Core Electrodynamics. Sandra C. Chapman. June 29, 2010 CORE ELECTRODYNAMICS COPYRIGHT SANDRA C CHAPMAN

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1 Core Electrodynamics Sandra C. Chapman June 29, 2010

2 Preface This monograph is based on a final year undergraduate course in Electrodynamics that I introduced at Warwick as part of the new four year physics degree. When I gave this course my intention was to engage the students in the elegance of electrodynamics and special relativity, whilst giving them the tools to begin graduate study. Here, from the basis of experiment we first derive the Maxwell equations, and special relativity. Introducing the mathematical framework of generalized tensors, the laws of mechanics, Lorentz force and the Maxwell equations are then cast in manifestly covariant form. This provides the basis for graduate study in field theory, high energy astrophysics, general relativity and quantum electrodynamics. As the title suggests, this book is electrodynamics lite. The journey through electrodynamics is kept as brief as possible, with minimal diversion into details, so that the elegance of the theory can be appreciated in a holistic way. It is written in an informal style and has few prerequisites; the derivation of the Maxwell equations and their consequences is dealt with in the first chapter. Chapter 2 is devoted to conservation equations in tensor formulation, here Cartesian tensors are introduced. Special relativity and its consequences for electrodynamics is introduced in Chapter 3 and cast in four vector form, here we introduce generalized tensors. Finally in Chapter 4 Lorentz frame invariant electrodynamics is developed. Supplementary material and examples are provided by the two sets of problems. The first is revision of undergraduate electromagnetism, to expand on the material in the first chapter. The second is more advanced corresponding to the remaining chapters, and its purpose is twofold: to expand on points that are important, but not essential, to derivation of manifestly covariant electrodynamics, and to provide examples of manipulation of cartesian and generalized tensors. As these problems introduce material not covered in the text they are accompanied by full worked solutions. The philosophy here is to facilitate learning by problem solving, as well as by studying the text. Extensive appendices for vector relations, unit conversion and so forth 1

3 2 are given with graduate study in mind. As SI units are used throughout here, conversions for units and equations are also given. There have been many who have contributed to the existence of this book. Thanks in particular go to Nick Watkins for valuable discussions and to George Rowlands for his insightful reading of the final draft. David Betts, the editor of this series, has also shown remarkable patience and tenacity as the many deadlines have come and gone. The completion of the book was also much assisted by a PPARC personal fellowship. Finally, my thanks go to the students themselves; their lively reception of the original course and their insightful questions were the inspiration for this book. If the reader finds that this book provides a shortcut to experience the beauty of electrodynamics, without sacrificing the rigour needed for further study, then I have suceeded. Sandra Chapman, June, 1999

4 Contents 3.5 Four Vectors and Four Vector Calculus Some mechanics, Newton s laws Some four vector calculus A Frame Invariant Electromagnetism Charge conservation A manifestly covariant electromagnetism The Field Tensors Invariant Form for E and B: The EM Field Tensor Maxwell s Equations in Invariant Form Conservation of Energy-Momentum Lorentz Force Manifestly covariant electrodynamics Transformation of the Fields Field from a Moving Point Charge Retarded Potential Four Vectors and Four Vector Calculus Some mechanics, Newton s laws Some four vector calculus A Frame Invariant Electromagnetism Charge conservation A manifestly covariant electromagnetism The Field Tensors Invariant Form for E and B: The EM Field Tensor Maxwell s Equations in Invariant Form Conservation of Energy-Momentum Lorentz Force Manifestly covariant electrodynamics Transformation of the Fields Field from a Moving Point Charge

5 4 CONTENTS 4.7 Retarded Potential A Brief Tour of Electromagnetism The Building Blocks Maxwell I and II Flux of a vector field Flux of E Flux of B Conservative and nonconservative fields Maxwell III and IV Faraday s law and Galilean invariance Ampère s law and conservation of charge Electromagnetic Waves Conservation Equations Field Energy and Momentum Tensors and Conservation Equations Momentum flux density tensor Momentum flux, gas pressure and fluid equations Cartesian tensors, some definitions Field Momentum and Maxwell Stress Energy conservation: Poynting s theorem Momentum conservation: Maxwell stress Radiation Pressure A Frame Invariant Electromagnetism The Lorentz Transformation The Moving Charge and Wire Experiment Maxwell in Terms of Potentials Generalized Coordinates Four Vectors and Four Vector Calculus Some mechanics, Newton s laws Some four vector calculus A Frame Invariant Electromagnetism Charge conservation A manifestly covariant electromagnetism The Field Tensors Invariant Form for E and B: The EM Field Tensor Maxwell s Equations in Invariant Form Conservation of Energy-Momentum Lorentz Force

