12/4/2003 Propagation of Electromagnetic Waves 1/5 ()()()()()()00r,xr,r,xr,r, 0r, 0ttttttttµε∂∇=∂∂∇=−∂∇⋅ =∇⋅ =EBBEEBPropagation of Electromagnetic Waves Maxwell’s equations were cobbled together from a variety of results from different scientists (e.g. Ampere, Faraday), whose work mainly was done using either static or slowly time-varying sources and fields. Maxwell brought these results together to form a complete theory of electromagnetics—a theory that then predicted a most startling result! To see this result, consider first the free-space Maxwell’s Equations in a source-free region (e.g., a vacuum). In other words, the fields in a region far away from the current and charges that created them:12/4/2003 Propagation of Electromagnetic Waves 2/5 ()()xr,xx r,ttt∂∇∇∇ =−∂BE()()()002002r,xx r,r,ttttttµεµε∂∂∇∇ =−∂∂∂=−∂EEE()0r∇⋅ =E() ()2xxrr=∇EE()()22002r,0tr,ttµε∂∇+ =∂EE()()00eˆr,t a sin t zωµε=−E radians/secω0ezˆˆaa⋅= Say we take the curl of Faraday’s Law: Inserting Ampere’s Law into this, we get: Recalling that if then ∇∇ , we can write the following differential equation, one which describes the behavior on an electric field in a vacuum: Among the many solutions to this differential equation is: In this case, the electric field is varying with time in a sinusoidal manner, with an angular frequency of . The direction of the electric field is orthogonal to the z-axis (i.e.,).12/4/2003 Propagation of Electromagnetic Waves 3/5 ()000tzφω µε=− =Lets plot this function at three different times: |E| t=1 z |E| t=2 z |E| t=3 z Here the red dot indicates a “phase” of zero, i.e., . Note that this dot appears to be moving forward along the z- axis as a function of time. The electric field is moving !12/4/2003 Propagation of Electromagnetic Waves 4/5 ()000tzωµε−=00tzµε=001pdzvdtµε==()( )00-7 -12114x10 8854x10meters secondpv.µεπ===83x10Q: How fast is it moving? A: Lets see how fast the red dot is moving! Rearranging , we get the position z of the dot as a function of time t: Its velocity is just the time derivative of its position: Hey we can calculate this! The electric field is moving at a velocity of: Q: Hey wait minute! 3 x 108 meters/second—that’s the speed of light!?! A: True! We find that the magnetic field will likewise move in the same direction and with the same velocity as the electric field.12/4/2003 Propagation of Electromagnetic Waves 5/5 We call the combination of the two fields a propagating (i.e., moving) electromagnetic wave. Light is a propagating electromagnetic wave! This was a stunning result in Maxwell’s time. No one had linked light with the phenomena of electricity and magnetism. Among other things, it meant that “light” could be made with much greater wavelengths (i.e., lower frequencies) than the light visible to us humans. Henrich Hertz first succeeded in creating and measuring this low frequency “light”. Since then, humans have put this low-frequency light to great use. We often refer to it as a “radio waves”—a propagating electromagnetic wave with a frequency in the range of 1 MHz to 20 GHz. We use it for all “wireless” technologies