Chapter 5 The Equations of Laplace and Poisson

Mathematicians love general tools, which can be applied to a variety of different problems. This has its great advantages, since much of the difficult work can be recycled from one field to the next. It also has its disadvantages, however, as it biases reasoning by analogy over other ways of solving physical problems. As a physicist, it is important for you to understand different ways of approaching similar physical problems. Some methods will be more mathematically efficient, others may better elucidate the physics of a problem. Often there are compelling reasons to pick one way over another. This is a textbook, so in the preceding chapters we chose methods of solving problems primarily to elucidate the physics, and somewhat to stress the scientific history of the subject. This chapter covers common mathematical techniques that we can use to solve similar problems more efficiently.

There were two formulations of electrodynamics in the nineteenth century, each based on its own analogy, which were: the fluid model of electrodynamics and the astronomical model of electrodynamics. These were not incompatible, but each had its own focus.

The fluid model not only applied the continuity equation to charge conservation, but also imagined the electric field vectors much like velocity flow in a liquid. This allowed for the recycling of the mathematics derived in the context of fluid flow to electrodynamics. Maxwell’s equations are the ultimate achievement of this analogy.

The astronomical model, on the other hand, emphasized the parallels between Newton’s law of gravity and Coulomb’s law of electrostatics. This allowed for the recycling of the prior century’s worth of work on celestial mechanics by such French powerhouses of mathematical physics as:   Joseph-Louis Lagrange), Pierre-Simon Laplace (1749-1827), Adrien-Marie Legendre (1752-1833), and Siméon Denis Poisson. Perhaps the discovery of gravitational waves is a recent success of this analogy, since they are often thought of as an analogy to electromagnetic radiation. Of course, Einstein developed his idea for gravitational waves in analogy to the way Maxwell’s equations predict electromagnetic waves, so the recent detection of gravitational waves can also be seen as a triumph for the fluid point of view!

We found in Chapter 3 that it is usually easier to solve for the electric potential, rather than the electric field directly. As long as we find the potential as a function of position, we can then determine the electric field by taking the gradient of the electric potential.

In Chapter 4 we discussed how Gauss combined these two analogies, by applying Lagrange’s divergence theorem to Newton’s law of gravity and Coulomb’s law of electrostatics. When written in potential form, Gauss’s law becomes:

-\vec{\nabla }\cdot \left( \vec{\nabla }\mathbb{V} \right)=\tfrac{1}{{{\varepsilon }_{0}}}\rho -\tfrac{1}{{{\varepsilon }_{0}}}\vec{\nabla }\cdot \vec{P}\approx \tfrac{1}{\varepsilon }\rho ,

This is called Poisson’s equation, and it can be numerically solved given the appropriate boundary conditions and knowing the charge density as a function of position. In a neutral medium, this simplifies to \vec{\nabla }\cdot \vec{\nabla }\mathbb{V}=0,, which is called Laplace’s equation. While Laplace’s equation is not easy to solve analytically, certain standard techniques to do so are known.

Solving differential equations comes down to making educated guesses as to the solution and then checking to see if one’s guess satisfies the equation and the initial or boundary conditions appropriate to the problem. But Joseph Fourier inaugurated a new phase of mathematical approaches to solving boundary value problems, such as those of electrostatics, in his famous work The Analytical Theory of Heat in 1822.[1] In particular, he introduced the technique of representing a function by an infinite trigonometric series to solve a partial differential equation describing heat flow. Fourier’s heat equation, as this partial differential equation is known, reduces to the mathematical form of Laplace’s equation in the case of a stationary temperature without a heat source or to the form of Poisson’s equation when heat sources are included. Fourier’s techniques are therefore capable of being extended to solve Laplace’s equation for the electrostatic potential.

Yet Fourier did more than provide new mathematical tools to solve partial differential equations with prescribed boundary values; his work on heat theory suggested a new approach to the subject of electricity. William Thomson for example, took Fourier’s theory of heat flow as an analogy to electricity.[2]

For instance, Thomson found that the electrostatic potential behaves like the temperature in Fourier’s theory for the case in which the temperature is time-independent. By assuming point sources of heat continuously distributed on the surface of a solid, he derived an expression for the temperature mathematically identical to the expression for the electrostatic potential due to a surface charge density. By considering the case of a solid body with heat sources distributed only on its surface, so that the surface of the body is isothermal, Thomson reasoned that if a solid body has a surface charge such that the potential is constant on the surface, then the potential is constant everywhere inside the solid and the electric field (which is just the gradient of the potential) equals zero there. Since a conductor forms a surface of constant potential (i.e. an “equipotential surface”), it follows that the electric field is zero everywhere inside a conducting body.

