James Clerk Maxwell’s equations, published in their final form in 1873, represent one of the greatest unifications in the history of physics. They unified electricity, magnetism, and optics into a single coherent framework — and in doing so, predicted the existence of electromagnetic waves traveling at the speed of light.

The Four Equations (Differential Form)

In their modern differential form using SI units, Maxwell’s equations are:

1. Gauss’s Law for Electricity

$$nabla cdot mathbf{E} = frac{rho}{varepsilon_0}$$

Electric charges create diverging electric fields. The total electric flux through any closed surface is proportional to the enclosed charge.

2. Gauss’s Law for Magnetism

$$nabla cdot mathbf{B} = 0$$

There are no magnetic monopoles (at least none that have been observed). Magnetic field lines always form closed loops.

3. Faraday’s Law of Induction

$$nabla times mathbf{E} = -frac{partial mathbf{B}}{partial t}$$

A changing magnetic field creates a circulating electric field. This is the principle behind electric generators, transformers, and wireless charging.

4. Ampère-Maxwell Law

$$nabla times mathbf{B} = mu_0mathbf{J} + mu_0varepsilon_0frac{partial mathbf{E}}{partial t}$$

Electric currents and changing electric fields create circulating magnetic fields. Maxwell’s key addition was the displacement current term (mu_0varepsilon_0frac{partial mathbf{E}}{partial t}).

Maxwell’s Insight: The displacement current term was Maxwell’s theoretical addition, not based on any experiment. He realized that Ampère’s law as originally formulated was inconsistent with charge conservation. Adding this term not only fixed the inconsistency but also predicted electromagnetic waves. The speed of these waves in vacuum is (c = 1/sqrt{mu_0varepsilon_0}) — exactly the speed of light.

Electromagnetic Waves

In vacuum ((rho=0, mathbf{J}=0)), Maxwell’s equations lead to wave equations:

$$nabla^2mathbf{E} – frac{1}{c^2}frac{partial^2mathbf{E}}{partial t^2} = 0$$

The solutions are traveling waves where (mathbf{E}) and (mathbf{B}) are perpendicular to each other and to the direction of propagation.

The Electromagnetic Potential Formulation

Since (nabla cdot mathbf{B} = 0), we can write (mathbf{B} = nabla times mathbf{A}). Then Faraday’s law gives (mathbf{E} = -nablaphi – partialmathbf{A}/partial t). This leads to the elegant four-potential formulation in relativistic notation:

$$A^mu = (phi/c, mathbf{A}), quad F^{munu} = partial^mu A^nu – partial^nu A^mu$$

The tensor (F^{munu}) contains all components of (mathbf{E}) and (mathbf{B}) in a single Lorentz-covariant object.

Conservation Laws from Maxwell’s Equations

Maxwell’s equations imply the conservation of energy (Poynting’s theorem):

$$frac{partial u}{partial t} + nabla cdot mathbf{S} = -mathbf{J} cdot mathbf{E}$$

where (u = frac{1}{2}(varepsilon_0|mathbf{E}|^2 + |mathbf{B}|^2/mu_0)) is the electromagnetic energy density and (mathbf{S} = frac{1}{mu_0}mathbf{E} times mathbf{B}) is the Poynting vector representing energy flux.

References

  • Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism. Clarendon Press.
  • Jackson, J. D. (1998). Classical Electrodynamics (3rd ed.). Wiley.
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