## Relativity And Maxwells Equations

You may be wondering how electromagnetics changes when special relativity is included. Or to put the question in historical context: How were Maxwell's equations modified to conform to Einstein's relativity? It is well known that Newton's basic tenets of physics and the concept of space and time developed mainly by Galileo and Newton had to be radically changed to accommodate relativity. You may be surprised to learn that there were no changes made to Maxwell's equations!* In fact, special relativity follows directly from Maxwell's equations.

I will now return to the story of Maxwell's equations and the aether. Maxwell's equations predict that electromagnetic waves exist and that their speed is ~3 x 108m/s, the speed of light. However, the equations

*Note that Maxwell's equations must be used in conjunction with the laws of relatavis-tic mechanics, as opposed to Newtonian mechanics, to be in accord with special relativity. Thus, change density and current density must be multiplied by the relatavistic stretch factor, g.

do not state what this speed is relative to. For example, when a car driving down the highway turns on its lights, the light travels away at the speed of light, relative to the car. The same is true for the radio waves sent out by a cellphone in the car. Assume the car is traveling at 60 miles per hour. Ordinary experience tells us that if you are standing still on the side of the highway, you would measure the speed of both the light waves from the headlights and the radio waves from the cellphone to be the speed of light plus 60 miles per hour. Maxwell's equations, however, state that the speed of radio waves and light waves is always the same under any measurement. How could the driver and you measure the speed of the light from the headlights to be the same? This question bothered the physicists of the late 19th century and early 20th century. Most physicists believed for this reason that Maxwell's equations were flawed and needed to be modified to match the prevailing view of space and time.

Einstein was an imaginitive thinker. He said that Maxwell's equations are correct as they were first written, and that it was Newton's laws of physics and the Newton-Galileo laws of space and time that needed to be modified. Obviously Einstein made a bold statement.* He also had the physics to back up his statement. It took decades, but Einstein's theory of relativity gradually gained full acceptance of the scientific community after countless experiments proved him to be correct. At this point in the 21st century, I think it is safe to say that Einstein was correct and perhaps his theory should now be called the law of relativity. Close to 100 years of scrutiny and experimentation have passed without a single failure. Of course, general relativity isn't the end of physics. Most notably, relativity and quantum physics have yet to be merged. There has been hope in recent years that string theory will provide the unified theory of physics, commonly called the Theory of Everything.

I cannot in one chapter teach the entirety of relativity, but I can highlight the most important aspects of relativity and mention the relation of relativity to electromagnetics. The basic premise behind Einstein's work is that we should be able to write the laws of physics such that any event or phenomenon can be described by the same laws, regardless of who does the measurement. In other words, there should be a universal set of laws to describe observations, regardless of whether the observer is stationary, moving at a constant speed, or accelerating. More

*In this abreviated history of relativity, I leave out the contributions of Michelson and Morley, Poincare, Lorentz, and FitzGerald. Refer to the books by Blatt (1989), Feynman, Leighton, and Sands (1963), and Moore (1995) for a more detailed history.

over, there must be a set of equations to translate measurements from one reference frame to another. When Einstein first introduced his theory in 1905, he covered the topics of stationary and constant speed observers. This subset of relativity is called special relativity. Later, in 1916, Einstein introduced his theory of general relativity, which not only provided the framework for any type of motion, including acceleration, but also provided a new theory of gravity. Gravity is not like other forces. Gravity manifests itself as the curvature (warping) of space-time. Mass causes space-time to curve, and the curvature of space-time determines how other masses move.

For the understanding of electromagnetics, special relativity is all that is really needed. The entirety of special relativity can be summarized in two simple statements:

• Light travels at the same speed when measured by any observer.

• The laws of physics are the same in every inertial (gravitational and acceleration free) reference frame.

From these two simple statements, all of special relativity can be derived. The first statement is a consequence of Maxwell's equations, so in some sense it is really only the second equation that defines special relativity. A reference frame is the coordinate system (including the three dimensions of space and one dimension of time) of the observer who is performing the measurement. The term inertial basically means that the observer is not being accelerated. Accelerated motion is very different from uniform motion. Newton's first law states that a body in uniform motion will continue in uniform motion unless a force acts upon it. Newton's second law states that the mass of an object times the acceleration of the object is equal to the net force acting on the object. The force provides the energy to change the object's velocity. You learned the electromagnetic consequence of Newton's laws in Chapter 5. Electromagnetic radiation can be produced only if there is a force acting on a charge. This force provides the energy of the radiation. If I were to hold a charged ball in my hand and wave it back and forth, it would radiate energy at the frequency at which I wave it. I provide the energy that is radiated away. In other words, I must burn calories to move my arm, some more calories to move the ball, and even more calories to move the ball's electric field. Hence, the electric field itself has inertia or mass, just as the ball has mass.

There are further differences between uniform straight-line motion (inertial motion) and accelerated motion (noninertial motion). If you are sealed inside a windowless compartment and are traveling in uniform motion, there is no way for you to determine how fast you are going or whether you are moving or not. Uniform straight-line motion is completely relative. For instance, if you are traveling in uniform motion in a car or plane, the laws of physics are no different. You don't feel any different when you are in a plane traveling several hundred miles an hour. If you drop a ball, it falls straight down in your frame of reference. On the other hand, if the plane is accelerating, you can feel it. The movement of liquid in your inner ear is the main source for determining acceleration. Furthermore, if you drop a ball in an accelerating airplane or car, it doesn't travel straight down. It curves toward the back of the vehicle as it falls because once it leaves your hand it is no longer accelerating, whereas you and the vehicle still are accelerating.

## Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

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