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) uG +u>w,@73 +u>w, (1a) uE +u>w,@3 (1b) 4 CE +u>w, uH +u>w,@ f Cw (1c) 4 CG +u>w, 7 uK +u>w,@ . M +u>w, f Cw f 3 (1d) In S.I units the equations are (see Physics 411 notes): uG +u>w,@ 3 +u>w, (2a-2d) uE +u>w,@3 CE +u>w, uH +u>w,@ Cw CG +u>w, uK +u>w,@ . M +u>w, Cw 3 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 3 +u>w, and the charge current density J+u> w, = Conservation of charge requires that the charge density and charge current density satisfy C u M +u>w,. +u>w,@3= 3 Cw 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,.7S +u>w, gaussian units (4a) K +u>w,@E +u>w, 7P +u>w, gaussian units (4b) G +u>w,@rH +u>w,.S +u>w, S.I units (5a) 4 K +u>w,@ E +u>w, P +u>w, S. I. units (5b) r where P+u>w, is the polarization ¿eld and M+u>w, is the magnetization ¿eld for the materials. u S +u> w,@s +u>w, the 1 Section 1.1 Maxwell’s Equations in Materials: D, H, P and M Now uMpdw . Cpdw @Cw @3therefore u^Mpdw Ms`@3= Since its divergence vanishes the current density ^Mpdw Ms` can be written as the curl of a vector ^Mpdw Ms`@fuP +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: 4 CH +u>w, 7 7 CS +u>w, uE +u>w,@ . M +u>w,. .7uP +u>w, f Cw f 3 f Cw (9) 4 C^H +u>w,.7S +u>w,` 7 u^E +u>w, 7P +u>w,` @ . M +u>w, f Cw f 3 becomes 4 CG +u>w, 7 uK +u>w,@ . M +u>w, f Cw f 3 or Eq. (1d ) which is consistent with the constituent equations (Equations 4 and 5). In gaussian (cgs) units the Lorentz force law is y I @ t H . E f gaussian (10) I @ t +H . y E, SI units which indicates that E, B, D, and H have dimensions of charge per length squared and that (charge/length)5 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) d) The properties of charge plasmas and electromagnetic waves in plasmas will be studied. (Chapter 10) e) Finally we will consider in more detail phenomenon involving moving charged particles. This will encompass collisions between charged particles, the energy loss and scattering (Chapter 13), and the potentials due to moving charges. The latter will involve both uniform motion and accelerated motion. (Chapter 14). Unless stated otherwise, we shall be using gaussian units. 3 Section 1.2 Field Energies 1.2 Field Energies From our experience with mechanical systems we deduce that the power density supplied to the charged particles of a charge current J3 +u>w, in an electric ¿eld E+u>w, is s +u>w,@H +u>w, M3 +u>w, = From Equations 1a and 1d and using the following identity u ^H +u>w, K +u>w,` @ H +u>w, u K +u>w,.K +u>w, u H +u>w, (11) we obtain, f 4 C C u ^H +u>w, K +u>w,` . K +u>w, E +u>w,.H +u>w, G +u> w, . H +u>w, M +u>w,@3 7 7 Cw Cw 3 (12) This has the form of a conservation law, . C u V +u>w,. x +u>w,.H +u>w, M +u>w,@3 Cw 3 (13) where the energy current density can be identi¿ed as f V +u>w,@ ^H +u>w, K +u>w,` 7 (14) and the time rate of change of the energy density stored in the ¿elds and the medium in the region of the ¿elds is C 4 C C x +u>w,@ K +u>w, E +u=w,.H +u>w, G +u> w, = Cw 7 Cw Cw (15) It is interesting to examine in detail the rate of change of the energy density. We ¿rst consider the energy density associated with the electric ¿eld, using G +u>w,@H +u>w,.7S +u>w, . C 4 C x +u>w,@ H +u>w, G +u>w, Cw H 7 Cw (16a) 4 C C @ H +u>w, H +u>w,.H +u>w, S +u>w, 7 Cw Cw (16b) C As previously noted u S +u>w,@s +u>w, >the polarization charge density and CwS +u>w, has been identi¿ed as the polarization charge current density Js +u>w, = The second term in Eq. 16b is then the power going into the ‘mechanical’ energy of the system. The relationship between the electric ¿eld and the polarization is generally very interesting. In the linear response approximation [6 ]]]] +H, 3 3 3 3 6 3 3 Sl +u>w,@ "lm +u u >w w , Hm +u >w, g u gw (17) m@4 where "+H, +u>w, is the electric susceptibility tensor for the system. The energy density associated with the magnetic ¿eld is C 4 C x +u>w,@ K +u>w, E +u>w, = Cw P 7 Cw (18) There are two points of interest. First, the magnetic force ‘does no work’ (in the Lorentz force law B is perpendicular to gu@gw) and, second, the time derivative operates on the source of the force, the magnetic induction vector B. Using Faraday’s 4 Section 1.2 Field Energies law and the relationship K +u>w,@E +u>w, 7P +u>w,, C 4 C C x +u>w,@ E +u>w, E +u>w,.P +u>w, E +u>w, Cw P 7 Cw Cw (19) 4 C @ E +u>w, E +u>w,.P +u>w, +fuH +u>w,, 7 Cw 4 C @ E +u>w, E +u>w,.fu^H +u>w, P +u>w,` . fH +u>w, uP +u>w, 7 Cw 4 C @ E +u>w, E +u>w,.fu^H +u>w, P +u>w,` . H +u>w, ^M M ` 7 Cw pdw s (19) The ¿rst term in Eq. 19 is the power transfer to the magnetic induction, the second term is the divergence of an energy Àux associated with the magnetization of the material, and the last term is the energy supplied to the rotational current density by the electric ¿eld generated by the time varying magnetic induction vector. In terms of the total charge current density, externally controlled (measured) and material response, the time rate of change of the total energy density change is C 4 C C 4 C x +u>w,@ H +u>w, H +u>w,.H +u>w, S +u>w,. E +u>w, E +u>w,. Cw 7 Cw Cw 7 Cw (20) fu^H +u>w, P +u>w,` . H +u>w, ^Mpdw Ms` 4 C @ ^H +u>w, H +u>w,.E +u>w, E +u>w,` . ; Cw C H +u>w, S +u>w,.^M M ` . fu^H +u>w, P +u>w,` Cw pdw s 4 C k l @ mE +u>w,m5 . mH +u>w,m5 . H +u>w, ^M .^M M `` . ; Cw s pdw s fu^H +u>w, P +u>w,` and the conservation of energy equation is written as f 4 C k l 3@ u ^H +u>w, +E +u>w, 7P +u>w,,` . mE +u>w,m5 . mH +u>w,m5 . 7 ; Cw (21) fu^H +u>w, P +u>w,` . H +u>w, ^Mpdw . M3` f 4 C k l u ^H +u>w, E +u>w,` . mE +u>w,m5 . mH +u>w,m5 . H +u>w, M +u>w,@3 7 ; Cw wrwdo (22) This is the microscopic equation for the conservation of energy known as Poynting’s theorem. The transfer of energy between the ¿elds and the mechanical system is carried out by the interaction of the electric ¿eld and the charge current density. The energy transferred to the current can reside in the translational kinetic energy of the system, the potential energy of interaction of the particles of the system, or thermal energy.