Historically the fields induced by the sources were called D =EoE+P =EE and H=B/Ho-M B/u,where P and M are the electric dipole moment density and the magnetic dipole moment density induced in the material by the presence of the fields. They were experimentally determined in materials where e and u varied from one material to another.The reason is that the sources on the right included only the "free"charges and currents which could be controlled in the lab,whereas materials were themselves made up of charged particles which moved internally causing currents.Then eo and uo were just the values of e,u in the vacuum. 1.3 Heaviside-Lorentz (HL)Units We put hats on the(HL)quantities (g is charge and m is magnetic dipole moment): E=VEoE,B=B/VH0, 9=q/o (7) 10B =-又×E. HL c Ot (8) 10E =V×B-), HL c Ot (9) 7.B=0, 7.它=p HL (10) F=(E+8xB), HL (11) In particle physics we go even further and choose units where h=c=1,which removes all coefficients from the equations. 1.4 Physical meaning of Maxwell's equations We first turn to the divergence equations which do not involve time derivatives.These have no direct analogy in particle mechanics and represent constraints on the fields which hold independently at each time. Constraint Equations (Gauss Laws) Consider a region of space R and integrate both sides of the divergence equations over this region Qenclosed dvp=Eo dW7·E=o dSn·E三eoΦE(R) (12) R JR 0 dWV·B=dSn.B=Φs(R) (13) The third equality in each of these equations is just one of the vector calculus analogues of the fundamental theorem of calculus (z1=x,2 =y,3 =2): ∂f dr- Tk dSnkf月 (14) 5 ©2010 by Charles Thorn
Historically the fields induced by the sources were called D = 0E + P = E and H = B/µ0 − M = B/µ, where P and M are the electric dipole moment density and the magnetic dipole moment density induced in the material by the presence of the fields. They were experimentally determined in materials where and µ varied from one material to another. The reason is that the sources on the right included only the “free” charges and currents which could be controlled in the lab, whereas materials were themselves made up of charged particles which moved internally causing currents. Then 0 and µ0 were just the values of , µ in the vacuum. 1.3 Heaviside-Lorentz (HL) Units We put hats on the (HL) quantities (q is charge and m is magnetic dipole moment): Eˆ = √ 0E, Bˆ = B/ √ µ0, qˆ = q/ √ 0, mˆ = m √ µ0 (7) 1 c ∂Bˆ ∂t = −∇ × Eˆ , HL (8) 1 c ∂Eˆ ∂t = ∇ × Bˆ − 1 c Jˆ, HL (9) ∇ · Bˆ = 0, ∇ · Eˆ = ρˆ, HL (10) F = qˆ Eˆ + v c × Bˆ , HL (11) In particle physics we go even further and choose units where = c = 1, which removes all coefficients from the equations. 1.4 Physical meaning of Maxwell’s equations We first turn to the divergence equations which do not involve time derivatives. These have no direct analogy in particle mechanics and represent constraints on the fields which hold independently at each time. Constraint Equations (Gauss Laws) Consider a region of space R and integrate both sides of the divergence equations over this region Qenclosed = Z R dV ρ = 0 Z R dV ∇ · E = 0 Z ∂R dSnˆ · E ≡ 0ΦE(R) (12) 0 = Z R dV ∇ · B = Z ∂R dSnˆ · B ≡ ΦB(R) (13) The third equality in each of these equations is just one of the vector calculus analogues of the fundamental theorem of calculus (x1 = x, x2 = y, x3 = z): Z R d 3x ∂f ∂xk = Z ∂R dSnkf (14) 5 c 2010 by Charles Thorn
(It's mathematics not physics!).Here dS is the element of surface area and n is the unit outward directed normal to the surface:nids=dz2dx3,nads dxsdr1,nads=dzid2. The physics (Gauss)is the connection between the total charge enclosed by a closed surface and the flux of the corresponding field through the surface.In the magnetic case, there is no such thing as magnetic charge (magnetic monopoles don't seem to exist!),so the flux of the magnetic field through any closed surface is always zero.This linkage between charge and flux is the physical content of the two constraint Maxwell equations.Note that this linkage is valid at all times. Dynamic equations The remaining two Maxwell equations govern the time dependence of the fields,i.e.their dynamics.They are first order in time,but second order equations can be obtained by taking the curl of one and substituting the other: OV×B 02E.0J =40+0元=-V×(V×E)=2E-V(V·E) (15) Ot ∂V×E B 三-02=VxG （×)-v×J=-号-v×J (16) Remembering c2=1/coto and Gauss law,a little rearrangement leads to /182 72 8J 2012- E=-Mt -2 (17)） E0 182 2那-2）B=oV×J (18) We recognize these as wave equations,showing that electromagnetic fields can form waves traveling at speed c,the speed of light.In the oscillator analogy -c2V2 plays the role of oscillating frequency.For a plane wave ek,this operator gives c22.Thus frequency and wavelength are linked w=ck =2c/A. 1.5 Charge conservation A striking consequence of Maxwell's equations is that the charge and current sources are not independent.To see why,take the divergence of the second equation,using V.