Equation(35. 1)is Faraday's induction law Equation (35.2)is the generalized Ampere's circuit law Equations (35.3)and(35.4)are Gauss' laws for magnetic and electric fields. Taking the divergence of (35. 2)and intro- ducing(35.4), we find that v·J(F,t)+p(7,t)=0 (35.5 at This is the conservation law for electric charge and current densities. Regarding (35.5) as a fundamental quation, we can use it to derive(35.4)by taking the divergence of (35.2).Equation(35.3 )can also be derived by taking the divergence of(35. 1)which gives a(v- B(, n))at=0 or that V. B(T, t)is a constant independent of time. Such a constant, if not zero, then implies the existence of magnetic monopoles similar to free electric charges. Since magnetic monopoles have not been found to exist, this constant must be zero and we arrive at (35.3) 35.2 Constitutive relations The Maxwell equations are fundamental laws governing the behavior of electromagnetic fields in free spad and in media. We have so far made no reference to the various material properties that provide connections to other disciplines of physics, such as plasma physics, continuum mechanics, solid-state physics, fluid dynamics, statistical physics, thermodynamics, biophysics, etc, all of which interact in one way or another with electro- magnetic fields. We did not even mention the Lorentz force law, which constitutes a direct link to mechanics It is time to state how we are going to account for this vast"outside "world. From the electromagnetic way point of view, we shall be interested in how electromagnetic fields behave in the presence of media, whether the wave is diffracted, refracted, or scattered. Whatever happens to a medium, whether it is moved or deformed of secondary interest. Thus we shall characterize material media by the so-called constitutive relations that can be classified according to the various properties of the media The necessity of using constitutive relations to supplement the Maxwell equations is clear from the following mathematical observations. In most problems we shall assume that sources of electromagnetic fields are given Thus and p are known and they satisfy the conservation law (35.5). Let us examine the Maxwell equations and see if there are enough equations for the number of unknown quantities. There are a total of 12 scalar unknowns for the four field vectors E, H, B, and D. As we have learned, Eqs. (35.3)and(35.4)are not dependent equations; they can be derived from Eqs.(35. 1),(35. 2), and (35.5). The independent equations are Eqs.(35. 1)and (35. 2), which constitute six scalar equations. Thus we need six more scalar equations. These are the constitutive relations The constitutive relations for an isotropic medium can be written simply as D=EE where e permittivity (356a) B=uH where u permittivity By isotropy we mean that the field vector E is parallel to D and the field vector His parallel to B In free space void of any matter, u=μand∈=∈ μ=4×107 henry/ meter ∈n=8.85×10-12 farad/ meter Inside a material medium, the permittivity e is determined by the electrical properties of the medium and the permeability u by the magnetic properties of the medium. c 2000 by CRC Press LLC© 2000 by CRC Press LLC Equation (35.1) is Faraday’s induction law. Equation (35.2) is the generalized Ampere’s circuit law. Equations (35.3) and (35.4) are Gauss’ laws for magnetic and electric fields. Taking the divergence of (35.2) and intro￾ducing (35.4), we find that (35.5) This is the conservation law for electric charge and current densities. Regarding (35.5) as a fundamental equation, we can use it to derive (35.4) by taking the divergence of (35.2). Equation (35.3) can also be derived by taking the divergence of (35.1) which gives ¶(— · ( ,t))/¶t = 0 or that — · ( ,t) is a constant independent of time. Such a constant, if not zero, then implies the existence of magnetic monopoles similar to free electric charges. Since magnetic monopoles have not been found to exist, this constant must be zero and we arrive at (35.3). 35.2 Constitutive Relations The Maxwell equations are fundamental laws governing the behavior of electromagnetic fields in free space and in media. We have so far made no reference to the various material properties that provide connections to other disciplines of physics, such as plasma physics, continuum mechanics, solid-state physics, fluid dynamics, statistical physics, thermodynamics, biophysics, etc., all of which interact in one way or another with electro￾magnetic fields. We did not even mention the Lorentz force law, which constitutes a direct link to mechanics. It is time to state how we are going to account for this vast “outside” world. From the electromagnetic wave point of view, we shall be interested in how electromagnetic fields behave in the presence of media, whether the wave is diffracted, refracted, or scattered. Whatever happens to a medium, whether it is moved or deformed, is of secondary interest. Thus we shall characterize material media by the so-called constitutive relations that can be classified according to the various properties of the media. The necessity of using constitutive relations to supplement the Maxwell equations is clear from the following mathematical observations. In most problems we shall assume that sources of electromagnetic fields are given. Thus and r are known and they satisfy the conservation law (35.5). Let us examine the Maxwell equations and see if there are enough equations for the number of unknown quantities. There are a total of 12 scalar unknowns for the four field vectors , , , and . As we have learned, Eqs. (35.3) and (35.4) are not independent equations; they can be derived from Eqs. (35.1), (35.2), and (35.5). The independent equations are Eqs. (35.1) and (35.2), which constitute six scalar equations. Thus we need six more scalar equations. These are the constitutive relations. The constitutive relations for an isotropic medium can be written simply as (35.6a) (35.6b) By isotropy we mean that the field vector is parallel to and the field vector is parallel to . In free space void of any matter, m = mo and e = eo , mo = 4p ¥ 10–7 henry/meter eo ª 8.85 ¥ 10–12 farad/meter Inside a material medium, the permittivity e is determined by the electrical properties of the medium and the permeability m by the magnetic properties of the medium. — × J r t + = t ( , ) (r,t) ¶ ¶ r 0 B – r – B – r – J E – H – B – D – D = eE where e = permittivity B = mH where m = permittivity E – D – H – B –