William Thomson developed the method of images, which we will present in Example 5.6, to solve electrostatic problems by applying an analogy with Fourier’s theory of heat. Thomson first reproduced Coulomb’s result that the electric force on a test charge near a charged closed surface is perpendicular to that surface and proportional to the surface charge density. He then applied this to Fourier’s theory of heat to deduce that the density of heat sources which sustain a constant temperature on the surface of a body is proportional to the heat flux across the surface. The temperature outside of any isothermal surface depends only on the temperature on that surface, and on the heat flux at every point of the surface, as long as all heat sources are within or on the surface. Applied to electrostatics, this result reads: “the electric force due to any distribution of electric charge is the same as the force due to a fictitious distribution of charge on a surface of constant potential containing all real charges, the surface density being proportional to the electric force created on the surface by the real charges”.[3] Thomson used this to show that the electric field due to a point charge placed above an infinite conducting plane is equivalent to the field of the point charge alone plus the field due to an imaginary equal but opposite mirror image charge placed at a specific distance below the plane.[4]

Gauss, whose contributions to the study of electricity and magnetism were substantial, independently discovered many of Thomson’s results as part of his program to make electrodynamics more rigorous. Having worked with German astronomer and mathematician Friedrich Wilhelm Bessel to improve precision and accuracy in astronomical measurements, Gauss was not impressed when he visited the magnetic observatory built by the scientist-philosopher Alexander von Humboldt in Berlin in the 1830s. Gauss, who was equally adept with theory and practical experimentation, demanded more rigorous standards of measurement in electricity and magnetism, and, collaborating with the physicist Wilhelm Weber, he soon did just that.[5]

To implement his research agenda, which would culminate in the first serious efforts at mapping the Earth’s magnetic field, Gauss first developed potential theory from the scalar function of Lagrange, Laplace, and Poisson, naming it potential independently of George Green, the English physicist who had coined the term in an obscure paper published in 1828.[6] In his earlier research into the gravitational force near a massive ellipsoid, Gauss had obtained the result that the potential on a closed surface surrounding all masses will determine the gravitational potential everywhere outside of the surface. He applied this result to electrostatics in his 1839 paper on potential theory.[7]

Certain arrangements of discrete charges are so common in physics and chemistry that they are standard, and we will derive the potential and field for some of these configurations such as the electric dipole, which consists of two particles of equal but opposite charge separated by a small distance, and the more complicated electric quadrupole. We then find an expression for the potential due to an arbitrary, but localized, distribution of charge in terms of equivalent expressions for the potential due to these standard arrangements: a dipole, quadrupole, octopole, etc. While the resulting multipole expansion is exact, its utility lies in allowing us to calculate a suitable approximation to the potential and field of an arbitrary charge distribution at large distances.

Although many of the fundamental ideas of electrostatics were formulated during the latter half of the eighteenth and the very beginning of the nineteenth centuries, astronomers, mathematicians, and physicists spent the first half of the nineteenth century developing highly specialized mathematical techniques to solve specific problems involving stationary charges and the effects that they produce. In the early development of such sophisticated techniques, investigators looked to the analogy between electrostatics and Newtonian gravity.

The mathematician and astronomer Joseph Louis Lagrange, for example, introduced the idea of a scalar potential in the context of studying the gravitational attraction of the moon. He demonstrated that if the potential of a body at a point external to that body were known, the gravitational force it exerted on other masses could be obtained directly from differentiation.[8] Legendre polynomials, which will be used in an example later in this chapter, were first introduced as coefficients of the Newtonian gravitational potential by the French mathematician Adrien-Marie Legendre.[9]

Shortly after Legendre introduced his polynomials, the famous mathematician Pierre-Simon Laplace generalized Legendre’s results to three dimensions (the co-called spherical harmonics), discussed the gravitational potential function of Lagrange, and presented the equation, which now bears his name, for the gravitational potential between masses.[10]