6 CONTENTS Manifestly covariant electrodynamics Transformation of the Fields Field from a Moving Point Charge Retarded Potential A Revision Problems 91 A.1 Static Magnetic Fields A.2 Static Electric Fields A.3 Conservation and Poynting s Theorem A.4 The Wave Equation: Linearity and Dispersion A.5 Free Space EM Waves I A.6 Free Space EM Waves II A.7 EM Waves in a Dielectric A.8 Dielectrics and Polarization A.9 EM Waves in a Conductor: Skin Depth A.10 Cavity Resonator B Solutions to Revision Problems 99 B.1 Static Magnetic Fields B.2 Static Electric Fields B.3 Conservation and Poynting s Theorem B.4 The Wave Equation: Linearity and Dispersion B.5 Free Space EM waves I B.6 Free Space EM Waves II B.7 EM Waves in a Dielectric B.8 Dielectrics and Polarization B.9 EM Waves in a Conductor: Skin Depth B.10 Cavity Resonator C Some Advanced Problems 111 C.1 Maxwell Stress Tensor C.2 Liouville and Vlasov Theorems: a Conservation Equation for Phase Space C.3 Newton s Laws and the Wave Equation under Galilean Transformation C.4 Transformation of the Fields C.5 Metric for Flat Spacetime C.6 Length of the EM field Tensor in Spacetime C.7 Alternative Form for the Maxwell Homogenous Equations C.8 Lorentz Transformation of the EM Field Tensor

7 6 CONTENTS D Solution to Advanced Problems 117 D.1 Maxwell Stress Tensor D.2 Liouville and Vlasov Theorems: a Conservation Equation in Phase Space D.3 Newton s Laws and the Wave Equation under Galilean Transformation D.4 Transformation of the Fields D.5 Metric for Flat Spacetime D.6 Length of the EM Field Tensor in Spacetime D.7 Alternative Form for the Maxwell Homogenous Equations D.8 Lorentz Transformation of the EM Field Tensor E Vector identities 129 E.1 Differential Relations E.2 Integral Relations F Tensors 133 F.1 Cartesian Tensors F.2 Special Tensors F.3 Generalized Tensors F.3.1 General properties of spacetime F.3.2 Flat spacetime G Units and Dimensions 139 G.1 SI Nomenclature G.2 Metric Prefixes H Dimensions and Units 143 H.1 Physical Quantities H.2 Equations I Physical Constants (SI) 147

8 List of Figures 4.1 Charge q is at rest at the origin of the S frame and is moving with velocity vˆx 1 w.r.t. frame S. The origins of the S and S frames coincide at t = Sketch of E 1 and E 2 at point P versus x 1 /γ = x 1 vt Sketch of the E field around a charge at rest, and the E field around a charge moving at relative velocity v, in the x 1,x 2 plane A volume element is located at rest at the origin of the S frame which is moving at +v in the x 1 direction w.r.t. the S frame. In the S frame the volume element contains charge ρ dx 1dx 2dx 3. The origins of the two frames are coincident at t = t = Charge q is at rest at the origin of the S frame and is moving with velocity vˆx 1 w.r.t. frame S. The origins of the S and S frames coincide at t = Sketch of E 1 and E 2 at point P versus x 1 /γ = x 1 vt Sketch of the E field around a charge at rest, and the E field around a charge moving at relative velocity v, in the x 1,x 2 plane A volume element is located at rest at the origin of the S frame which is moving at +v in the x 1 direction w.r.t. the S frame. In the S frame the volume element contains charge ρ dx 1 dx 2 dx 3. The origins of the two frames are coincident at t = t = Surface element ds on surface S spanning curve C with line element dl Surface element ds on volume V The right hand rule

9 8 LIST OF FIGURES 1.4 Location of charges q 1 and q The electric field due to charge q i = ρ(r )dv Flux across ds is maximal when v is parallel to ds, and zero when v is perpendicular Volume vdt containing particles that cross ds in time dt Surfaces enclosing single charge q. The surface element ds lies on the arbitrary surface S and forms the end of the cone. There is no flux of E across the sides of the cone Side view of cone enclosing single charge q. The surface element vector ds and field E lie in the plane of the paper Cone enclosing single charge q in cartesian and cylindrical polar coordinates The closed surface S has zero nett flux through it E dl is integrated around the closed curve C in the vicinity of a point charge. The radial sections of the path give contributions of equal size and opposite sign and the sections along contours of the potential (dotted lines) give zero contribution. The integral around the closed curve is zero The observer is at rest in frame 1 (top) and moves with the wire in frame 2 (bottom) The observer is at rest and the wire moves on conducting rails Wire element dl sweeps out surface element ds in time dt Charge flows out of volume V An ideal discharging capacitor Forces acting on fluid element dv form the components of a pressure tensor Vector r in the x, y, z and x,y,z coordinate systems The light clock is oriented perpendicular to its direction of motion. Top: observer rest frame ie the clock moves past us; bottom: clock rest frame ie we move with the clock The light clock is oriented parallel to its direction of motion. Top: observer rest frame ie the clock moves past us; bottom: clock rest frame ie we move with the clock In frame S 1 the current in the wire is carried by the electrons, the test charge q moves at u. In frame S 2 the current is carried by the protons, the test charge q is at rest The (dashed) loop over which we integrate B dl is a circle of radius r centred on the wire, with ˆr transverse to the direction of motion u. Also shown is the direction of u B. 53

