Maxwell’s Paradox of Attraction

In 1877 Maxwell wrote a short paper entitled “On a Paradox in the Theory of Attraction”, in which he considered the gravitational force of attraction exerted by a uniform rod of matter on any particle of the rod itself. We can placing the particle in question at the origin, and let a₁ and a₂ denote the positions of the two ends of the rod, as shown below.

Note that a₁ is negative and a₂ is positive. We wish to define a variable point x₁ to the left of the origin, and a corresponding point x₂ to the right of the origin, in such a way that as the point x₁ moves from a₁ to zero the point x₂ moves from a₂ to zero. This could be done in many different ways. For example, we could set x₂ = −(a₂/a₁)x₁. However, Maxwell instead defines the relationship between x₁ and x₂ by requiring their harmonic mean to be constant. (Since one value is positive and the other negative, the harmonic mean falls outside the two values.) Thus we have

The differential of this expression immediately gives

so the incremental segments to the left and right of the origin are directly proportional to the squares of their distances from the origin. It follows that each increment exerts an equal and opposite force on the origin. This might seem to imply that the net force on the origin is zero. Indeed, since we chose the origin arbitrarily, the same would be true of every point on the rod. This is Maxwell’s “paradox of attraction”, because the conclusion contradicts the obvious fact that the force of attraction must be greater in the direction of the longer segment. By symmetry, it’s clear in the above figure that the net force exerted on the origin by the portion of the rod between +a₁ and –a₁ must be zero, so the net force of the total rod must equal the force exerted by the “excess” segment from –a₁ to a₂. Maxwell resolved the paradox by noting that, as x₁ and x₂ approach zero, the net force exerted by the segment (of the complete rod) between them does not approach zero. In fact, the portions of the rod on either side of, and adjacent to, the origin each exert infinite force on the origin, as shown by the divergence of the integral of 1/x² from x = 0 to 1. Hence we can’t actually evaluate the two components individually, we can only argue by symmetry that equal segments adjoining the origin exert equal (infinite) forces, and so the net force exerted by two segments of different length is simply equal to the force exerted by the excess portion of one of the segments over the other. In other words, the force exerted on a particle at the origin by the segment shown in the figure below is the same as the net force exerted by the entire rod in the previous figure.

This excess segment becomes progressively smaller as it approaches the origin (always relating a₁ and a₂ by their constant harmonic mean). The combined effect is a constant force.

Equation (1) can also be written in the form

where α = (a₁a₂)/(a₁ + a₂).  This makes it clear that the locus of (x₁, x₂) pairs is a hyperbola, as shown below for the case a₁ = −1, a₂ = 2, α = −2.

Equation (3) also shows that the magnitudes of (x₁ – a) and (x₂ – a) can be represented as complementary parts of the hypotenuse of a right triangle, as depicted below.

It’s worth noting that Maxwell’s paradox and its resolution are based on the premise that the rod is one-dimensional, i.e., with zero thickness. If a continuous medium of finite volumetric density actually produced infinite attraction at each point, this would have significant implications for the physical constitution of bodies. Every extended object would instantly collapse under its own weight to a single point. However, we are assured by Poisson’s formula   that the potential is well-behaved for any finite volumetric density in three-dimensional space. In contrast, the rod in Maxwell’s paradox is singular, because it has finite mass with zero volume. Hence, in the absence of any countervailing force of repulsion, such an object would instantly collapse.

If we consider, instead, a realistic rod in the shape of a cylinder with non-zero radius, the force exerted on a point at the end of the rod is finite, as shown by the integral

This represents the horizontal force exerted by a segment of a cylinder on a point as shown below.

In contrast, Maxwell’s paradox is based on an unrealistic singular distribution of matter, not quite as singular as a point-like mass, but singular nevertheless. Finite continuous objects in three dimensions with finite volumetric density do not yield infinite forces at any point, so Maxwell’s paradox is based on an unrealistic physical situation.

It’s also worth noting that Maxwell’s paradox does not depend on the supposition of an inverse square force. If the force of attraction was posited to be inversely proportional to any other power of the distance, we could construct the same paradox simply by selecting a suitable relationship between x₁ and x₂. In general, if the force varies inversely as the nth power of the distance, we can stipulate that x₁ and x₂ are related according to

The differential of this expression gives

so the incremental segments to the left and right of the origin are directly proportional to the nth powers of their distances from the origin, leading again to Maxwell’s paradox.

