Notes+6+ Fall 2004

Notes+6+ Fall 2004 - Notes 6 Fall 2004.doc The Uniform...

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Notes 6 Fall 2004.doc 1 The Uniform Plane Wave Electromagnetic wave propagation 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 Maxwell’s first equation we can take the curl of both sides to get t × = × × B - E G G G G G Using the vector identity (see Notes 2) () A A A 2 G G G G G G G G = × × then pulling out the t operator from the right-hand side and using H B G G o μ = yields () t B - E E × = G G G G G G G 2 () H - E E G G G G G G G × = t μ o 2 Noting that free space is a sourceless region ( ) 0 , 0 = = ρ c v J G , we can substitute on the right-hand side with + = × c J G G G H t t o o ε = ε E E G G from Maxwell’s second equation and also substitute on the left- hand side with Maxwell’s fourth equation 0 E E D = = ε = G G G G G G 0 o to get ε = t t μ o o 2 E - E G G G ε = 2 2 o o 2 t μ E - E G G G 0 1 t 2 o o 2 2 = ε µ 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 vector wave equation for free space. Note that it was derived directly from Maxwell’s equations.
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Notes 6 Fall 2004.doc 2 The vector wave equation is a mathematical description of the wave phenomenon. The quantity o o 1 ε µ has an interesting meaning and history. It has units of velocity and a quick calculation shows that s m 8 o o 10 3 1 × = ε µ which one recognizes as the measured velocity of light in free space, c. It was this recognition by Maxwell that finally showed visible light to be only one small part of a much larger continuum called the electromagnetic spectrum. With this, we can now write (for free space) 0 c t 2 2 2 2 = E E G G G In an analogous fashion we can derive a wave equation for the magnetic field as well: 0 c t 2 2 2 2 = H H G G G As a simplification we will select an electromagnetic wave that is propagating in the direction and that has an oscillating electric field whose vector direction is in the direction. It is of utmost importance that one recognizes and distinguishes these two directions associated with any electric field (or magnetic field) traveling wave. The one-dimensional wave equation becomes z ˆ
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Notes+6+ Fall 2004 - Notes 6 Fall 2004.doc The Uniform...

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