10 LIST OF FIGURES The (dashed) surface over which we integrate E ds is a cylinder of radius r centred on the wire, with ˆr transverse to the direction of motion u Charge q is at rest at the origin of the S frame and is moving with velocity vˆx 1 w.r.t. frame S. The origins of the S and S frames coincide at t = Sketch of E 1 and E 2 at point P versus x 1 /γ = x 1 vt Sketch of the E field around a charge at rest, and the E field around a charge moving at relative velocity v, in the x 1,x 2 plane A volume element is located at rest at the origin of the S frame which is moving at +v in the x 1 direction w.r.t. the S frame. In the S frame the volume element contains charge ρ dx 1dx 2dx 3. The origins of the two frames are coincident at t = t = Volume element dv and point P located w.r.t. a single fixed origin O A.1 Problem 1a: A straight wire carrying a steady current I A.2 Problem 1b: A straight wire carrying a steady current I C.1 A pair of wires D.1 Surface S enclosing element of a wire of length dl D.2 Curve C enclosing the wire with element along the curve dl. 122

11 10 LIST OF FIGURES

12 Chapter 1 A Brief Tour of Electromagnetism. So far you will have encountered various expressions from electrostatics and magnetostatics which we will show can then be synthesised into four equations, the Maxwell equations which, with the Lorentz force law are a complete description of the behaviour of charged particles and electromagnetic fields. Electromagnetism is usually presented in this way for two reasons, first this is how electromagnetism was first discovered experimentally and second, these expressions (Coulomb s law, Lenz law, Faraday s law, Biot Savart law and so on) are useful in particular circumstances. Here we will first take a look at how the Maxwell equations are constructed, from the experimentally determined expressions and by using the mathematics of vector fields. The Maxwell equations are a unification of electric and magnetic fields that are inferred experimentally. The unified equations yield an important prediction: the existence of electromagnetic waves, which then compels us to develop a formalism that is consistent with special relativity. This leads to a form for the Maxwell equations, the Lorentz force law and the laws of mechanics that are frame invariant and thus consistent with the requirement that physical laws are the same in all frames of reference, anywhere in the universe. 1.1 The Building Blocks. We will use the experimentally determined: Coulomb s Law 11

13 12 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. ds dl C Figure 1.1: Surface element ds on surface S spanning curve C with line element dl. ds V Faraday s Law Ampère s Law and some mathematics: Figure 1.2: Surface element ds on volume V Stokes Theorem (for any vector field A): C A dl = S A ds (1.1) where surface S spans curve C as shown in figure 1.1 Gauss or Divergence Theorem: S A ds = V AdV (1.2) where surface S encloses volume V as shown in figure 1.1. and the right hand rule î ĵ = ˆk

14 1.2. MAXWELL I AND II 13 i k j v F B Figure 1.3: The right hand rule which is built in to the way the cross product is defined, so that in the Lorentz force law the magnetic force F M = qv B is positive for protons (figure 1.1). If we had used a left handed cross product so that î ĵ = ˆk then Lorentz force law would be written F M = qv B for protons. We have also defined the electric (vector) field to point away from positive charge so that F E = qe for protons. 1.2 Maxwell I and II Coulomb experimentally determined the force between two static charges to be: F 21 = 1 q 1 q 2 4πɛ 0 r 1 r 2 3 (r 1 r 2 ) (1.3) where the force is in Newtons (N), charge q is in Coulombs (C), r in meters and ɛ 0 = C 2 N 1 m 2 is the permittivity of free space (see figure 1.2). Coulomb s law embodies three empirically known properties of F: 1. the force is proportional to the charges F q 1,q 2 2. the force points radially away from the positive charges 3. the force is inverse square F 1/r 2 In addition we can show that the force obeys the principle of superposition so that for a collection of n charges; the force on the jth charge q j is

15 14 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. q 1 r 2 0 r-r 1 2 r 1 q 2 Figure 1.4: Location of charges q 1 and q 2 F j = 1 4πɛ 0 n q i q j r i r j 3 (r i r j ) (1.4) i j We then define the electric field at the position of the jth charge as E = F j (1.5) q j that is, the Lorentz force law for electric field only, so that E is in NC 1 (we will see that it is also in Vm 1 from energy considerations). The vector field E(r j ) is then defined at the position r = r j of the test charge q j : E(r j )= 1 q i 4πɛ 0 r i r j 3 (r i r j ) (1.6) i j This can be expressed in terms of a scalar field, the charge density ρ(r) provided that the collection of point charges can be treated as a smoothly varying function, that is q i = ρ(r )dv (1.7) This description will therefore be valid on length and timescales over which 1.4 to 1.7 hold. When we consider small length and timescales there are two distinct considerations. First, we require that the charge density 1.7, and all other quantities that will be described by scalar and vector fields here, such as electromagnetic fields, energy and momentum densities and so forth, are still describable by continuous functions. Second, we need to use the correct mechanics in the equations of motion for the charges and hence the Lorentz force. Here we will develop electrodynamics in terms

16 1.2. MAXWELL I AND II 15 E(r) r 0 r-r r dv Figure 1.5: The electric field due to charge q i = ρ(r )dv of mechanics that is consistent with special (and general) relativity, but is classical. At some length and timescale quantum mechanics is needed to replace classical mechanics. The field equations may also need to be quantized (to give Quantum Electrodynamics) 1. In this sense, the Maxwell equations, the Lorentz force and the mechanics of special relativity that we will discuss are classical. With these caveats, we can write the electric field in terms of a volume integral over the charge density: E(r) = 1 ρ(r )(r r ) 4πɛ 0 r r 3 dv (1.8) V hence the electric field retains the experimentally determined properties: it points radially away from an element of positive charge, its magnitude is inverse square with distance, it is proportional to the charge and it obeys the principle of superposition. We will now take these properties and phrase them in a more profound form, in terms of the flux of E Flux of a vector field Flux is mathematically and conceptually the same for any vector field. We can explore the concept with a simple example; a cold gas 2 comprised of 1 The need for a field theory that permits both a continuous, classical limit, and discrete quanta (photons in the case of electromagnetic fields) was highlighted by the discovery of the photoelectric effect 2 We will make use of the cold gas model several times in this book, in all cases it is defined as here: a population of identical particles which all have the same velocity v(r,t) at any given position and time.