Since the mathematical form of the gravitational potential due to a point mass is identical to the electrostatic potential due to a point charge, it is natural that Lagrange’s potential, Legendre’s polynomials, and Laplace’s partial differential equation were applied to electrostatic problems. In 1812 Siméon Denis Poisson, who had been a student of Lagrange and a disciple of Laplace, took over the scalar potential from Laplace’s and Lagrange’s studies of gravitation and applied to it in an electrostatic context. Poisson extended Laplace’s equation to include the charge density and solved it for several simple cases.[11]

The introduction of Bessel functions, which can be used to solve Laplace’s equation when expressed in cylindrical coordinates, provides yet another instance in which the study of gravity supplied tools for handling electrostatic problems. These functions were developed and studied by Friedrich Bessel in 1817 during an investigation of the three-body problem: to determine the motion of three masses acting only under the influence of their mutual gravitational attraction. Bessel later extended his work on these functions in a study of planetary perturbations, and the optics of telescopes. This last application allowed Bessel to make the greatest astronomical discovery since Galileo.

For two millennia astronomers failed to measure stellar distances using geometry. The idea, called stellar parallax, predicts that as the earth moves around the sun the position of starts will shift because of our perspective.   The Greek astronomer Aristarchus proposed that the Earth orbits the sun, but his idea was rejected largely because the expected stellar parallax was not observed. Unless the universe is unfathomably huge, and the parallax to the nearest star is therefore so small as to be undetectable, it was difficult to account for this fact.

Eighteen centuries later, Galileo’s observations of the phases of Venus vindicated Aristarchus’s view, but still no one had observed stellar parallax. In 1729 the astronomer James Bradley tried in vain to measure stellar distances, but instead discovered the aberration of star light, which at first supported Newton’s corpuscular model and later helped lead to the downfall of the aether. Finally, in 1838, Bessel measured the parallax of the star 61 Cygni to be 0.3 seconds of arc, which corresponds to a distance of 10 light years.[12]

In practice, specialized functions such as the Bessel functions and the Legendre polynomials were defined by infinite series and numerically tabulated. In the absence of modern computers, which we now take for granted, scientists like Bessel and Legendre developed numerical methods to solve differential equations for problems where the solution could not be expressed in the simple closed form of a simple polynomial, exponential or sine function. These special functions are still tabulated, or calculated on the fly, and included in modern data analysis packages.

In this chapter we present analytical methods of solving Laplace’s equation, worked out by mostly French mathematicians who were solving problems using Newton’s law of gravity.

Given a known distribution of charge, or a fixed potential, prescribed along the boundary to a region, the potential throughout the region may be obtained as a solution to Poisson’s equation, if there is charge in the region of interest, or Laplace’s equation for those regions that are free of charge.

As with any differential equation, the primary method used in solving either Poisson’s or Laplace’s equation is that of “guess and check.” The procedure is simple enough to describe: make an educated guess of the functional form of the solution, given the nature of the boundary conditions, and then verify that the equation is satisfied by our guess.

There are, thankfully, a few standard techniques, such as separation of variables and the method of images, to aid us in solving these problems and we will work through several classic examples to illustrate them. Placing a spherical conductor in a uniform electric field, for example, induces a dipole and is a physically interesting problem in its own right.

We solve this particular boundary value problem in two ways: first by using the results of the previous section and modeling the sphere as a physical dipole, and then by employing the technique of separation of variables to transform Laplace’s equation from one partial differential equation in three variables to three ordinary differential equations that are fairly straightforward to solve.

In both approaches, we combine physical intuition with the formal mathematics to set conditions that restrict the class of plausible solutions. We cannot emphasize too much the importance of the physical intuition. Mathematical formalism is fundamentally one big if-then statement. If the postulates are true, then the conclusion must also be true. If the postulates are based on false assumptions, then correct mathematics will yield physically wrong results.

Subtle problems arise when false assumptions produce correct results. This is often the case with applications of Laplace’s equation, since it can apply to many different physical situations, and mathematical solutions of Laplace’s equation are already well known. When this happens, scientists can follow the wrong path for some time before noticing contradictions with the physics of what they are studying. Since so much had been invested in the old way of doing things, and it mostly worked anyway, scientists and engineers usually only switch to the new theory when forced to in order to accomplish their particular goals.

