Chapter.1.F09 - Chapter 1 Maxwells Equations Conservation...

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Chapter 1 Maxwell’s Equations, Conservation Laws 1.1 Maxwell’s Equations in Materials: D, H, P and M Our starting point will be the experimentally deduced Maxwell’s Equations These consist of the four differential equations (in gaussian, cgs units) u ± G + u >w , @ 7 ± ² + u > w , (1a) u ± E + u > w , @ 3 (1b) u ² H + u > w , @ ³ 4 f C E + u > w , Cw (1c) u ² K + u > w , @ 4 f C G + u > w , Cw . 7 ± f M + u > w , (1d) In S.I units the equations are (see Physics 411 notes): u ± G + u >w , @ ² + u > w , (2a-2d) u ± E + u > w , @ 3 u ² H + u > w , @ ³ C E + u > w , Cw u ² K + u > w , @ C G + u > w , Cw . M + u > w , for the displacement ¿ eld D + u > w , > the magnetic induction B + u > w , > the electric ¿ eld E + u > w , > and the magnetic ¿ eld H + u > w , = The sources of the ¿ elds are the charge density ² + u > w , and the charge current density J + u > w , = Conservation of charge requires that the charge density and charge current density satisfy u ± M + u > w , . C Cw ² + u > w , @ 3 = (3) This set of equations require information concerning the properties and responses of the materials in the region of the ¿ elds. These are isolated in the constituent equations G + u > w , @ H + u > w , . 7 ± S + u > w , gaussian units (4a) K + u > w , @ E + u > w , ³ 7 ± P + u > w , gaussian units (4b) G + u > w , @ ³ H + u > w , . S + u > w , S.I units (5a) K + u > w , @ 4 ´ E + u > w , ³ P + u > w , S. I. units (5b) where P + u > w , is the polarization ¿ eld and M + u > w , is the magnetization ¿ eld for the materials. u ± S + u > w , @ ³ ² + u > w , the 1
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Section 1.1 Maxwell’s Equations in Materials: D, H, P and M Now u ± M . @Cw @ 3 therefore u ± ^ M ³ M ` @ 3 = Since its divergence vanishes the current density ^ M ³ M ` can be written as the curl of a vector ^ M ³ M ` @ f u ² P + u> w , = P + u > w , is identi ¿ ed as the magnetization of the materials. In terms of P and S the Ampere-Maxwell law is in gaussian units: u ² E + u > w , @ 4 f C H + u > w , Cw . 7 ± f M + u > w , . 7 ± f C S + u > w , Cw . 7 ± u ² P + u > w , (9) u ² ^ E + u > w , ³ 7 ± P + u > w ,` @ 4 f C ^ H + u > w , . 7 ± S + u > w ,` Cw . 7 ± f M + u > w , becomes u ² K + u > w , @ 4 f C G + u > w , Cw . 7 ± f M + u > w , or Eq. (1d ) which is consistent with the constituent equations (Equations 4 and 5). In gaussian (cgs) units the Lorentz force law is I @ t ± H . y f ² E ² gaussian (10) I @ t + H . y ² E , SI units which indicates that E, B, D, and H have dimensions of charge per length squared and that (charge/length) has the dimensions of force (dyne). Outline of topics : a) We will ¿ rst investigate some general properties of the equations. Symmetry properties and conservation laws for the ¿ elds will be examined. The solutions of the equations in a uniform isotropic medium will be developed using scalar and vector potentials. And ¿ nally the wave equation in a uniform, isotropic medium will be obtained. (Chapter 6) b) Next we will investigate the propagation of electromagnetic waves in various systems. This will include the propagation of waves in dissipative and non-uniform systems. We will also consider the conditions placed by causality on the response of a system to applied electromagnetic ¿ elds. (Chapter 7 and 8) c) The interaction of the electromagnetic waves and matter will be investigated through the scattering of the waves and through radiation by simple systems, oscillating electric and magnetic dipoles. (Chapter 9)
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