17 16 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. v(r) ds v(r) Figure 1.6: Flux across ds is maximal when v is parallel to ds, and zero when v is perpendicular. v(r) vdt! l l cos! ds v(r) Figure 1.7: Volume vdt containing particles that cross ds in time dt particles with number density n(r) per unit volume all of which have the same velocity v(r) at any position r. We then define: Flux through surface element ds = number of particles crossing ds per second. Notice that the flux depends upon the angle between ds and v(r) as in figure To find the flux across ds we just need to identify the volume containing all particles that will cross the surface element per second. This volume is sketched in figure 1.2.1, where we have rotated our point of view such that both v and ds are in the plane of the paper (we can always do this, since the two vectors will define a plane). The projection of ds in this plane is l and perpendicular to the plane is a so that ds = la. The angle between v and ds is θ. If the surface element ds is sufficiently small that n and v are constant across it, then in time dt all particles in the cold gas ds

18 1.2. MAXWELL I AND II 17 r q S ds Figure 1.8: Surfaces enclosing single charge q. The surface element ds lies on the arbitrary surface S and forms the end of the cone. There is no flux of E across the sides of the cone. in volume la cos θvdt = ds cos θvdt = v dsdt cross the surface element ds. The number of particles crossing ds in time dt is then nv dsdt, so that a flux of nv ds particles crosses ds per second. Over an arbitrarily large surface S, where across the surface n(r) and v(r) now vary with r the total flux of particles is obtained by the surface integral: Flux = nv ds (1.9) The flux of the vector field nv across the surface S is given by equation Flux of E S Now we can write down the flux of any vector field. For the electric field we will find the flux due to a single positive charge q which is located somewhere inside the closed surface S Flux of E = E ds (1.10) We will utilize the fact that we can choose any convenient S as long as q is located inside, and will build in the three experimentally determined properties of E = F/q from Coulomb s law. We start by taking a surface element ds on an arbitrary surface enclosing q. We can then consider the cone shaped surface with sides formed by radius vectors from q and an end formed by ds as shown in figure From Coulomb s law: S

19 18 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. ds q r " ds n E(r) Figure 1.9: Side view of cone enclosing single charge q. The surface element vector ds and field E lie in the plane of the paper. 1. E points radially out from the charge: that is, E points in the r direction. There is then no flux of E out of the sides of the cone. The flux of E must emerge from the end of the cone ds. We can again sketch the surface element ds choosing the plane of the paper to be the plane defined by the vectors ds and E and the cone is shown cut by this plane in the figure 1. The flux of E out of the cone is then E ds = E ds n (1.11) where ds n is the projection of ds parallel to E (i.e. E is normal to the plane in which surface ds n lies) as shown in figure 1. The element ds n has a very useful property: it lies on the surface of a sphere centred on the charge q. The element ds n is sketched in cylindrical polar coordinates with q at the origin in figure 1, where the element on the surface of the sphere at radius r is ds n = rdθ r sin θdφ (1.12) We can now exploit 2. E is inverse square and 3. E is proportional to q so that E = 1 q 4πɛ 0 then the flux of E through the cone E ds = E ds n = 1 q rdθ r sin θdφ 4πɛ 0 r2 r 2

20 1.2. MAXWELL I AND II 19 x # z! d! d# rd! r sin! d# Figure 1.10: Cone enclosing single charge q in cartesian and cylindrical polar coordinates. y = q dθ sin θdφ 4πɛ 0 which is independent of r (E is inverse square and the surface area of the sphere is proportional to r 2 ). Now the evaluation of the flux of E over the entire closed surface is simply the integral over the solid angle 4π: E ds = q 4πɛ 0 π 0 sin θdθ S 2π 0 dφ = q ɛ 0 For a collection of charges we just add the electric field from each one, by principle of superposition. So the flux through S from n charges will be E ds = (E 1 + E 2 + E ) ds S S = E 1 ds + E 2 ds + E 3 ds +... S S S = q 1 + q 2 + q ɛ 0 ɛ 0 ɛ 0

21 20 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. = n q i = Q ɛ 0 Thus to obtain the total flux of E (unlike E itself from Coulomb s law), we don t need the locations of the charges, just the total charge enclosed. If we now write the total charge in terms of the charge density integrated over the volume V enclosed by S Q = ρdv (1.13) then we have Gauss law in integral form E ds = 1 ɛ 0 S V V i=1 ρdv (1.14) The differential form follows immediately from the Divergence theorem 1.2 EdV = 1 ρdv (1.15) ɛ 0 which we can write as: V V MAXWELL I : E = ρ ɛ 0 (1.16) In going from 1.14 to 1.16 we are implicitly treating the fields as classical, and our field theory will not hold on the quantum scale. It is in this sense that the fields E(r), E(r) are defined Flux of B Similarly we can write a Maxwell equation for magnetic flux. If we define the magnetic flux through any surface S as B ds =Φ B (1.17) S Again, experimentally, it can be shown that the magnetic flux through any closed surface is zero so that B ds = 0 (1.18) which immediately from Divergence theorem 1.2 gives S MAXWELL II B = 0 (1.19)

