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FWLec18

# FWLec18 - A F Peterson Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Fields & Waves Note #18 Electromagnetic Waves Objectives: Continue the study of electromagnetic waves by illustrating the process by which a sheet of surface current density acts as a source of uniform plane waves. Introduce phasor notation and show that the phasor form of Maxwell’s equations can be used directly to obtain electromagnetic waves. In the previous Note, it was demonstrated that electromagnetic plane waves that have a functional dependence of the form ft z v p () ± (18.1) can be valid solutions of Maxwell’s equations, as long as the propagation velocity of the wave is the speed of light v p = 1 me (18.2) in the medium. In this Note, we connect waves of this form with a source. We also consider the sinusoidal steady state, and the description of fields in terms of phasor quantities. Sources of electromagnetic plane wave An infinite current sheet can be used as a source of electromagnetic plane waves. Suppose such a current sheet resides in the z = 0 plane, and is infinite in extent along the x and y directions. A surface current density of the form Jy f t s = ˆ () (18.3) exists on the sheet, where f ( t ) is defined by Figure 1. This source produces field components E y and H x propagating away from the sheet. Assume that the medium is air ( m = 0 , e = 0 ). We would like to find and plot E y versus t at z = 200 m, H x versus z t = 3 s, and H x versus t at z = –300 m. To develop a solution, we first label the fields in the region z > 0 as Ez t y + (, ) and Hz t x + (,) , and the fields in the region z < 0 as t y - t x - (, ). We also know that EH ¥ points in the direction of propagation (away from the sheet) and that

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 EH yx = h (18.4) where = 0 @ 377 ( W ). It follows that Ez t Hz t z ++ =- > (,) , 0 (18.5) t t z -- =< , 0 (18.6) The standard boundary conditions ˆ () nE E z ¥- = +- = 0 0 (18.7) ˆ nH H J z s = = 0 (18.8) can be used to connect the fields E y + and H x + with E y - and H x - , and with the source. Imposing these conditions at z = 0 yields Et yy = 00 (18.9) Ht Htf t xx -= (18.10) Comibining equations (18.5)–(18.6), evaluated at z = 0, with equations (18.9)–(18.10) produces the result f t x + = 0 1 2 (18.11) f t x - 0 1 2 (18.12) f t y + 0 2 (18.13) f t y - 0 2 (18.14) These are the fields in the z = 0 plane. Since these fields propagate away with the functional dependence of (18.1), we may write the full solution as t f t z v z y p + - > ( ) , 2 0 (18.15)
A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Hz t f t z v z x p + =- > (,) ( ) , 1 2 0 (18.16) Ez t f t z v z y p - + < ( ) , h 2 0 (18.17) t f t z v z x p - + < ( ) , 1 2 0 (18.18) where v p 31 0 8 m /s = 300 m / s m .

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FWLec18 - A F Peterson Notes on Electromagnetic Fields...

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