3.4.2 Maxwell’s Original Equations

How Twenty Equations Became Four

We’ve accounted for three missing equations in Ohm’s Law. The other four “missing” Maxwell’s equations are the relations for the continuity of charge and the “electric elasticity.” The continuity of charge is a mathematical recognition that if the amount of charge increases or decreases somewhere, then there must be a corresponding current bringing or taking away, respectively, the appropriate amount of charge. Electric elasticity is the relation between the vector electric field (E) and the vector displacement (D). In modern notation, the two are related by the electric permittivity, ε, as follows: D = ε E. More about this “permittivity” which keeps popping up in our equations in a moment.

Figure 3.19: Maxwell’s equations [[iii]] top, in his original notation from “A Dynamical Theory of the Electromagnetic Field”. The modern and original variables correspond as follows: E ↔ (P,Q,R); D ↔ (f,g,h); H ↔ (α,β,γ); B ↔ µ(α,β,γ); J ↔ (p,q,r); ρ ↔ e; Ψ is the electric potential; A ↔ (F,G,H) is the magnetic potential. The Heaviside-Maxwell equations [[iv]], bottom.

Figure 3.19 compares Maxwell’s own version of his famous equations (top) to Oliver Heaviside’s streamlined four-equation version (bottom). The mathematical details involved in applying these complicated equations are best left to a formal course in electromagnetism. You don’t need to be proficient in vector calculus, however, to appreciate the difference in complexity and the simplification Heaviside’s vector formulation wrought.

For instance, Maxwell wrote the three Cartesian components of the electric displacement as {f, g, h}, and the charge density as “e,” instead of “ρ.” A modern scientist or engineer, using vector notation, would write it as D, instead, and combine the x, y, and z derivatives in the “gradient,” or “nabla” sign (∇).

In addition, Maxwell intermingled potential and field relationships in a way that is confusing to more modern sensibilities. The technical details of the various ways Maxwell derived his equations are in a paper by Alfred M. Bork (1896–1975) [[i]], and the Engineering and Technology History Wiki offers an excellent overview [[ii]].

The following sections consider the four “Maxwell’s equations:” Gauss’s Laws for electricity and magnetism, Faraday’s Law, and Ampère’s Law in further detail. Maxwell may not have worked out each equation all by himself, but he took the individual pieces and fit them together in a way that fully explains and describes electromagnetism.

Next time: 3.4.3 Gauss’s Laws: The Famed Geophysicist Was Also Reputed to be an Excellent Mathematician

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References

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[[i]]         Bork, Alfred M., “Maxwell and the Electromagnetic Wave Equation,” American Journal of Physics, vol. 35, no. 9, pp. 844-849. doi:10.1119/1.1974263 

[[ii]]        See: https://ethw.org/Maxwell%27s_Equations 

[[iii]]       See: https://ethw.org/File:Maxwell_image_05.jpg.

[[iv]]        See: https://ethw.org/File:Maxwell_image_02.jpg.