22 1.2. MAXWELL I AND II 21 S E Figure 1.11: The closed surface S has zero nett flux through it. Field lines are drawn passing through a closed surface S in figure The electric field has nonzero divergence so that closed surfaces can be found that yield zero or nonzero nett flux from 1.14 depending upon whether charge is enclosed in the surface. The magnetic field has zero divergence so that 1.18 always yields zero nett flux as far as we are able to determine experimentally. Lines of E end on electric charge whereas lines of B are continuous. No particle has yet been identified that is a source of magnetic field, but as we shall see in Chapter 4 section 2, the Maxwell equations themselves and special relativity does not preclude the existence of such particles, that is, of magnetic monopoles Conservative and nonconservative fields. Maxwell I was obtained from Coulomb s law by considering the force on a test charge and then using this to define the electric field as E = F/q NC 1. Another definition is in terms of the work done on the charge as it is moved around in the electric field. The energy gained by charge q from the field as it is moved along path C is W = F dl = q E dl (1.20) C from the Lorentz force law (this is minus the work done by the field on the charge). The work done per unit charge W/q (in units JC 1 ) is defined as the potential of the electric field (in V ) giving units of E as Vm 1. Coulomb s law then reveals an interesting property of the electrostatic field via Knowing that the field is radial and is any function of r, for B C S B

23 22 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. the field from a single point charge we have r2 E dl = E(r)dr (1.21) C r 1 Now we can choose C(r, θ,φ ) (in spherical polar coordinates) to have any θ,φ dependence, 1.21 will yield the same result which will depend only on the endpoints r 1 and r 2. It then follows that 1.21 is zero if r 1 = r 2, that is, if the path C is closed. The electrostatic field is conservative. This will also hold for the field from any collection of charges, since by the principle of superposition we can write 1.20 as a sum of integrals due to the electric field from each point charge. For the electrostatic field, taking 1.21 around a closed path and using Stokes theorem 1.1 then immediately gives then since for any scalar field φ E = 0 (1.22) ( φ) =0 we can write the conservative electrostatic field in terms of a potential: E = φ (1.23) For magnetic fields, Maxwell II implied that magnetic field lines form closed loops. From figure one might expect the magnetic field to have curl whereas the electrostatic field does not; this is what is found experimentally and Ampère s law of magnetostatics is: B dl = µ 0 I (1.24) C where the current I is in Amperes (A C/s) from which we can now define the units of B as Tesla (T ) and the permeability of free space µ 0 =4π.10 7 T msc 1. Again, if the collection of (moving) charges can be treated as a smoothly varying function, the current that they carry I flowing across surface S can be expressed in terms of a vector field, the current density J: I = J ds (1.25) This, along with 1.1 immediately gives, for magnetostatics: S B = µ 0 J (1.26)

24 1.3. MAXWELL III AND IV 23 +q E # dl C Figure 1.12: E dl is integrated around the closed curve C in the vicinity of a point charge. The radial sections of the path give contributions of equal size and opposite sign and the sections along contours of the potential (dotted lines) give zero contribution. The integral around the closed curve is zero. Then B is nonconservative unless there are no currents. The special case of current free systems (or regions) can be treated by defining a magnetostatic scalar potential in analogy to Generally we define a magnetic vector potential B = A (1.27) Since for any vector field ( A) = 0 this satisfies B = 0. The potentials φ and A are an equivalent representation for the electrostatic and magnetostatic fields. In section 1.3 we will complete this representation by explicitly considering the general case where the fields vary with time. 1.3 Maxwell III and IV We will now complete the set of Maxwell equations by explicitly considering systems that change with time Faraday s law and Galilean invariance. It is found experimentally that if the magnetic flux passing through a loop of wire changes for any reason, a voltage is induced across the wire. This

25 24 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. v B ^ v B B Figure 1.13: The observer is at rest in frame 1 (top) and moves with the wire in frame 2 (bottom). is Faraday s law: C v E dl = dφ B = d dt dt S B ds (1.28) where the magnetic flux is through surface S spanned by the wire loop forming curve C. For this to be true, it has to hold in all frames of reference. To check Faraday s law, we will consider a wire moving with respect to the magnetic field, so that the magnetic flux changes, and we will look at what happens to the charges in the wire in two frames 3. The two frames are shown in We are in a frame where the magnetic field is time independent, and the wire moves through the field. 2. We transform frames to move with the wire, so that the magnetic field depends on time. 3 Note that this is a non relativistic, or Galilean, frame transformation. The relativistic treatment is in Chapter 3

