James Clerk Maxwell was a young, mathematically brilliant Scottish physicist. In 1855, barely 24 years old, he presented a paper to the Cambridge Philosophical Society entitled “On Faraday’s Lines of Force.” It had been 30 years and more since the major discoveries of Oersted, Ampère, Faraday, and others, but electromagnetism remained a loose collection of facts and “rules of thumb” without a consistent theory to link these ideas together.
Maxwell’s goal was to synthesize this body of knowledge and to form a theory of electromagnetic fields. The critical step along the way was his recognition of the need to include a displacement-current term in Ampère’s law.
Maxwell’s theory of electromagnetism is embodied in four equations that we today call Maxwell’s equations. These are
Maxwell’s claim is that these four equations are a complete description of electric and magnetic fields. They tell us how fields are created by charges and currents, and also how fields can be induced by the changing of other fields. We need one more equation for completeness, an equation that tells us how matter responds to electromagnetic fields. The general force equation
is known as the Lorentz force law. Maxwell’s equations for the fields, together with the Lorentz force law to tell us how matter responds to the fields, form the complete theory of electromagnetism.
Maxwell’s equations bring us to the pinnacle of classical physics. When combined with Newton’s three laws of motion, his law of gravity, and the first and second laws of thermodynamics, we have all of classical physics—a total of just 11 equations.
While some physicists might quibble over whether all 11 are truly fundamental, the important point is not the exact number but how few equations we need to describe the overwhelming majority of our experience of the physical world. It seems as if we could have written them all of this book and been finished, but it doesn’t work that way. Each of these equations is the synthesis of a tremendous number of physical phenomena and conceptual developments. To know physics isn’t just to know the equations, but to know what the equations mean and how they’re used. That’s why it’s taken us so many chapters and so much effort to get to this point. Each equation is a shorthand way to summarize a book’s worth of information!
Let’s summarize the physical meaning of the five electromagnetic equations:
Gauss’s law: Charged particles create an electric field.
Faraday’s law: An electric field can also be created by a changing magnetic field.
Gauss’s law for magnetism: There are no isolated magnetic poles.
Ampère-Maxwell law, first half: Currents create a magnetic field.
Ampère-Maxwell law, second half: A magnetic field can also be created by a changing electric field.
Lorentz force law, first half: An electric force is exerted on a charged particle in an electric field.
Lorentz force law, second half: A magnetic force is exerted on a charge moving in a magnetic field.
These are the fundamental ideas of electromagnetism. Other important ideas, such as Ohm’s law, Kirchhoff’s laws, and Lenz’s law, despite their practical importance, are not fundamental ideas. They can be derived from Maxwell’s equations, sometimes with the addition of empirically based concepts such as resistance.
It’s true that Maxwell’s equations are mathematically more complex than Newton’s laws and that their solution, for many problems of practical interest, requires advanced mathematics. Fortunately, we have the mathematical tools to get just far enough into Maxwell’s equations to discover their most startling and revolutionary implication—the prediction of electromagnetic waves.