rich

Electric Dipoles

4 posts in this topic

My girlfriend is in her second year of medical school, and the material she is currently studying involves electrocardiograms. Her class is given printed notes on ECG, and the notes give a basic introduction to electric dipoles. Here is how the notes define an electric dipole (almost verbatim):

"An electric dipole consists of two equal, but oppositely charged, point charges or "poles" separated by a distance d. Since a battery features two equal, oppositely-charged poles, a battery is an example of a dipole."

I was very surprised to read this. Being careful not to sound too pedantic, I told my girlfriend that most physicists would definitely not agree that a battery can be thought of as a dipole. For one, the separation distance of the battery's poles is not small with respect to the distance to your typical origin of coordinates. If a person's eye is taken as the origin, the person can almost always distinguish the two poles; resolution of the two poles is nearly perfect. Also, the charges of a dipole are typically so close together that nothing material exists in the space between the charges. The poles of a battery are separated by battery fluid, etc., and so the field of a battery does not resemble that of a physical dipole.

Of course, there is technically no such thing as a pure dipole, because a pure dipole represents two charges being separated by no distance at all. All real dipoles are physical, that is, separated by a "finite" distance. But what is meant by "finite"? Is there a rigorous definition of a physical dipole other than that the charges are separated by a "small" distance?

I doubt the true definition of a dipole makes much difference to a doctor's performance anyhow, and I know my girlfriend certainly has more important things to worry about in her studies. But as a physics student, I am curious to know when the concept of a physical dipole breaks down.

Share this post


Link to post
Share on other sites
...the charges of a dipole are typically so close together that nothing material exists in the space between the charges. The poles of a battery are separated by battery fluid, etc., and so the field of a battery does not resemble that of a physical dipole. Of course, there is technically no such thing as a pure dipole, because a pure dipole represents two charges being separated by no distance at all. All real dipoles are physical, that is, separated by a "finite" distance. But what is meant by "finite"? Is there a rigorous definition of a physical dipole other than that the charges are separated by a "small" distance? ... I am curious to know when the concept of a physical dipole breaks down.

You need to distinguish between three technical concepts: the actual "electric dipole", the "dipole moment", and the "dipole moment density" (and the related mathematical abstraction of the "dipole distribution" in particular).

A physical dipole consists of two opposite charges with a separation of any magnitude. The expression for a simple electrostatic potential due to an electric dipole depends explicitly on the separation distance, no matter how small or large (within the classical theory).

If you approximate that expression for distances far from the dipole compared to the charge separation (r>>d), the lowest order dependency is on the dipole moment consisting of the combination p=qd (as a vector), so even there the separation is not ignored. The "dipole moment" is often called a "dipole" as a shorthand, but you have to remember the distinction between the dipole and the dipole moment (or strength). In this case you are expressing the potential due to a dipole solely in terms of the physical concepts of the dipole moment, the distance away from the dipole, and the angles.

The accuracy of this mathematical representation for the far-field potential in terms of the dipole moment depends not on the smallness of the absolute separation itself, but on how far away you are from the dipole relative to the separation (i.e., in the form r>>d). The meaning of d being "small" is that d is small enough in comparison with r for the accuracy you need when you use the pure dipole moment formulation of the potential. That is a relative comparison and is entirely a matter of simplification of the mathematics which may be possible for a particular degree of precision.

That is how you decide what a "small separation" means. It doesn't mean that d itself is negligible as a physical parameter by itself or is somehow "ideally" 0 in reality or even that there is some absolute physical limit on the size of separation.

Physically, you regard the charges as a dipole if the potential depends essentially only on the dipole configuration (regardless of distances or any approximate formulation you may use). If something else important is going on (like a field leaking out of the inside of a battery), you can't usefully abstract it as a "dipole" in trying to describe its effects because the dipole configuration is no longer the essence of the behavior. Any two opposite charges in a configuration are still a dipole, but that may not help for what you are trying to do in a more complex context and in such analysis you can't rely only on the fact that a pair of charges is a dipole.

For an arbitrary charge distribution of any extent you can express the potential using a generalization of the approximation mentioned above for a pure dipole. Mathematically this is a truncated power series in the distance r from the charges, with coefficients depending on the angles and on the geometry of the complex of charges. It results in the classical multipole expansion in which the lowest order term is a Coulomb potential, the next order is the net dipole moment for the collection of all the pairs of charges at their different separations, the next order term is a quadrapole moment, etc.

The net dipole moment in that expansion gives a contribution to the potential as if there were a physical dipole at a great distance. The relative size of that term tells you how much of the field has the characteristics of a dipole. It characterizes the polarization, but does not specify a physical separation (although it results from whatever separations there are in the pairs of charges within the collection).

