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15: Maxwell's Equations

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

We describe these four equations in this chapter, and, in passing, we also mention Poisson's and Laplace's equations. We also show how Maxwell's equations predict the existence of electromagnetic waves that travel at a speed of $$3 \times 10^8\, \text{m} \,\text{s}^{-1}$$. This is the speed at which light is measured to move, and one of the most important bases of our belief that light is an electromagnetic wave.

• 15.1: Introduction
One of Maxwell's great achievements to show that all of the phenomena of classical electricity and magnetism – all of the phenomena discovered by Oersted, Ampère, Henry, Faraday and others whose names are commemorated in several electrical units – can be deduced as consequences of four basic, fundamental equations.
• 15.2: Maxwell's First Equation
Maxwell's first equation, which describes the electrostatic field, is derived immediately from Gauss's theorem, which in turn is a consequence of Coulomb's inverse square law. Gauss's theorem states that the surface integral of the electrostatic field D over a closed surface is equal to the charge enclosed by that surface.
• 15.3: Poisson's and Laplace's Equations
Regardless of how many charged bodies there may be an a place of interest, and regardless of their shape or size, the potential at any point can be calculated from Poisson's or Laplace's equations.
• 15.4: Maxwell's Second Equation
Unlike the electrostatic field, magnetic fields have no sources or sinks, and the magnetic lines of force are closed curves. Consequently the surface integral of the magnetic field over a closed surface is zero.
• 15.5: Maxwell's Third Equation
Maxwell's third equation is derived from Ampère's theorem, which is that the line integral of the magnetic field H around a closed circuit is equal to the enclosed current.
• 15.6: The Magnetic Equivalent of Poisson's Equation
A alternative for static magnetic fields can be constructed to mimic how Poisson's equation addresses static electrostatic fields.
• 15.7: Maxwell's Fourth Equation
Maxwell's Fourth Equation is derived from the laws of electromagnetic induction.
• 15.8: Summary of Maxwell's and Poisson's Equations
• 15.9: Electromagnetic Waves
Maxwell predicted the existence of electromagnetic waves, and these were generated experimentally by Hertz shortly afterwards.  In addition, the predicted speed of the waves was $$3 \times 10^{8}\, m \,s^{-1}$$, the same as the measured speed of light, showing that light is an electromagnetic wave.
• 15.10: Gauge Transformations
Electric and magnetic fields can be written in terms of scalar and vector potentials. However, there are many different potentials which can generate the same fields. We have come across this problem before. It is called gauge invariance.
• 15.11: Maxwell’s Equations in Potential Form
• 15.12: Retarded Potential
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past.