Chapter 4 Gauss’s Law

In 1813, the great German physicist, mathematician, and astronomer, Carl Friedrich Gauss, reformulated Newton’s law of gravity into an equivalent and elegant statement, which relates the total mass enclosed by an arbitrary closed surface to the surface integral of the gravitational field through the same surface,[1] and in 1839 he applied this to inverse square laws in general. [2] In 1861, Maxwell applied Gauss’s law to electricity,[3] relating the total charge enclosed within an arbitrary surface to the electric field through the same surface. The divergence theorem of vector calculus, which Gauss also discovered, states that the surface integral of a vector field (such as the electric field) through a closed surface equals the volume integral of the divergence of that vector field taken over the volume bounded by the closed surface. Gauss’s law can thus be stated locally as well as globally: the divergence of the electric field at a point is proportional to the charge density at that point. Thus, despite being physically equivalent to Coulomb’s law, Gauss’s law is mathematically similar to the continuity equation.

To see Gauss’s law in action, consider the electrostatic generator, shown Robert van de Graaff’s patent application[4]. Since the electric charge distributes itself on the outside of the metal sphere, a Gaussian surface matching the inside of the sphere contains no charge, so it is easy to show, with Gauss’s law, that the electric field inside of the sphere is zero. Thus, the inside of the sphere can accumulate more and more charge, so long as it remains insulated from the ground. Moreover, even though the electric potential of the sphere is 1.5 MV above ground, it is still safe for both people and electronics to be working inside it even when it is charged.

Not only is Gauss’s law usually more useful in the practical matter of directly computing the electric field, it is almost always more useful in formulating general statements of principle in electrodynamics. We will use Gauss’ law to determine the electric field between the surfaces of a charged capacitor and a coaxial cable, and introduce the notion of capacitance in an example on charging a parallel plate capacitor through a resistor.

Recall that in Chapter 1 we made a distinction between the global principle of the conservation of charge, and the local condition of charge being conserved on all scales. Similarly, we use the integral formulation of Gauss’s law to find the electric field surrounding a known charge distribution, but its differential form applies continuously to every point in space. Just as marrying Franklin’s experiments to Euler’s mathematics produced the continuity equation, the marriage of Coulomb’s experiment to Gauss’s mathematics leads to Gauss’s law.

[1] C.F. Gauss, “Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,” Werke, (Gottingen, 1867) 5: 1-22.

[2] C.F. Gauss, “General Propositions relating to Attractive and Repulsive Forces acting in inverse ratio of the square of the distance,” Scientific Memoirs, Selected from the Transactions of Foreign Academies and Learned Societies and from Foreign Journals, ed. R. Taylor (London: R. and E. Taylor, 1841), Vol. III, Part X, 151-196.

[3] Gauss’s law in differential form is equation (115) in Maxwell’s 1861 paper “Lines of Physical Force.” Maxwell, subsequently, states the integral version of Gauss’s law in Chapter II, articles 75 and 76 of his Treatise on Electricity and Magnetism, Vol. I (Oxford: Clarendon, 1873), 76-79, and uses the differential version of the law in the course of deriving Poisson’s equation (section 77, page 79 of Volume I of the Treatise).

[4] Figure from patent US1991236A filed by R.J. van de Graaff on Dec. 16, 1931, and awarded to the Massachusets Institute of Technology. The MIT van de Graaff machine is on permanent display at the Museum of Science in Boston.