603: Electromagnetic Theory I CONTENTS Maxwell’s Equations: Introduction; units; boundary conditions. • Electrostatics: Uniqueness theorem; Green’s theorem; gauge potentials; energy • Boundary value problems in electrostatics: Method of images; separation of variables • in Cartesian, spherical polar and cylindrical polar coordinates. Dielectric media • Multipole expansion • Magnetostatics • Time-dependent fields • Contents 1 Introduction 3 1.1 Maxwell’sequations .............................. 3 1.2 Gaussianunits .................................. 4 1.3 Macroscopicmedia ................................ 6 1.4 Boundary conditions at media interfaces . ....... 8 1.5 Gaugepotentials ................................. 12 1.6 Electric field of a point charge; Coulomb’s law . ........ 14 1.7 Gauss’slaw .................................... 15 1.8 Electrostatic potential . 17 1.9 Electrostaticenergy ............................. 18 1.10Capacitance.................................... 22 2 Uniqueness Theorem, Green Functions and Method of Images 23 2.1 Uniquenesstheorem ............................... 23 2.2 Green’stheorem ................................. 27 2.3 Green functions and the boundary-value problem . ........ 28 2.4 Dirichlet and Neumann Green functions for infinite planar conductor . 32 2.5 Dirichlet Green function for spherical conductor . ........... 35 3 Separation of Variables in Cartesian Coordinates 41 3.1 Introduction.................................... 41 3.2 Separation of variables in Cartesian coordinates . ........... 42 4 Separation of variables in spherical polar coordinates 45 4.1 Series solution of the Legendre equation . ....... 47 4.2 Rodriguesformula ................................ 52 4.3 Thegeneratingfunction . 53 4.4 ExpansioninLegendrepolynomials. ..... 54 4.5 Azimuthally-symmetric solutions of Laplace’s equation............ 56 4.6 Some useful techniques for azimuthally-symmetric problems......... 59 4.7 Thesphericalharmonics . 64 4.8 General solution of Laplace’s equation without azimuthalsymmetry . 70 4.9 Another look at the generating function . ...... 72 4.10 Dirichlet Green function expansion . ....... 79 1 4.11 Inversionsymmetryrevisited . ..... 80 5 Separation of Variables in Cylindrical Polar Coordinates 82 5.1 Solutions of Bessel’s equation . ..... 84 5.2 PropertiesoftheBesselfunctions . ...... 88 5.3 A boundary-value problem in cylindrical polar coordinates.......... 92 5.4 ModifiedBesselfunctions . 93 5.5 Green function in cylindrical polar coordinates . .......... 95 6 Multipole Expansion 98 6.1 Index notation for Cartesian vectors and tensors . ......... 98 6.2 Multipole expansion in Cartesian coordinates . ......... 101 6.3 Multipole expansion using spherical harmonics . ......... 104 6.4 Another construction of the spherical harmonics . ......... 105 6.5 Multipole expansion of the energy in an external field . ......... 107 7 Dielectric Media 108 7.1 Microscopicdescription . 108 7.2 Examplesofdielectricmedia . 113 7.3 Boundary-value problems with dielectric interfaces . ............ 115 7.4 Electrostatic energy in dielectric media . ......... 123 8 Magnetostatics 126 8.1 Amp`ere’s law and the Biot-Savat law . 127 8.2 Magnetic field of a circular current loop . 131 8.3 LocalisedCurrentDistribution . 137 8.4 Force on a current distribution in an external B~ field............. 141 8.5 Magnetically permeable media . 143 8.6 Boundary conditions at medium interfaces . ....... 145 8.7 Techniques for solving boundary-value problems in magnetostatics . 145 9 Electromagnetism and Quantum Mechanics 153 9.1 The Schr¨odinger equation and gauge transformations . ........... 153 9.2 Magneticmonopoles ............................... 155 9.3 Dirac quantisation condition . 157 2 1 Introduction 1.1 Maxwell’s equations The equations now known as Maxwell’s equations were obtained over an extended period, principally during the early nineteenth century. Here, we shall take as our starting point the set of four differential equations as they were presented by Maxwell in about 1861. It was Maxwell who completed the process of constructing the equations, thereby achieving the first unification of fundamental theories in physics. Prior to Maxwell, there were two essen- tially independent theories, one describing electricity and the other describing magnetism, and it was he who brought about the synthesis that unified them into a single theory of electromagnetism. It was only later, after Einstein developed the theory of Special Relativity in 1905, that the magnitude of Maxwell’s achievement really became clear. Especially, a quite remarkable feature of Maxwell’s 1861 equations is that they are already completely compatible with special relativity, with no need for modification of any kind.1 Aside from changes in notation and units, Maxwell’s equations have remained otherwise unaltered since 1861. Let us begin by considering Maxwell’s equations in free space, by which is meant that the space outside of any conducting surfaces is assumed to be a vacuum. Using the SI system of units, Maxwell’s equations are: ′ ~ ′ ′ ρ ′ ∂E ′ ~ E~ = , ~ B~ µ0ǫ0 = µ0J~ , ∇ · ǫ0 ∇× − ∂t ∂B~ ′ ~ B~ ′ = 0 , ~ E~ ′ + = 0 . (1.1) ∇ · ∇× ∂t Observe that I have written these equations with a “prime” on the electric field E~ the magnetic field B~ , the electric charge density ρ and the electric current density J~ . This is to indicate that these quantities are all expressed in the SI system of units. (The SI system typically maximises the number of “redundant” dimensionful constants, and so one might say that it is Super Inconvenient.) The remaining quantities appearing in (1.1) are the constants ǫ0 and µ0, which are, respectively, the permitivity of free space and the permeability of free space. They have the values ǫ 8.85419 10−12 Farads/metre , µ = 4π 10−7 Henries/metre (1.2) 0 ≈ × 0 × 1This contrasts with the case of Newtonian mechanics, which is not compatible with special relativity and therefore did not survive as a “fundamental” theory after 1905. 3 1.2 Gaussian units SI units have their virtues for some purposes, but they can also be quite inconvenient in practice. This seems to be especially true in electromagnetism, and for this reason it is often more convenient to stick with an earlier system, known as Gaussian units. In this system, Maxwell’s equations in free space take the form 1 ∂E~ 4π ~ E~ = 4πρ , ~ B~ = J~ , ∇ · ∇× − c ∂t c 1 ∂B~ ~ B~ = 0 , ~ E~ + = 0 , (1.3) ∇ · ∇× c ∂t where c is the speed of light. Observe that here, we are writing the equations using the unprimed quantities E~ , B~ , ρ and J~ , and it will probably therefore come as no surprise that it is this Gaussian system of units that I prefer to use. It should already be evident upon comparing (1.1) and (1.3) that Gaussian system is somewhat simpler, in that one needs only one “fundamental constant” (i.e. the speed of light) rather than two (the permittivity and the permeability of free space).2 The introduction of the 4π factors in (1.3) may perhaps seem tiresome, but the advantage of doing so will become apparent in due course. (There can be no escape from having 4π factors somewhere, because of the fact that a unit sphere has area 4π.) In order to ensure that we can, whenever desired, revert to SI units, it is useful to work out explicitly the relation between the Gaussian quantities (denoted without primes, as in (1.3)) and the SI quantities (denoted with primes, as in (1.1)). In order to do this, we first need to understand a very important property of the Maxwell equations, namely that they imply the existence of electromagnetic waves that propagate at the speed of light. Consider the Maxwell equations (1.1) in a completely empty region of space, where there ′ is no charge density ρ′ and no current density J~ . Taking the curl of the last equation, and using the vector identity ~ (~ V~ )= ~ (~ V~ ) 2V~ , we obtain ∇× ∇× ∇ ∇ · −∇ ∂ 2E~ ′ + ~ (~ E~ ′)+ ~ B~ ′ = 0 , (1.4) −∇ ∇ ∇ · ∂t∇× and hence, since the partial derivative ∂/∂t commutes with the spatial partial derivatives 2Actually, one can do even better by changing the units in which one measures length from the metre to the light second, or alternatively, changing the unit of time to the light metre (the time light takes to travel 1 metre). In either of these systems of units, the speed of light becomes equal to 1. 4 in ~ , ∇ ∂B~ ′ 2E~ ′ + ~ (~ E~ ′)+ ~ = 0 . (1.5) −∇ ∇ ∇ · ∇× ∂t ′ Using the first equation in (1.1) (with ρ′ = 0) and the second equation (with J~ = 0) then gives ∂2 2E~ ′ µ ǫ E~ ′ = 0 . (1.6) ∇ − 0 0 ∂t2 Analogous manipulations show that B~ ′ satisfies an identical equation. We see, therefore, that the electric and magnetic fields satisfy an equation for waves that propagate at the speed 1 c = 2.99792 108 metres/second . (1.7) √µ0ǫ0 ≈ × This is precisely the speed of light in vacuo, and these wave solutions describe the propa- gation of radio waves, light, etc. With this preliminary, we are nearly ready to establish the relation between the SI units used in (1.1), and the Gaussian units used in (1.3). The procedure for doing this is to introduce constant factors α, β, γ and δ that relate the primed to the unprimed quantities, ′ E~ ′ = αE~ , B~ ′ = βB~ , ρ′ = γρ , J~ = δJ~ , (1.8) and to fix the values of these constants by demanding that plugging (1.8) into (1.1) should give (1.3). It is essential, in doing so, that we have the relation (1.7) between c, µ0 and ǫ0. Elementary algebra then gives γ γ µ α = , β = 0 , δ = γ . (1.9) 4πǫ 4π ǫ 0 r 0 Observe that the value of the constant γ has not yet been determined. This means that we can choose any value for γ, and we may use this freedom in order to make some other equation as nice as possible by removing superfluous constant factors. Consider Coulomb’s law, giving the force between two electric charges separated by a distance R. We again need ′ ′ to distinguish between the charges q1 and q2 expressed in SI units, and the charges q1 and ′ q2 expressed in Gaussian units. Since we have the relation ρ = γρ between charge densities in the two systems, and since the unit of volume is the same in the two systems, it follows that the charges will also be related by the same factor of γ: q′ = γq .