Macroscopic vs microscopic electric fields

Question

What exactly is the difference between macroscopic and microscopic electric fields? Is the macroscopic one just the average of the micro over some not-to-small volume?

Answer 1

What exactly is the difference between macroscopic and microscopic electric fields?

Microscopic electric fields are the electric fields at the atomic and molecular level, e.g, the fields between single electrons or protons responsible for chemical bonding.

Macroscopic electric fields are those associated with macroscopic objects. Examples are the field associated with parallel plate capacitors and charged spheres.

Is the macroscopic one just the average of the micro over some not-to-small volume?

It depends. A neutral molecule may have no net electric field around it, but has electric fields between its individual electrons and protons. On the other hand, the electric field is a parallel plate capacitor is the sum of the field contributions of the negative and positive charge on the plates.

Hope this helps.

Answer 2

The microscopic electric field is the electric field at a very small scale. This electric field is 1) wildly varying in space, and 2) wildly varying in time. Dealing with such a field is really messy!

We introduce the concept of a macroscopic electric field to average out the insignificant wild variations of the microscopic electric field without losing large scale variations in the electric field. It is in effect averaging out the microscopic noise.

Griffiths explains it as such:

[The macroscopic field] is defined as the average field over regions large enough to contain many thousands of atoms (so that the uninteresting microscopic fluctuations are smoothed over), and yet small enough to ensure that we do not wash out any significant large-scale variations in the field. (In practice, this means we must average over regions much smaller than the dimensions of the object itself.)

Answer 3

What exactly is the difference between macroscopic and microscopic electric fields? Is the macroscopic one just the average of the micro over some not-to-small volume?

Not exactly.

The point in introducing and distinguishing microscopic EM field, as opposed to macroscopic field, is that we are assuming a somewhat different theory.

The original EM theory, formulated by Maxwell and others, is about macroscopic bodies, charged or neutral, conductors or isolators. The characteristic properties of various materials are accounted for by some functions of position or wave vector, such as resistivity, permittivity, and so on. Here, there is no concept of material media being composed of microscopic charged particles which have immensely strong electric field near them, on microscopic scales.

Later with development of molecular theory of matter, people came to the idea that all those continuous material media from above theory are made of molecules and atoms, which are very small and contain charged particles that are even smaller. Applying macroscopic laws like the Coulomb law or the Ampere law to these lead to the idea that there are actually very strong electric and magnetic fields oscillating and changing directions on the length scale of molecules. Naturally Maxwell's equations and the force equation got adapted to these small scales. The resulting theory - often known as the Lorentz theory of electrons - works with microscopic rapidly varying and very strong EM fields, but their relation to macroscopic EM fields used for description of the same systems is complicated.

Sometimes one encounters the idea that macroscopic EM fields are "just the microscopic fields, averaged in some way, maybe over small volume of space, maybe over some probability distribution". This is not always tenable, especially if the averaging process proposed is conservative. In other words, the averaging process like integrating field over some compact volume often preserves some of the details of the microscopic fields and is sometimes even reversible, i.e. one can get the microscopic field from the averaged field. This shows that naive averaging over compact volumes isn't appropriate for deriving the macroscopic theory from the microscopic theory.

Another example, when one defines averaged field of many charges interacting with external EM wave, by integrating microscopic field over some small compact volume, one does not get the simple smooth wave known in the macroscopic EM theory of wave propagation in dielectric media. One can devise specific averaging for this problem, involving either integrating over infinite plane to get rid of variations, or integrating additionally over some probability distribution of position of the particles in the infinite plane normal to the direction of the wave propagation. Since an infinite domain is used, a smooth result for the average may be obtained that does not reveal the microscopic variations.