By Ganesh

2011-04-07 18:59:12 8 Comments

When determining the gravitational attraction between 2 solid bodies, we can simplify computations by taking their masses to be concentrated at their respective centres of mass. However, had they been electrically charged bodies and we needed to compute electrostatic attraction, there is no "centre of charge" notion available to us. ( We would evaluate a 2-dimensional integral, or equivalently, apply Gauss's law. )

Is there any simple reason for the absence of such a centre of charge? Frequently, I've got the answer: "charge just doesn't work that way".


@Emilio Pisanty 2017-04-17 11:56:13

We don't normally talk about the center of charge of distributions because most useful charge distributions are neutral, with total charge $Q=0$, and therefore it doesn't even make sense to define $$ \mathbf r_\mathrm{cc}=\frac1Q \int \mathbf r\rho(\mathbf r)\mathrm d\mathbf r. $$ This means that charged and neutral systems behave very differently in this regard.

  • For neutral systems, the equivalent quantity of interest is the electric dipole moment, $$\mathbf d=\int \mathbf r\rho(\mathbf r)\mathrm d\mathbf r,$$ which has the property that the leading term in the electrostatic potential at points $\mathbf r$ far away from the charge distribution is given by $$\varphi(r) = \frac{\mathbf d\cdot\mathbf r}{r^3} + O(1/r^3).$$ As opposed to the centre of charge, we do talk about electric dipole moments, all the time.

  • For a charged system, the dipole moment still plays a role, but it is secondary: far away from the distribution, the electrostatic potential can be better described in the form $$\varphi(r) = \frac{Q}{r}+\frac{\mathbf d\cdot\mathbf r}{r^3} + \frac{p_2(x,y,z)}{r^5} + O(1/r^4), \tag{$*$}$$ where $p_2$ is a homogeneous polynomial of degree 2; these are the first three terms in the multipole expansion of that potential. As you can see, the dipole moment is now relegated to the subleading term, with the leading term given by the charge.

    Moreover, for charged systems, the dipole moment becomes less useful as a quantity, because it becomes dependent on the position of the origin; thus, if you displace your coordinate system by $\mathbf r_0$, the dipole moment changes as $\mathbf d\mapsto \mathbf d-Q\mathbf r_0$. This means, therefore, that the centre of charge has the very specific property that

    the centre of charge of a charge distribution with nonzero total charge is the position where the distribution's electric dipole moment vanishes.

    In terms of the multipole expansion $(*)$, this is important, because the subleading term (which goes as $\sim 1/r^2$) vanishes, so you get a monopole contribution (which goes as $\sim 1/r$) corrected by a quadrupole term which goes as $\sim 1/r^3$, one order of magnitude than the dipole correction.

@Mark Eichenlaub 2011-04-07 19:32:27

There is no "center of charge" that simplifies calculations completely because there is no center gravity that does this, either.

Two rigid bodies feel gravity that is quite similar to that of point masses, but not exactly the same. Unless the objects are perfect spheres, they feel tidal forces, which depend in second-order and higher derivatives of the potential.

We can only calculate gravitational interactions as points at the center of mass if we want to ignore these second-order effects. On a day-to-day basis, using gravity for things like playing baseball with your nephew, that's fine. The tidal forces are usually very small in applications that we encounter because the phenomena are small compared to the characteristic length scales involved (e.g. a baseball's trajectory is small compared to the radius of Earth, Earth's diameter is fairly small compared to the Earth-Sun distance, etc.)

We can indeed make the same first-order approximation in electrostatics that we do in Newtonian gravity. In practice, a slightly different procedure is normally used, called the multipole expansion. First, a reference point is fixed. The total charge, called the monopole moment, is independent of this reference point, so if you want your first-order approximation to be good, it is wise to choose a reference point near the center of your charge distribution. Higher-order effects are then calculated relative to the reference point by computing dipole, quadrupole, and higher moments of the charge distribution. These higher orders are more likely to come into play in everyday life in electrostatics than in gravity because the size of the charge distributions is similar to the separation between charged bodies (e.g. two balloons you rubbed on your head and are playing with are not very far separated from each other, compared to their diameters).

You might also be interested in the hard sci-fi novel "Incandescence", in which a species of beings living inside a large asteroid orbiting a black hole observe tidal effects to discover general relativity without ever seeing the outside world. It's an interesting demonstrations of what higher-order gravity effects look like beyond the center-of-mass approximation.

The author, Greg Egan, has a web page explaining the tidal effects described in his novel here.

@becko 2011-08-19 00:45:05

I could be wrong, but I'd say that the analogue of center of mass is the dipole moment, not the monopole moment. The monopole moment is the analog of the total mass, but total mass is not the same as center of mass. Just compare the formulas for the center of mass and the dipole moment: they are the same, but with charges instead of masses.

@Mark Eichenlaub 2011-08-19 00:48:13

@becko The analog of the monopole moment is the total mass located at the center of mass. Both are involved; I considered it implicit the the amount of mass to be located at the center of mass was the total mass. Center of mass itself is simply a location and could not have a well-defined potential.

@becko 2011-08-19 00:57:45

Just as the center of mass is a location without a mass and so it does not by itself define a potential, the monopole moment of the charge (this is not the same as the monopole term in the power series of the potential) is a charge without a location and by itself it doesn't define a potential. What I'm saying is that the dipole moment of the charge, divided by the total charge, gives a location that is analogous to the location given by the center of mass.

@Mark Eichenlaub 2011-08-19 01:17:08

@becko Counterexample: the dipole moment is well-defined and usually non-zero even if the total charge is zero.

@Mark Eichenlaub 2011-08-19 01:21:51

@becko I see your point. The moments of a charge distribution are reletive to some fixed point. I'll rework the answer some to be more clear.

@Vladimir Kalitvianski 2011-04-07 19:12:12

Charges for electrostatic interaction determine the interaction force. In this respect they are similar to masses that determine the gravitational force.

However when we look at the particle dynamics, it is the particle mass that determines the particle inertia whatever the force nature is. In this respect the masses are different from charges. According to the Newton equations, the center of mass makes sense as a quasi-particle position (the system as a whole).

Instead, charge distribution can be described and is often described as system charge form-factors.

@Nic 2011-04-07 19:35:27

Defining a centre of mass relies on the symmetry of particles with a mass charge. Namely that they are all of the same sign. This often allows the problem to be reduced to a geometrically simple description. For electric charge there are two signs and therefore this simple symmetry doesnt exist at short distances, though as mentioned above it can be approximated to at large distance.

@Lagerbaer 2011-04-07 19:27:30

In the same way that bodies that are far away from you can be approximated as point masses, charge distributions that are sufficiently far away can be approximated as point charges, with total charge $Q$, and then you would, indeed, compute something like the center of charge.

This is the first part of the so-called multipole expansion: You begin by approximating a charge distribution $\varrho(x)$ by a point charge of charge $Q = \int dx \varrho(x)$. The next step would be to look at the dipole moment, $p = \int dx x\varrho(x)$, followed by the Quadrupole moment etc

@stl_piznaul 2011-04-07 19:14:14

You can define something like the center of mass for charge, but it can be undefined, when the total charge is equal to zero.

Take a look at the definition of the dipole moment as well.

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