biology occurs inside complex matter, often in a fluid. If the charges we are considering are electrons and ions and we look at them in the context of other electrons, ions, atoms, and molecules, all of which have lots and lots of charges, things become unbearably complex. 

This isn't new with our treatment of electric forces and energies. We have a similar experience with concepts like temperature, pH, and chemical concentration. On the molecular level (at the nanometer scale) they make no sense. They will fluctuate wildly depending on whether a molecule happens into the tiny volume (pixel) we are considering. But if we are not talking about individual molecules, but about larger structures such as membranes (at the micrometer scale), any small volume (pixel) we might consider will have thousands of atoms in it. At that scale, we are happy to 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 to 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 the ions that make the matter neutral; (2) an ionic fluid, where the movable charges tend to be ions, for example Na⁺ and Cl^- in a salt solution.

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 below.

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 below.

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 x 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.

The fields in an insulator: the dielectric constant

Now for many materials, charges can move a little, but not freely -- not all the way to the edge. 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, but not all the way to 0. What is the exact value depends on the details of the properties of the material. As a result, the reduction of the field is typically measured (by measuring how much the voltage goes down in a capacitor when you slip a block of the material in).

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

E_{inside material} = (1/κ) x E_{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.

Notice: Sometimes in physics the dielectric constant is combined with the Coulomb constant. Since the E field in matter is reduced by a factor of kappa, the Coulomb constant is treated as if it is reduced by a factor of kappa:

Regrettably, in biology, instead of using this epsilon, the notation "epsilon" is often used to mean "kappa". Since epsilon-0 and kappa have different units, this violates a strong physics tradition that the label cues the kind of quantity involved. (We use "L" or "D" for length or distance. We do NOT use "L" to represent a time!) We will not use this unfortunate notation in this class.

Value of the toy model

This analysis shows the value of our toy model as a way of thinking about looking at a uniform field example and therefore being able to define our parameters simply. These definitions work even over short distances when the forces are NOT uniform (down to micrometers, anyway).

Joe Redish 2/22/12