(A large scale, i.e, not microscopic, example is a dipole antenna with oscillating opposite charges flowing down the two ends and which results in a time dependent strong dipole moment -- but this goes beyond static fields.)

This has all been framed so far in terms of discrete sums over individual charges. At a higher level of abstraction you regard the mass of charges as a continuum and the potentials, forces, fields, densities, etc. as continuous (or piecewise continuous) functions. As with any classical macroscopic physical continuum theory, the relevant degree of precision and minimum magnitudes of distances that can be analyzed do not include the sizes of microscopic statistical fluctuations or consideration of discrete atoms and molecules, let alone electrons.

But unlike other classical continuum theories, including classical gravitation and the continuum mechanics of solids and fluids, the existence of two kinds of charge (+ and -) in electrical theory means that a simple scalar density is not enough; you also need to represent the polarization by the dipole moment density.

In the context of the mathematically formulated continuum you can represent a highly localized charge at a "point" using a delta function for the density, and in accordance with this use of the delta function, localized dipole moments are mathematically represented using the derivative of the delta function for the dipole moment density.

This involves more than a limit as the separation distance tends to 0 as in ordinary derivatives of functions because the charge magnitude in the dipole configuration simultaneously increases to infinity to keep the local value of the moment at a fixed number. In its mathematically correct and most abstract form, such a dipole moment density, like the delta function itself, is formulated as a "distribution" in the sense of "generalized functions". (The limit process must be explained in the context of how the dipole moment density acts as an integral operator, i.e., as a linear functional.)

In this context the dipole is the dipole distribution, which is based on, but is a distinct mathematically formulated concept from, either "dipole" or "dipole moment". For a clear explanation of how the dipole distribution works see the introduction to the classic Fourier Analysis and Generalised Functions by Lighthill, 1958, and the sections on distributions in volume 2 of Stackgold's Boundary Value Problems of Mathematical Physics, 1968.

This limiting process for a dipole moment with "0 separation" is, like the delta function for "point" charges, mathematically meaningful as a method for dealing with extreme localized charges within the context of precision possible for the macroscopic continuum theory, and does not imply that dipoles have no separation or a minimum separation between charges.

Representing polarization by a macroscopic density function -- whether continuously or as a "distribution" using delta functions -- implies that whatever dipoles cause it are within the limits on minimum sizes and precision of the continuum, and that certainly limits the size of the charge separation magnitude. But in this context those dipoles are not explicitly represented as physical dipoles -- only their polarization effects at the macroscopic level are explicitly accounted for. As in the simple far-field approximation for the potential due to a physical dipole, that mathematical approach does not mean that a dipole moment, let alone all physical dipoles, somehow "ideally" require no separation between charges or a minimum separation in physical reality. There is only a mathematical cutoff for which separations are explicit or implicit.

Share this post


Link to post
Share on other sites

Thank you for the very informative response, ewv. I have a lot to think about now.

I have never heard of the term "dipole distribution", but I will read up on it.

Any two opposite charges in a configuration are still a dipole, but that may not help for what you are trying to do in a more complex context and in such analysis you can't rely only on the fact that a pair of charges is a dipole.

Any two opposite charges in any configuration? Is a charged capacitor then technically a dipole?

Share this post


Link to post
Share on other sites
Thank you for the very informative response, ewv. I have a lot to think about now.

I have never heard of the term "dipole distribution", but I will read up on it.

Yes. It doesn't mean a "distribution of dipoles". "Distribution" in this context is a technical mathematical term pertaining to "generalized function" (like the delta function) which has meaning as a linear functional. The delta function is a "distribution" in that sense. The dipole distribution is relevent to your question because it directly involves the limit as the dipole separation goes to zero. The best explanation I know of to conceptually understand "functions" like the delta function in terms of actual functions and limits and how it all actually works, not just "abstractly", is in Stakgold.
Any two opposite charges in a configuration are still a dipole, but that may not help for what you are trying to do in a more complex context and in such analysis you can't rely only on the fact that a pair of charges is a dipole.

Any two opposite charges in any configuration? Is a charged capacitor then technically a dipole?

A capacitor has a large number of charges spread out on each plate, not just two opposing opposite charges. But the whole charge distribution (in the ordinary sense of distribution now) on both plates has a "dipole moment" as part of the mulitpole expansion which gives the relative strength of the equivalent dipole.

I think you already know all of this technically, except for the part on the mathematics of the delta function and dipole distributions; it's just a matter of better organizing it conceptually in your mind so you have a better understanding of what it all means.

Share this post


Link to post
Share on other sites