gauss's law

The equations shown on the t-shirt are the key to understanding how electric and magnetic fields interact to create light. The person credited with linking them together in one compact bundle of equations is a Scottish man named James Clerk Maxwell. His famous equations are known as the “Maxwell Equations”. 

The tensor form of the equations on the t-shirt are referred to as the “macroscopic” form of Maxwell’s Equation. The letter D refers to the electric displacement field; whereas the B refers to the magnetic flux and E is the electrical field. The letter H refers to the magnetic field strength. The letter J , which is used more often for this form of the equation, is the same as the letter i shown on the t-shirt, both referring to the amount of electrical current.

The “microscopic” tensor form of the Maxwell equations looks like this:

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You readily notice two new symbols in this microscopic version of the equations. The symbol µ0 is a constant representing the permeability of free space (also called the “magnetic constant”). The symbol ε0 is a constant representing the permittivity of free space (also called the “electric constant”). I will use this microscopic version of the Maxwell equations for attempting to explain the thought processes that led up to Maxwell’s discovery. The relationship between the magnetic and electric constants were what brought Maxwell to the conclusion that light is connected to the interaction of magnetism and electricity, as well as to conclude that the speed of light must be constant. 

The first equation shown in the above microscopic form of Maxwell’s equations is Gauss’ Law. It is merely a compact way of showing mathematically that the vectors in an electric field diverge from positive charges and converge to negative charges.

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The second equation says that magnetic fields never diverge or converge. They always form closed loops. The amount going out is always the same as that going in, and the net result is zero (that is, divergence equals convergence).

 

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The third equation is Faraday’s Law that mathematically expresses that the vectors in an electric field curl around any changing magnetic field; and at the same time, changing magnetic fields induce electric fields. 

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The fourth and final equation says that the vectors of a magnetic field curl around electric currents.

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Geometrically, the electric field (E) and the magnetic field (B) are orthogonal to each other and “traveling” at the speed of light ©:

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 Maxwell’s discovery of the relationship that light has with electricity and magnetism, was almost by accident. It happened when he was applying what I call “fudge factors”. Actually the real name for what he was doing is “applying proportionality constants”.

You know those tables that wobble if you don’t put a folded piece of paper (a shim) under one of the legs so that it rests evenly on the floor? Proportionality constants are kind of like shims. Without the shim under the table leg, it is quite distracting when a student is erasing something aggressively on a paper and the resulting rattling and shaking sounds like the entire table is going to fall to the ground any moment. The shim is needed to even up the relationship between the table leg and the floor.

Maxwell had to use “shims”, that is proportionality constants, to “even up” the relationship mathematically between magnetism and electricity. It took a total of two such constants for his original quaternion equations, and these are included in the modern versions that are expressed in tensor calculus form. One was the electric constant (ε0 = 8.8542 x 10-12) and the other was the magnetic constant (µ0 = 4π x 10-7). These are called today, permittivity (ε0) and permeability (µ0), respectively. 

By playing around with these constants, Maxwell discovered the following relationship:

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Wow!

The implication was startling. the relationship of these two constants not only linked light to the interaction of magnetism and electricity, but it also indicated that since the speed of light is the reciprocal of the square root of magnetic and electric constants multiplied, the speed of light must be a constant also.  

When Maxwell saw the relationship between magnetism, electricity and light, he suggested that maybe there was also a link to gravity as well. However, he never pursued that hunch. I found this little comic strip out on the internet to commemorate that place in history with a tongue-in-cheek conversation that really did not occur, but it makes a point with a some subtle humor.
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And a video that explains it all one more time in a catchy song!  http://www.youtube.com/watch?v=YLlvGh6aEIs