User:Brews ohare/Sandbox13

Discrete formulation

For an array of point charges {q_j} the dipole moment of the array is defined as:

$\mathbf {d} =\sum _{j}\ \mathbf {r} _{j}q_{j}\ ,$

where the vectors {r_j} locate the charge from some origin. Assuming only pairs of opposite charges are present, the overall charge is zero:

$\sum _{j}\ q_{j}=0\ ,$

which has the interesting consequence that the polarization doesn't depend upon the choice of origin. To show this fact, suppose the origin is shifted so all the charges are now located at r_j − R. The dipole moment is then:

$\mathbf {d} =\sum _{j}\ \left(\mathbf {r} _{j}-\mathbf {R} \right)q_{j}\ =\sum _{j}\ \mathbf {r} _{j}q_{j}-\sum _{j}\ \mathbf {R} q_{j}\ ,$

and because of overall neutrality the last sum is zero, so the dipole moment is the same as originally.

Some authors introduce the notion of an induced dipole moment, defined in terms of the equilibrium positions of charges by:

$\mathbf {d} _{ind}=\sum _{j}\ \left(\mathbf {r} _{j}-\mathbf {r} _{j,eq}\right)q_{j}=\mathbf {d} -\mathbf {d} _{eq}\ ,$

with r_j,eq the equilibrium location of charge q_j, and r_j its actual position, the difference caused by displacement of the charge by the force from an applied field.

Suppose we pair positive and negative charges that are close to each other, and express the dipole moment int terms of these pairs. For example,

$\mathbf {d} =\sum _{j}\ \mathbf {r} _{j}q_{j}=\sum _{k}\ q_{k}\left(\mathbf {r} _{k+}-\mathbf {r} _{k-}\right)=\sum _{k}\ q_{k}\mathbf {s} _{k}=\sum _{k}\mathbf {d} _{k}\ ,$

where subscripts refer to the plus and minus charges of a particular dipole, and the last three sums are over pairs of opposite charges that are separated by vector displacements {s_k}, so each pair k has dipole moment d_k = q_k s_k.

To introduce the polarization density, we imagine the volume containing the charges to be divided into subregions, such that any given dipole is entirely in one subregion or another, and none straddle regions. Then the dipole moment of each subregion can be found using the above formula, and for a subregion of volume ΔV the polarization density is defined as:

$P_{\Delta V}={\frac {1}{\Delta V}}\sum _{j\in \Delta V}\ \mathbf {r} _{j}q_{j}\ .$

If as above we confine ourselves to suregions composed of charge arrays consisting only of pairs of opposite charges with dipole moments {d_k} we find:

$P_{\Delta V}={\frac {1}{\Delta V}}\sum _{k\in \Delta V}\ \mathbf {d} _{k}\ ,$

where k now labels pairs of opposite charges, and not individual charges.

In the event that different subregions contain different arrays of dipoles; differing in orientation, strength or number; the polarization of each subregion will be different, and the polarization density at any given position will correspond to the particular polarization density at that location.

Integral formulation

The polarization density in a volume Ω often is described in terms of the charge density per unit volume defined at any location r in that region ρ(r) using:

$\mathbf {P} ={\frac {1}{\Omega }}\int _{\Omega }\ d^{3}\mathbf {r} \ \mathbf {r} \rho (\mathbf {r} )\ .$

The charge density in this integral can be taken to be the true microscopic charge density, or alternatively as a macroscopic charge density if less detail is wanted. Evaluation of the formula involves both the volume itself and the surface surrounding the volume. It should be cautioned that a clearly defined separation of the bulk and surface contributions can be problematic. In particular, a molecular dipole moment d_m for a molecule m inside volume ΔV surrounding a point r composed of charges ρ_m can be defined as:

$\mathbf {d} _{m}(\mathbf {r} )=\int _{\Delta V(\mathbf {r} )}\ d^{3}\mathbf {r_{o}} \ \mathbf {r_{o}} \rho _{m}(\mathbf {r_{o}} )\ ,$

and the dipole moment of a volume of molecules ΔV as:

$\mathbf {d} _{\Delta V}(\mathbf {r} )=\sum _{m\in \Delta V(\mathbf {r} )\ }\ \mathbf {d} _{m}(\mathbf {r} )$

leading to a polarization density:

$\mathbf {P} (\mathbf {r} )\ =\ {\frac {1}{\Delta V(\mathbf {r} )}}\mathbf {d} _{\Delta V}(\mathbf {r} )\ .$

In a crystalline solid, this formulation often is taken using for Ω the volume of a unit cell. (The unit cell contains a representative group of atoms or ions that is repeated to form the entire solid. The cell can become distorted by strain, and this strain can vary with position inside a sample. For example, normally strain varies with position as the surface of the sample is approached.) Some ambiguity is introduced by the cell location unless the cell boundary lies in a region with zero charge.

