Maxwell's Equations

From GHN

Maxwell's Equations

London, England. In 1865 James Clerk Maxwell published the set of equations now bearing his name.

Maxwell’s four equations by themselves, define the entire field of electromagnetics.

In the 1800s theoretical physics expanded far beyond mechanics as thermodynamics, electromagnetic theory, optics, and other studies made great advances. The transition from the relatively straightforward experimentation with magnets, batteries, and other devices to more complex, theoretical ideas, such as Einstein’s Theory of Relativity, was the work of many people, but the greatest single advance was the formulation of what are now known as Maxwell’s Equations, named after the 19th century physicist James Clerk Maxwell (1831-1879). The eponymous equations, shown here in one of their several modern forms, were actually developed by several physicists. Maxwell was able to get the equations to work by adding one term, a displacement current. The equations are named after Maxwell because he showed how they work together to define completely the field of electromagnetics. Although this may seem simple today, in Maxwell’s time, working within the bounds of what was then known, it was a stroke of genius.

Maxwell’s Equations are called “first principles” because they were not derived from other equations or theories. Throughout most of the 19th century physicists experimented with magnets, batteries, and such. Then, by trial and error, they developed mathematical equations that produced the same answer as their experiments.

Maxwell’s four equations are laws that magnetic fields and electric fields must always obey. For example, if you wrap an insulated wire around a nail and connect a battery to the wire, you make a magnet. The magnetic field created by the magnet must obey Maxwell’s first equation, Ampère’s Law. According to the second equation, Faraday’s Law, you can take a magnet and spin it around and around to make an electric generator. This is where the electricity running your computer comes from (unless it is running on batteries!). The third equation is Gauss’ Law. Gauss’ Law says static electricity (electric charge) must generate an electric field (voltage). You can see this on any dry day by shuffling across a rug and touching a doorknob. The fourth equation says that magnetic charge does not exist, a great mystery that physicists are still trying to understand.

As mentioned earlier, one of Maxwell’s great insights was the inclusion of displacement current, the last term of Ampère’s Law, in the first equation. This has to do with what happens to electric fields when they change, such as when the static electricity from your finger arcs to the doorknob (and gives you a shock). The electric field around your finger suddenly drops to zero. This sudden change in the electric field generates a magnetic field. Faraday’s Law, the second equation, says the reverse is true. If you change a magnetic field, (say, by turning an electromagnet on and off) you generate an electric field. In fact, electric and magnetic fields can keep working together, all by themselves, each changing from one into the other, sometimes for millions of years. The result is what we know as light, or radio waves, or gamma rays, or, in general, electromagnetic radiation. If you want to see electromagnetic radiation that’s millions of years old, look up into the sky some starry night.

At the end of the 19th century, a young Swiss patent clerk named Albert Einstein learned about Maxwell’s Equations and wondered, what happens if a source of electromagnetic radiation (like a light bulb) were moving and he stood still? Then, he wondered, What would happen if he were moving and the source stood still? Einstein discovered that if the movement speed approaches the speed of light, some really strange things happen. But, more importantly, he found that he could develop his theory assuming everything depended only on relative movement. Which one was actually moving and which one actually stood still was of absolutely no concern. Thus was born Einstein’s Theory of Relativity.

Even today, physicists can spend an entire lifetime finding new solutions to Maxwell’s Equations. But we sometimes lose sight of the fact that Maxwell’s Equations do not explain why, for example, two magnets attract or repel. They only allow us to calculate how strongly and in what direction the magnets are pulled. The question, “Why?” has yet to be answered.