Applied mathematicians and engineers will look to you, the physicist, to know which laws to apply to which problem. This ability to keep the fundamental principles in mind, while also understanding the gist of the details, will be your relative strength in the modern diverse workplace. Always keep in mind Mark Twain’s remark: “To a man with a hammer, everything looks like a nail.” Your job will be to select the correct tool to use for the problem under investigation.

[1] Note the watercolor portrait of French mathematicians Adrien-Marie Legendre and Joseph Fourier: Boilly, Julien-Leopold. (1820). Album de 73 Portraits-Charge Aquarelle’s des Membres de I’Institute (watercolor portrait #29). Biliotheque de l’Institut de France.

[2] William Thomson, later Lord Kelvin, (under the pseudonym P.Q.R), “On the uniform motion of heat and its connection with the mathematical theory of electricity”, Cambridge Mathematical Journal 3, (1842), 71–84. Maxwell would later replace Thomson’s analogy of heat flow with “an imaginary incompressible fluid” because it would make for a less abstract analogy.

[3] Olivier Darrigol, Electrodynamics from Ampère to Einstein, (London: Oxford University Press, 2000), 114-15.

[4] Lord Kelvin, Reprint of Papers on Electrostatics and Magnetism, (London: Macmillan, 1872), 52-85.

[5] Olivier Darrigol, Electrodynamics from Ampère to Einstein, (London: Oxford University Press, 2000), 49.

[6] Ibid., 49-54, and G. Green, “An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism,” reprited in Mathematical Papers of the Late George Green, ed. N.M. Ferrers, (London: McMillian & co., 1871).

[7] “General Theory of Terrestrial Magnetism,” and “General Theory of attraction and repulsion forces acting inversely proportional to the square of the distance,” (1839) published in Scientific Memoirs, Selected from the Transactions of Foreign Academies and Learned Societies and from Foreign Journals, ed. R. Taylor (London, 1841), Vol. II, Part VI, 184–251, and Vol. III, Part X, 151-196, respectively. Elizabeth Sabine, an accomplished linguist who happened to be the wife of physicist Edward Sabine, translated both papers from the original German. A new translation of the first paper has been recently given by Glassmeier and Tsurutani, Hist. Geo Space Sci., 5, (2014): 11–62. Gauss emphasized that potential theory could be applied to gravitational, electrostatic, and magnetic forces despite the apparent differences between each phenomenon.

[8] J.L. Lagrange, “Sur l’Equation Séculaire de la Lune”, L’Académie Royale des Sciences de Paris, VII (1773); Prix pour l’année 1774; Oeuvres, VI, 335; “Theorie de la Libration de la Lune”, Mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin, (1780) in Oeuvres, V, 5; and A. S. Hathaway, “Early History of the Potential”, Am. Bull. New York Math. Soc. 1, no. 3, (1891): 66-74. The name “potential” was apparently first used in an essay by George Green in 1828.

[9] Adrien-Marie Le Gendre, “Recherches sur l’attraction des sphéroïdes homogènes,” Mémoires de Mathématiques et de Physique, présentés à l’Académie royale des sciences (Paris) par sçavants étrangers, vol. 10, pages 411-435.

[10] Simon-Pierre Laplace, “Théorie des attractions des sphéroïdes et de la figure des planets”, Paris Mémoires but is also reprinted in the third volume of Laplace’s Méchanique céleste (1799).

[11] J.L. Heilbron, Electricity in the17th and 18th centuries: a study of early Modern physics (Berkley: University of California Press, 1979), 499.

[12] Credit for the first successful measurement of stellar parallax should perhaps belong to F. von Struve, who published his measured value for the parallax of Vega a few years before Bessel’s announcement. Struve’s result for Vega was preliminary, and he later refined it in light of Bessel’s work on 61 Cygni, and ended up with a reported value that was twice his initial result. Interestingly, Struve’s original value of 0.125 seconds of arc for the parallax of Vega is nearly identical with the modern, accepted value of 0.129 seconds of arc, which corresponds to a distance of about 25 light-years. See Suzanne Débarbat, “The First Successful Attempts to Determine Stellar Parallaxes in the Light of the Bessel/Struve Correspondence” in Mapping the Sky: Past Heritage and Future Directions, Volume 133 of the IAU Symposium (Dorddrecht: Kluwer, 1988).