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Unformatted text preview: Notes 6 Spring 2005.doc 1 The Uniform Plane Wave Electromagnetic wave propagation Maxwells equations show us that in a source-free region of free space ( ) c = = J , a time-varying magnetic field results in a non-zero time-varying 1 circulation or curl of the electric field ( ) t H E = G G G and that a time varying electric field results in a non-zero time-varying circulation of the magnetic field ( ) t E H = G G G . One can imagine for instance an antenna where a circuit sinusoidally drives a current up and down (via a changing electric field, of course) causing the magnetic field in the vicinity to vary sinusoidally resulting in a varying electric field a little farther out which induces a varying magnetic field a little farther out still and the effect propagates outward at the speed of light c. We say that an electromagnetic wave has been created which moves out in all directions (Figure 1). d l d c p = v E d P l d H Wire filament (antenna) H i x o r y = 0 ( ) sin r o Figure 1 d z Any wave can be represented by a function of both space and time. Wave motion occurs when a disturbance at one place and time is related to an effect at another place and at a later time. Waves carry energy and have a definite velocity . Starting with Maxwells first equation we can take the curl of both sides to get t = B- E G G G G G 1 One can see from these equations that a monotonically changing magnetic field would produce a constant curl value for the electric field and similarly for the other case. However we are looking ahead a little where we will be assuming sinusoidal variation in time of our fields thereby producing a time-varying curl of the corresponding field implying a similar variation in the field value itself (recall, for example, that t sin t cos dt d = ). Notes 6 Spring 2005.doc 2 Using the vector identity (see Notes 2) ( ) A A A 2 G G G G G G G G = gives us ( ) t B- E E = G G G G G G G 2 and then, substituting with , we move the del operator inside the time derivative operator on the right-hand side to get H B G G o = ( ) H- E E G G G G G G G = t o 2 We can further substitute on the right-hand side with Amperes law, + = c J G G G H t t o o = E E G G (Maxwells second equation), and also substitute on the left-hand side with the electric law of Gauss (Maxwells fourth equation) to get o = = = E E D G G G G G G = t t o o 2 E- E G G G = 2 2 o o 2 t E- E G G G 1 t 2 o o 2 2 = E E G G G This is a second order partial differential equation in E G involving both the spatial and temporal dimensions. This equation is called the 3-dimensional...
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This note was uploaded on 01/20/2012 for the course EE 4460 taught by Professor Czarnecki during the Fall '10 term at LSU.
- Fall '10