After dispensing with the original paradox based on a singular one-dimensional rod (with an inverse-square force of attraction), Maxwell goes on to consider the condition of equilibrium of a fluid confined within a slender cylinder, on the assumption that the elements of the fluid repel each other with a force inversely proportional to some power of the distance. This leads him to a more subtle “paradox”, but still implicitly based on a one-dimensional model. (Maxwell actually refers to a non-zero cross-sectional area for the cylinder, but still treats the problem one-dimensionally, and the cross-section plays no role in his analysis.) Let ρ(x) denote the variable density of the fluid as a function of position. If each of the elements of a fluid confined to a segment of a straight line repel each other with a force that varies inversely as the nth power of the distance, then the condition for an element (placed at the origin of our coordinates, for convenience) to be subjected to zero net force due to the incremental elements at x₁ and x₂ is

Using equation (2), this implies

In the special case of an inverse-square law, we have n = 2, and hence ρ₁ = ρ₂. From this it might seem to follow (again) that a sufficient condition for the fluid at every point of the (one-dimensional) segment to be in equilibrium is that the density be constant. But as in the “paradox of attraction”, we know that only the element at the central point of the segment would really be in equilibrium. As before, the fallacy is in the assumption that the net force exerted on a particle at the origin goes to zero as the lengths of the adjoining segments go to zero. Oddly enough, Maxwell doesn’t acknowledge that this second “paradox” is just a repetition of the first. After arriving at the conclusion that uniform density is sufficient for equilibrium of every point, he writes

We have already shewn that when the density is uniform a particle not at the middle of the rod cannot be in equilibrium, but on the other hand any finite deviation from uniformity of density would be inconsistent with equilibrium. We may therefore assert that the distribution of the fluid when in equilibrium is not absolutely uniform, but is least at the middle of the rod, while at the same time the deviation from uniformity is less than any assignable quantity.

This is a rather strange “resolution”, claiming that the density is not uniform, but differs from uniformity by less than any non-zero amount. Even setting aside the fact that the net forces on particles away from the center of the rod are computable and not vanishingly small, the claim that a physical variable is positive but “less than any assignable quantity” is curious. Maxwell’s conclusion here is somewhat reminiscent of his claim (in his great treatise on electromagnetism) that the speed of propagation of electric force by diffusion is greater than any assignable quantity – after he has just shown that electromagnetic disturbances propagate at a definite speed (the speed of light).

From considering Maxwell’s paradox (and other similar conundrums arising from singular charges) we can appreciate the deftness of Newton’s original formulation of “the quantity of matter” as the product of density and volume, thereby precluding any singularities for any finite densities. Of course, we could still posit infinite densities, but Newton’s approach at least represents one way of re-normalizing the field, by imagining that mass is distributed continuously over a non-zero volume. Whether or not field singularities exist in nature remains an open question. Modern “string theory” seeks to avoid singular point-like masses and charges by positing one-dimensional “strings” (or higher-dimensional membranes in more recent speculations) as the basic elements.

An interesting related problem – one the Maxwell didn’t consider – is to determine the distribution of a set of identical particles on a line, assuming each point-like particle is attracted to each other particle by a force inversely proportional to the nth power of the distance, and is repelled from each other particle by a force inversely proportional to the mth power of the distance, where m and n are two positive integers. This could be seen as a particular case of the theory of matter proposed by Roger Boscovitch in the 1700s. It is also a generalization of what is sometimes called “Thomson’s Problem”, which was to determine the equilibrium configurations of particles confined to certain regions (such as the surface of a sphere) under the influence of some specified force law. Here we specify two distinct forces with different dependencies on distance. Clearly the exponent in the law of repulsion must be greater than the exponent in the law of attraction, because otherwise the particles would expand indefinitely, with no static equilibrium. Four particles in three-dimensional space under the influence of a pair-wise attractive force of the form A/rⁿ and a pair-wise repulsive force of the form B/r^m, would arrange themselves at the vertices of a tetrahedron, with edge length equal to r = (B/A)^m−n.

For larger numbers of particles, or for particles confined to manifolds of fewer dimensions, the equations for the equilibrium configurations are more difficult to solve. Consider, for example, three identical particles on a line, as shown in the figure below.

We know by symmetry that the distances from the middle to the outer particles are equal, so we call this distance r, and the equilibrium equations for the two outer particles are both of the form

Simplifying and re-arranging, this gives

Given four identical particles on a line, we can argue by symmetry that the two outer separations are equal, so we have a configuration of the kind shown below.

The equilibrium equations for the two left-most particles are

Cross-multiplying gives a homogeneous equation of reciprocal degree m+n, so in terms of the ratio ϕ = R/r we get

This equation can be solved for ϕ, which can then be used to compute r from the formula

It’s interesting that the ratio ϕ = R/r depends only on the exponents m and n of the repulsive and attractive forces, not on the strength coefficients A and B. As an example, suppose the attractive force is of the form A/r² and the repulsive force is of the form B/r³, so we have n = 2 and m = 3. In this case, the equation for ϕ reduces to the condition

which has the root ϕ = R/r = 0.941406365849572…  The equation can often be simplified by making the substitution ω = 1 + ϕ, in terms of which the above polynomial is

This septic polynomial is irreducible over the rationals, as shown by the fact that it takes on prime or unit values for the eight values of ω = −43, −36, −16, −9, −6, −1, 2, and 36, each of which differs from the others by more than 2.

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