Way back when in th 19th centuryeveryone's good fiend James Clerk Maxwell summarized Electromagnetic phenomena in 4 different equations, known ironically enough as Maxwell's equations. These describe the electric field as defined by carge, (like coulomb's law but better), then an equation defining magnetic field, the equation relating an electric field to a changing magnetic one, and lastly an equation relating a magnetic field to a changing electric one. An important realization of Maxwell's was that if electric field and magnetic field are related thus, then they can be oscillated. He thought of an experiment involving a DC current, with two conducting rods connected to opposite terminals of a an electrochemical cell/battery.
Positively charged rod
|+_ _ _ _ _ _ _
|+_ _ _ _ _ _ \
^ |+_ _ _ _ _ \ \
| |+_ _ _ _ \ \ \
_____/ \ \ \ \
/ \ \ \ \
| \ \ \ \
_|_ \ \ \ \
- DC current | | | | Electric Field lines
| / / / /
| / / / /
\_____ / / / /
|-_ _ _ _/ / / /
|-_ _ _ _ _/ / /
^ |-_ _ _ _ _ _/ /
| |-_ _ _ _ _ _ _/
Negatively charged rod
Now use your imagination, and also thing that the Magnetic field lines are perpendicular the Electric field lines, and are coming out of your screen. So now if we change the power supply to AC, we can see that the electric field will all of sudden change direction. We are rapidly decelerating the electrons, or charge, and thus changing the directiong of the two fields. Inbetween we will get areas of zero field or charge, we can observe that the field itself will oscillate. Now what will be the speed of this oscillation, this wave. We can represent the strength of the magnetic field and the electric field as two sinusoidal graphs of a function of displacement r. We can conclude at any point the magnetic field is equal to the electric field.
Now onto the speed: Imagine a conductive rectangular loop (rectangular is easier). The rectangle is at points a, b, c, d. The wave travels through it with velocity v. While the position of the loop is unchanging relative to us, relative to the wave it goes to position a', b', c', d'. We let y0= ab. Going to Maxwell's third equation, the EMF(work done per unit charge) is equal to the derivative of the rate of change of Magnetic Flux, PHIB. that is:
EMF=d(PHIB)/dt
EMF=Bd(A)/dt
EMF=B y0vdt/dt (recall that A=length times width, and the width will be equal to velocity times change in time) (recall that A=length times width, and the width will be equal to velocity times change in time)
EMF=B y0v
Going back to the fact that EMF is the work per unit charge, or electric field times distance parallel.EMF=EMFab+ EMFbc+EMFcd+EMFad. Because they are perpendicular to the elctric field EMFad=EMFbc=0 Also luckily for us the Electric field hasn't travelled through segment ab, EMFab=0. Thus EMF= EMFcd=Ey0
Ey0= B y0v
E= B v
E/B=v
d' d a'
__ __ __ _ _ __________ ____ a
/ \ / \ / \ ! | ! |
/ \ / \ / \ ! | ! |
/ \ / \ / \ ! | ! |
/ \__/ \__/ \ !_ _ |__________!_____| b
c' c b'
Now we'll use the sqame diagram as above, except now the wave refers to the Magnetic field (remember in both incidence the other field is going through the computer screen). Using Ampere's Law that has been processed using Gauss' Law we come up with the equation:
sum(B||dl)=mu0e0d(PHIE)/dt
We let dc=z0, thus:
sum(B||dl)=mu0e0Ez0v
Now the part of B parallel(||) to dl is going to be zero for all but dc as discussed above for electric field. so:
Bdc=mu0e0Ez0v
B=mu0e0vBv
1/(mu0e0)=v2
Then we just sqrt(not a typo, but abrv) it, and we get the speed of the wave. But we need to know mu and e, (0's ommitted to save time and HTML formatting). These two constants are permeability of free space and permittivity of free space respectively. These two are used to describe the force of a magnetic and electric field repectively. However the sqrt of the reciprocal is equal to about 300000 km/s. and indeed experiments later should that the wave produced by this were similar to light waves only of a lowever frequency. Indeed it was shown light was an electromagnetic wave, through experimentation, and also with more accurate values for the constants the value for c is exactly what 3.9x10^-43 mentioned above. I think that's nifty
Addition
The speed of light can be very easily determined by easy manipulation of Maxwell's equations. One need only to take the curl of Faraday's law, and then insert Ampere's law in the place of the time derivative of the curl of magnetic field. Biot-Savart's law states that the divergence of magnetic field is zero, and Gauss' law states that the divergence of electric field is charge density over epsilon zero. Then for a system without charge or current density the equations simplify to the wave equation for electric field, which propagates at a speed of 1/(mu0epsilon0). | 