Relation to bound charge

The concepts of bound and free charge densities are introduced when Maxwell's equations are applied to electrical media like dielectrics, semiconductors and so forth. The total charge density that includes all the microscopic charges is defined by the electric field. In SI units:

$\mathbf {\nabla \cdot E} ={\frac {\rho _{t}}{\varepsilon _{0}}}\ ,$

where ε₀ is the electric constant. In material media the displacement field D and polarization density P are introduced and their divergences are supplied by the so-called free charge density ρ_f and bound charge density ρ_b:

$\mathbf {\nabla \cdot D} =\rho _{f}\ ,$

$\mathbf {\nabla \cdot P} =-\rho _{b}\ .$

Using the definition of D,

$\mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} \$

the above divergence relations produce:

$\rho _{t}=\rho _{f}+\rho _{b}\ ,$

indicating the total charge is split into two populations, called free and bound charge. Because bound charge depends upon inhomogeneity in P (the divergence is zero unless P has a spatial dependence), bound charge cannot be interpreted simply as charges tied to a site, but instead is related to charges whose relative positions vary throughout the sample. For example, in a charge array of dipoles, the bound charge is not the individual, constituent charges forming each dipole, but the variation in orientation and/or strength of the dipoles from position to position. A particularly striking example is at the boundary separating a dielectric from a classical vacuum, where the polarization steps down from a finite value inside the dielectric where dipoles are located to a value of zero outside the dielectric, causing a step function variation of the polarization with position that results in a sheet of bound charge on the interface. (If the outside region is a quantum vacuum, the polarization in the vacuum is not exactly zero, but still it is very small.)

Some authors refer to bound charge as "charges of equal magnitude but of opposite signs that are held in close proximity and are free to move only atomic distances (roughly 1Å or less)." One may reasonably inquire whether this definition is equivalent to the div P definition. Looking at the definition of polarization above, a material composed only of a spatially independent array of dipoles will have a spatially independent polarization and hence a polarization with zero divergence and zero bound charge. In fact, even if the dipoles change strength or orientation under the influence of an applied electric field, but all shift by the same amount so the dipole moments still are distributed uniformly in space, there will be zero bound charge. That is the case, for example, when a dielectric sphere is introduced into a uniform electric field: the bound charge is everywhere zero except at the surface of the sphere where a surface charge appears because of the step in dipole moment upon leaving the dielectric. These results are incompatible with this paragraph's introductory word definition.

The original meaning of free and bound charges began with the observation of charges induced upon a metal electrode. When a charge was induced in the ends of a metal rod by bringing one end near an external charge, the charge on the far end left the rod when the far end was grounded. That was free charge. However, the charge at the near end next the inducing charge remained and was called bound charge.

References

http://books.google.com/books?id=FBwVY10FXpUC&pg=PA15&dq=define+polarization+charge+distribution+density&hl=en&ei=h9_PTviDBKKtiAKEwIXrCw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEsQ6AEwBA#v=onepage&q=define%20polarization%20charge%20distribution%20density&f=false Clemov Stratton

A discussion by Clemmow explicitly introduces an averaging procedure using a continuously differentiable weighting function W(r) that is shaped so the first two terms of its Taylor expansion are an adequate approximation over dimensions comparable to a molecule. In particular, W doesn't exhibit any steps. He determines the polarization as P = Σ_m p_mW(r−r_m), with m the index of each molecule in an assembly of molecules. Then div P is related to grad W, which exists as W is continuously differentiable. The bound charge would then be given by −div P = ρ_b = −Σ_m p_m∘grad_r W(r−r_m) and the charge on a molecule at r_m is given by ρ(r−r_m)=q_m W(r−r_m)-p_m∘grad_rW(r−r_m). The first term is zero if the molecules are charge neutral making q_m=0.

§Pfhorrest has famed the present Intro around constraints. Given our intuition that we can make decisions, it seems natural to suppose the topic of freely making decisions would lead to discussions about what interferes with that freedom. From a common-sense standpoint things like addiction, brainwashing, inhibition, where they arise, how they can be dealt with, and so forth, would be the content of this article.

Unfortunately, this kind of limitation upon our freedom is of almost no importance to philosophers. Rather, their attention is directed to consistency with the rest of our knowledge. So the early Greeks noticed that in nature things happened in two ways: some things always happened when other things happened first, like a stone falling to the ground when it is dropped. Other things were acts of God, or rather the gods, and occurred more or less "out of the blue". So where did our decisions fit in?

This same dilemma remains today: the phenomena explained by science occur according to laws, some mathematically formulated, and these laws suggest the future, while not to the most minute detail, in very large measure is tied to the past. Our brains appear to be a subject suitable to be studied by by science, so events occurring in the brain are seemingly governed by discoverable laws, possibly similar to those governing what science has already found. If our thoughts are governed by events in our brains, then they are governed by these laws too, and if so thoughts are not initiated by our own decisions, but by earlier events of an impersonal nature. We do know that there is a subconscious that influences our decisions, but we are not inclined intuitively to put all control out of our hands. We intuit that there is some free will, although it is somewhat bounded.

A way toward some freedom is to suppose our thoughts are not entirely a result of activity confined inside our own brain. One obvious point is that we are influenced by our culture and what happens in our brain when placed in some situation varies from one culture to another, and varies also as our culture evolves. For example, in earthquake prone regions we construct buildings differently today than decades ago, because our culture has taken our understanding of earthquakes to a more practical level. Obviously, science helped to uncover these principles, but science does not dictate that we research this phenomenon, nor that we use what we learn.

So from whence comes these cultural activities, and does this question bear upon free will? This question is connected to the interplay between individuals and the plasticity of their culture, fields studied by historians, novelists, sociobiologists, and even philosophers. My guess is that there is far to go before this interplay is understood, and the question of free will await this development and the understanding of how individual creativity and social development are coupled.