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.

Answer 4

We are talking here about averaging over a physically small volume (also called coarse graining or envelope approximation in other contexts.) The point is that we are dealing with a situation where our spatial resolution is limited to the scale that are much greater than the atomic size (which characterizes the rate of spatial change of the true electric and magnetic fields.) In the same time, we would like to have nice differential equations and be able to use all the associated mathematical machinery (which means formally/mathematically taking limits of length differences going to zero, when calculating the derivatives.) Hence we made an approximation - assuming that the materials do not consist of atoms/molecule,s but that they are continuous, and their properties can be characterized by empirically determined constants, such as susceptibility/permeability, Young modulus, etc.

In electrodynamics or mechanics of continuous media one usually does not need to go any further than the verbal discussion above. In solid state physics one usually goes a step further, performing expansion of the energy near the band minimum/maximum (effective mass approximation), and stating that we are interested only in the processes occurring near these extrema (which means that wave vectors are small, but the wavelengths are big - much bigger than the interatomic spacing.)

One could however arrive at this approximation formally, but defining averaged quantities, like (taking one-dimensional case for simplicity): $$ U(x)=\frac{1}{2\Delta}\int_{x-\Delta}^{x+\Delta}u(x) $$ where $\Delta$ is the characteristic physically small scale, which is much larger than interatomic spacing.

Another approach is transfomring to Fourier space and cutting off the higher harmonics: $$ u(x)=\int_{-\infty}^{+\infty} \frac{dk}{2\pi}\tilde{u}(k)e^{ikx}\rightarrow U(x)=\int_{-k_0}^{+k_0} \frac{dk}{2\pi}\tilde{u}(k)e^{ikx}, $$ where $\Delta=k_0^{-1}$ is much larger than the interatomic spacing.

I we now write the equationns (e.g., Maxwell equations) for these average quantities, there will appear small corrections due to the approximations, which we neglect. It is a somewhat tedious task... but anyone studying physics is recommended to go through it at least once in their life.

Finally, in the theory of strongly correlated systems, coarse graining takes on a new life, producing various scaling approaches and the celebrated renormalization group, providing completely new insights.

It is also worth mentioning scale-free approaches in hydrodynamics/acoustics, although these have been made obsolete by the modern computing power.)

Related: Is there a general math term for the idea behind the WKB and similar methods that assume slowly varying sources?