26 1.3. MAXWELL III AND IV 25 d c dl a b v B Figure 1.14: The observer is at rest and the wire moves on conducting rails. In frame (1) the Lorentz force acts on the charges (electrons) in the wire so that F e = ev B (when a steady state is reached the ends of the wire become charged, and a back e.m.f is induced, that is, an electric field that acts opposite to F e ). When we transform to frame (2) we (the observer) see a stationary wire. The electrons still respond to the Lorentz force however, so we now conclude that F e = ee. So for the Lorentz force law to work in both frames, v B in one frame is just equivalent to E in another. E and B are frame dependent and form a single quantity, the electromagnetic field; a form of the Maxwell Equations must therefore also exist that is frame invariant, and we will derive it later. This electric field that is implied by the frame transformation modifies the conservative, curl free electrostatic field. We will now calculate its curl for the moving wire. To stop the charges piling up at the ends of the wire we will complete the circuit by running the wire on conducting rails as sketched in figure 1.3.1; the rails and the rest of the circuit are at rest w.r.t. the observer. The wire and rails then form a closed loop shown in figure The work done on the electrons around the loop is: b c d a F dl = F dl + F dl + F dl + F dl (1.29) a b Now in this frame only the wire is moving, so that all the terms in the path integral 1.29 are zero except between a and b (note that in the presence of an additional electrostatic field, the contribution to the integral around the closed loop would still be zero). This leaves F dl = b a F dl = e c b a b (v B) dl (1.30)

27 26 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. v dt ds dl Figure 1.15: Wire element dl sweeps out surface element ds in time dt. in the observer s rest frame. To evaluate 1.30 consider the small element of wire dl shown in The area element swept out by wire element dl in time dt is ds = vdt dl. Rearranging the r.h.s. of 1.30 gives b b F dl = e B (v dl) =e B ds dt = e d b B ds = e dφ B (1.31) dt dt a a since in the observer rest frame the magnetic field is time independent, so that work is done by the rate of change of magnetic flux. Now we can make the following assertion: the Lorentz force law yields the same force on the electrons in both frames. Hence the work done on the electrons in the moving wire in frame 1 (equation 1.31) must be equivalent to that done by an electric field in frame 2 where the wire is at rest: e E dl e d B ds (1.32) dt ie, Faraday s law The electric field in the l.h.s. of 1.32 and the magnetic field in the r.h.s. are in different frames. To obtain a relationship between E and B we need to work in a single frame: so let us consider a loop of wire a rest w.r.t the observer so that ds is fixed, and B changes with time. In this frame ds is fixed and v = dr/dt = 0 so from chain rule: S a db dt = B t + B r v (1.33) then 1.32 becomes B E dl = ds (1.34) S t Using Stokes theorem 1.1 this is just Maxwell III in differential form: MAXWELL III E = B t (1.35)

28 1.3. MAXWELL III AND IV 27 Maxwell III ( 1.35) gives the nonconservative part of the electric field that arises when the electromagnetic fields are time varying. It also expresses the equivalence of E and B implied by the Galilean frame transformation. Implicit in Maxwell III is the frame transformation: E 2 = E 1 + v B 1 (1.36) where the subscripts refer to frames 1 and 2 and v is the transformation velocity. Using the principle of superposition we have added the (arbitrary) electric field in frame 1, E 1, which to simplify the above discussion we assumed to be zero. The nonrelativistic 1.36 was needed to make the Lorentz force law Galilean frame invariant; in section 4.5 equation 1.36 will be generalized for Lorentz frame invariance. Since E is no longer zero we cannot describe this field as the gradient of a scalar potential. To retain B = A we can use E = φ A t which is consistent with 1.35 and B = A Ampère s law and conservation of charge. (1.37) Recall from section that when the fields and currents are steady we have Ampère s law 1.26, B = µ 0 J (1.38) which will now be amended to apply to time dependent situations. To work out what happens to the fields when charge and current densities change with time we will make one important assumption, that nett charge is conserved that is, single charges are neither created or destroyed. Experimentally this has been verified to high precision, but we will assume that it is exactly true and later, that it is true in all frames of reference. Charge and current densities can then be related if we consider an arbitrary finite volume containing some total charge that is varying with time Q(t), as charges flow out of the volume as in figure There is a total current flowing out of volume V : I = Q t = ρ (1.39) t and as I flows across the surface S which encloses V we can also write from 1.25: ρ I = J ds = (1.40) t S V V

29 28 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. q q V Q(t) q S q ds q q Figure 1.16: Charge flows out of volume V. This can be written in differential form using 1.2: J = ρ (1.41) t Equation 1.41 immediately shows the problem with 1.26 in time dependent situations; the divergence of 1.26 gives J = 0, that is, currents must close in steady state. When we obtained 1.41 we allowed nett current to flow out of a closed surface, so that figure shows a divergence of J. The correction needed for 1.26 can be found using 1.16 to rewrite 1.41 as J + ρ ( ) t = E J + ɛ 0 =0 t equating this with ( B) = 0 then gives: E MAXWELL IV B = µ 0 J + µ 0 ɛ 0 (1.42) t which is Ampère corrected for time dependent fields. We have added a displacement current to the r.h.s. of How does this work in practice? A simple example of the fields around an ideal capacitor that is discharging are sketched in figure 1.3.2, where B is given by the conduction current flowing in the circuit outside of the capacitor plates, and by the displacement current due to the time dependent electric field between the plates. 1.4 Electromagnetic Waves. It is now straightforward to show that Maxwell s equations support free space waves. We use the vector relation: ( A) = ( A) 2 A (1.43)

