Solutions to Maxwell's Equations


If we can find a solution to Maxwell’s Equations for a particular problem, such as the best way of making a microwave oven, then we have a chance to improve its design.

Maxwell’s Equations are laws that electricity and magnetism must obey. When you use a TV, a computer, a cell phone, a microwave oven, or a magnifying glass, Maxwell’s equations are always obeyed.

For nearly all engineering purposes, Maxwell’s Equations are obeyed everywhere and all the time. The problem is in determining where electromagnetic fields are strong and where they are weak. Some real world questions answered by Maxwell’s Equations are, Will the TV picture be clear or fuzzy? Will your computer work or will it crash? Can you call home, or is your phone inundated with static? Will our lunch be hot or cold?

The answers to these questions are found in solutions to Maxwell’s Equations. For any given situation, like in designing a microwave oven, we guess where the fields will be strong and where they will be weak. We write down an equation for our guess. Then we try our guess in Maxwell’s Equations. If the guess obeys all the equations exactly, then we have a solution and we know where the fields are strong and where they are weak. If not, we try another guess. After a while, we can actually become quite good at guessing solutions.

One example situation is a rectangular box. If we take one specific box, say made of copper and 10 cm on a side, we can successfully guess a solution to Maxwell’s Equations for all the electromagnetic fields on the inside of the box. Then, we can do it for another box, say 20 cm on a side. Pretty soon, we see a pattern, and we can write an equation for the electromagnetic fields on the inside of any rectangular box. The inside of a microwave oven is just a rectangular box. Now we can look at the solution to Maxwell’s equations for a rectangular box and we can tell if our lunch will be hot or cold.

A cell phone uses a kind of electromagnetic wave called radio waves. A solution to Maxwell’s Equations gives clues on how to design an antenna so that we get the best possible signal instead of static.

The speed of a multigigahertz computer means the electric and magnetic fields inside switch back and forth like crazy while we play the latest video game. Maxwell’s Equations tell us that the faster the fields change, the bigger the fields they generate. All these changing fields twist their way through our computer. The computer designer keeps that incredible mess from destroying the next byte of data by finding solutions to Maxwell’s Equations.

Solutions to Maxwell’s Equations are often close to exact. This is one of the few areas of engineering where such exact solutions are common. For example, ask an engineer how much weight a bridge can take before it crashes into the river and you’ll get a general answer such as, “10, maybe 15 tons.” If you ask how well a new airplane will fly in a difficult situation you will only receive a vague answer, because factors that affect the outcome, such as turbulence, can only be approximately calculated. But ask an engineer how fast a beam of light will travel 1 million kilometers, and you get an answer like 3.33564095 seconds.

In spite of this exactness, we must keep in mind that even Maxwell’s Equations can break down. In fact, they break down with very weak radio waves, very dim light, etc. Light, and all electromagnetic radiation, is composed of small packets of energy called photons. When light is so dim that you can start counting photons (for example, with a photo-multiplier), a new theory, called Quantum Electrodynamics or QED, takes over.

We talk about electric fields and magnetic fields as though they are real. Sure, you have seen iron filings move around, but no one has ever seen or touched a field. Like lines of force, fields are just a mathematical convenience that allows us to predict what happens when we do an experiment. QED says that electromagnetic fields are really trillions and trillions of photons that, together, that just happen to act like fields. Physicists describe photons in terms of wave functions, which are closely related to probability functions. (One example of a probability function is the “bell curve,” which is used to adjust test scores.) This model of the photon just happens to work incredibly well in predicting the results of many experiments.