30 1.4. ELECTROMAGNETIC WAVES. 29 B(t) I(t) -Q(t) B(t) E(t) +Q(t) B(t) I(t) Figure 1.17: An ideal discharging capacitor. and take the curl of Maxwell IV 1.42 to give ( B) 2 B = µ 0 J + µ 0 ɛ 0 E (1.44) t which, with Maxwell II 1.19 and III 1.35 and in free space where there are no currents J = 0 gives 2 B = 1 2 B c 2 t 2 (1.45) where µ 0 ɛ 0 = 1 c. If instead we take the curl of Maxwell III 1.35 a similar 2 procedure gives (with no charges in free space ρ = 0) 2 E = 1 2 E c 2 t 2 (1.46) Equations 1.45 and 1.46 are wave equations for E and B, and are linear. This means that any wave with frequency ω and wavenumber k of the form: B, E f(ωt k r) (1.47) is a solution to 1.45 and These waves will propagate at phase speed ω/k = c. Crucially, we identify these with light waves in free space. Since 1.45 and 1.46 are linear, we can superpose any solutions of the form 1.47 into a wave group or packet, and this will be nondispersive. The wavegroup will have speed c and will carry energy and momentum through free space 4. 4 The properties of these waves are explored in the revision problems

31 30 CHAPTER 1. A BRIEF TOUR OF ELECTROMAGNETISM. In the next chapter we will discuss momentum flux in the cold gas (defined in section 1.2.1), in terms of distributed bulk properties (such as the momentum flux density tensor) rather than the motions of individual particles. Electromagnetic fields can be quantized, that is, treated as a collection of photons which carry energy and momentum. We would then expect the free space electromagnetic fields to have equivalent energy and momentum flux (the latter given by the Maxwell stress tensor). 1.5 Conservation Equations. Equation 1.41 is a conservation equation and perhaps not surprisingly, all conservation equations are of this form; they embody the premise that the particles and the quantity that they carry (in this case, charge) is neither created nor destroyed. If we recall the cold gas from section of number density n(r,t) with each particle moving with the same velocity v(r,t) carrying charge q, then ρ = nq and J = nqv and 1.41 will become (cancelling q from both sides): (nv) = n t (1.48) Since 1.48 is linear we can use as many different populations of particles as necessary, each with a different q and/or v(r), to represent a gas with finite temperature composed of several particle species of different charge. The nett result will still be an equation of the form of If all the equations describing our system (ie the Maxwell equations) are also linear then any quantity, such as energy, mass, momentum, that can be envisaged as being carried by particles will have a conservation equation of the form At this point it could be argued that the electromagnetic fields are known to be particulate, ie composed of photons, then conservation equations can be found to include field energy and momentum as well as that of the charges. However, to obtain 1.48 from its integral form we represented the ensemble of particles in the gas at position r and time t with the number of particles in elemental volume n(r,t)dv and the flux of particles across an elemental surface n(r,t)v(r,t).ds. The assumption of smoothness 1.7 has been made, that is, we are on spatiotemporal scales over which fields and charge densities behave as smoothly varying functions. So in our field theory here, it is the charges that have been smoothed out, rather than the fields treated as photons.

32 Chapter 2 Field Energy and Momentum. So far we have discussed the free space macroscopic electromagnetic fields and the field equations: the Maxwell equations that describe how the fields evolve in space and time. Charges are included in this description as macroscopic charge density and current density. However point charges carry energy and momentum, the Maxwell equations have wave solutions and we might expect this to imply that the fields carry energy and momentum also. Indeed, the energy and the radiation pressure of waves, and of photons has been measured experimentally. In this chapter we will formalize the concept of the energy and momentum of the electromagnetic fields. This is most easily achieved through conservation equations, and we will use the cold gas model from Chapter 1 in this context. In a gas, individual particles carry momentum which is a vector quantity, the gas as a whole, when described by macroscopic or fluid variables has a corresponding tensor pressure. We will first use the simpler cold gas model to introduce Cartesian tensors, and their role in equations for conservation of momentum flux. We will then find the Maxwell Stress Tensor that is, the ram pressure tensor for the electromagnetic field. The cold gas model will allow us to treat a system with free charges and electromagnetic fields (ie E and B). This is readily generalized to linear media 1. 1 The Maxwell equations and conservation equations for free space and charges discussed here are linear. Our approach then generalizes to media in which these equations remain linear. This is the case if the fields induced in the medium are linearly proportional to those in the surrounding free space. See the revision problems for examples. 31

33 32 CHAPTER 2. FIELD ENERGY AND MOMENTUM. x P xz P xx P xy z P yx P yz P yy Figure 2.1: Forces acting on fluid element dv form the components of a pressure tensor 2.1 Tensors and Conservation Equations Momentum flux density tensor. Tensors arise in macroscopic or bulk descriptions such as gases, fluids, and solids where we have vector fields describing distributed bulk properties rather than properties at a vanishingly small point (such as at a particle). A cube shaped elemental volume dv in a gas is sketched in figure and the possible forces that can act on the three faces of the cube are labelled. If we just consider the x faces, we can compress the cube by exerting forces normal to the surface, in the ±ˆx direction (P xx ), we can also twist the cube by exerting shear forces tangential to the surface, in the ±ŷ direction (P xy ), and in the ±ẑ direction (P xz ). The same is true of the y and z faces, so that in our three dimensional gas we have nine numbers describing the forces on the gas. These nine elements constitute the pressure tensor for the gas and we can write them as a matrix: y P xx P xy P xz P ij = P yx P yy P yz (2.1) P zx P zy P zz If there were no shear forces the tensor has three independent elements

34 2.1. TENSORS AND CONSERVATION EQUATIONS. 33 and becomes: P ij = P xx P yy 0 (2.2) 0 0 P zz and if these normal forces on all sides of the cube are equal, the pressure is isotropic: P ij = P P 0 (2.3) 0 0 P We will now obtain the momentum flux tensor for the cold gas in which all particles move with the same velocity v(r) at a given position r. In a given direction ˆr the momentum of one particle is p r = mv r = mv ˆr (2.4) and so momentum flowing across surface element ds in time dt due to this component is ρ(v ˆr)(v dsdt) =ρv r v n dsdt (2.5) where the term in the first bracket of the l.h.s. of 2.5 is the momentum density due to the ˆr component, and the term in the second bracket is the volume containing the particles that flow across ds in time dt. The subscripts r and n denote components along ˆr, and the normal to ds, ˆn respectively. The momentum flow per unit area per unit time from equation 2.5 is the scalar P rn = ρv r v n (2.6) We could write the momentum flux flowing in ˆn due to all components of v by summing 2.6 over r =1, 3 (that is, r =1, 2, 3 2 the x, y, z directions) as the vector P n = ρ(v xˆx + v y ŷ + v z ẑ)v n = ρvv n (2.7) If we then sum 2.7 over n =1, 3 (again, the x, y, z directions), we have the momentum flux flowing in all directions due to all components as the tensor: P = ρ(v xˆx + v y ŷ + v z ẑ)(v xˆx + v y ŷ + v z ẑ)=ρvv (2.8) Tensors written as vector outer products as in equation 2.8 are known as dyadics. 2 Throughout this book we will use the index notation i, j to mean all values between i and j inclusive.

35 34 CHAPTER 2. FIELD ENERGY AND MOMENTUM. The dyadic vv can be written out in full: v x v xˆxˆx +v x v yˆxŷ +v x v zˆxẑ vv = +v y v x ŷˆx +v y v y ŷŷ +v y v z ŷẑ =(ˆxŷẑ) v ˆx xv x v x v y v x v z v y v x v y v y v y v z ŷ +v z v x ẑˆx +v z v y ẑŷ +v z v z ẑẑ v z v x v z v y v z v z ẑ (2.9) For any orthogonal coordinate system we can use the following shorthand: vv = v xv x v x v y v x v z v y v x v y v y v y v z (2.10) v z v x v z v y v z v z For most purposes we don t really need to write out all the elements of the matrix Instead, we can simply write v i v j where it is assumed in this three dimensional, Cartesian world i =1, 3 and j =1, 3. The momentum flux density tensor is then written as: P ij = ρv i v j (2.11) The notation in 2.11 is useful, for example we can immediately deduce from 2.11 that vv and hence P is symmetric as v i v j = v j v i (there are only 6 independent components for the cold gas) without writing out all nine elements of the matrix We can now calculate the force on the cube due to the momentum flux of the cold gas. The gas particles can deliver momentum to the cube either by flowing out of the cube (the rocket effect), or by slowing down in the cube. The force on the cube of arbitrary volume V due to flow out of particles is minus the rate at which momentum flows out. The force due to particles slowing down in the cube is again minus the momentum inflow rate as ds points outwards on the surface. If we consider one component of v in the ˆr direction, the rate at which momentum is delivered through ds is given by df r = ρ(v ˆr)(v ds) (2.12) where the first bracket is the momentum density of the ˆr component. So we have from all three components: Thus the force on the whole volume is F v = df = P ds = ρvv ds = S df = ρvv ds = P ds (2.13) S V P dv = V ρvvdv (2.14)

36 2.1. TENSORS AND CONSERVATION EQUATIONS Momentum flux, gas pressure and fluid equations. The above then implies that the momentum flux density corresponds to a force per unit volume: F v V = 1 df = P (2.15) V V We can now obtain a conservation equation for momentum in the gas. To do this we need to relate the l.h.s. of 2.15 to the rate of change of momentum in the gas. For this we will make an important restriction: we will consider the force over a small volume dv in which the total number of particles is constant. This means we are considering the rate of change of momentum due to a local ensemble of particles slowing down, or speeding up, and we will choose dv accordingly; if this dv contains a constant mass M of particles then it experiences a force Then the force per unit volume F V = dmv = M dv dt dt F V dv = ρ dv m dt (2.16) (2.17) One possibility is to choose dv to contain the same particles by moving with the fluid at velocity v, then 2.17 becomes (using chain rule) [ ] dv v ρ m dt = ρ m t +(v. )v (2.18) This is equivalent to Liouville s theorem 3 The momentum flux density tensor that we have derived is not the same as the pressure tensor P th. The pressure of a warm gas is the momentum flux due to the thermal or random motions of the gas particles in the rest frame of the gas. In the warm gas, a particle will have total velocity u = v +c where c is the random velocity relative to the average or bulk velocity v(r,t) with which the gas as a whole moves, so that the average < u >= v and < c >= 0. 3 Liouville s theorem expresses conservation of probability density along a trajectory in phase space. In a system with no sources or sinks of particles, we can follow the phase space trajectory r(t), v(t) of any particle and along that trajectory the probability of finding the particle in elemental phase space volume drdv is constant. See advanced problem 2.