(Vx B)=0 identically: ∂7.E 证✉-VJ (19) Integrating both sides over a region of space shows that the rate of decrease (increase)of the total charge in the region is exactly equal to the charge flowing out (in)through the boundary of the region. Historically,the time derivative of E appearing on the left of the second equation was much too small to be seen experimentally.The empirically well-supported Ampere law 6 ©2010 by Charles Thorn
(It’s mathematics not physics!). Here dS is the element of surface area and n is the unit outward directed normal to the surface: n1dS = dx2dx3, n2dS = dx3dx1, n3dS = dx1dx2. The physics (Gauss) is the connection between the total charge enclosed by a closed surface and the flux of the corresponding field through the surface. In the magnetic case, there is no such thing as magnetic charge (magnetic monopoles don’t seem to exist!), so the flux of the magnetic field through any closed surface is always zero. This linkage between charge and flux is the physical content of the two constraint Maxwell equations. Note that this linkage is valid at all times. Dynamic equations The remaining two Maxwell equations govern the time dependence of the fields, i.e. their dynamics. They are first order in time, but second order equations can be obtained by taking the curl of one and substituting the other: ∂∇ × B ∂t = µ00 ∂ 2E ∂t 2 + µ0 ∂J ∂t = −∇ × (∇ × E) = ∇2E − ∇(∇ · E) (15) 0 ∂∇ × E ∂t = −0 ∂ 2B ∂t 2 = ∇ × ∇ × B µ0 − ∇ × J = −∇2 B µ0 − ∇ × J (16) Remembering c 2 = 1/0µ0 and Gauss law, a little rearrangement leads to 1 c 2 ∂ 2 ∂t 2 − ∇2 E = −µ0 ∂J ∂t − ∇ ρ 0 (17) 1 c 2 ∂ 2 ∂t 2 − ∇ 2 B = µ0∇ × J (18) We recognize these as wave equations, showing that electromagnetic fields can form waves traveling at speed c, the speed of light. In the oscillator analogy −c 2∇2 plays the role of oscillating frequency. For a plane wave e i·✁ , this operator gives c 2k 2 . Thus frequency and wavelength are linked ω = ck = 2πc/λ. 1.5 Charge conservation A striking consequence of Maxwell’s equations is that the charge and current sources are not independent. To see why, take the divergence of the second equation, using ∇ ·(∇ × B) = 0 identically: 0 ∂∇ · E ∂t = ∂ρ ∂t = −∇ · J (19) Integrating both sides over a region of space shows that the rate of decrease (increase) of the total charge in the region is exactly equal to the charge flowing out (in) through the boundary of the region. Historically, the time derivative of E appearing on the left of the second equation was much too small to be seen experimentally. The empirically well-supported Ampere law 6 c 2010 by Charles Thorn
did not need the term.Maxwell realized,however,that without such a term,one would have V.J=0 implying the absurd conclusion that charge could never build up in a localized region.He resolved the absurdity by adding the time derivative term.This is a stunning example of world-class theoretical physics.Not only did it resolve an inconsistency of the equations,but it also implied the necessity of electromagnetic waves,which were experimentally confirmed only later. 1.6 Potentials and Gauge Invariance Returning to the 4 Maxwell equations,we see that two of them (the first and third)do not involve sources.They are completely linear,and we should be able to solve them once and for all.We first note that if B=Vx A,then V.B =0 for any A.The converse is also true:if V.B =0,then one can always find a vector potential A such that B=V x A. So by using A instead of B to describe the magnetic field,the third Maxwell equation will automatically be satisfied.Plugging this into the first Maxwell equation leads to (+ =0 (20) But any vector function with a vanishing curl can be expressed as the gradient of a scalar function,so we can write ∂A E+ =-7φ， or E-- Ot aA-Vφ 8t (21) In summary we can put E=-V6- 6A B=V×A, Ot (22) and then forget about the first and third Maxwell equations.This reduces the number of independent fields from 6 to 4.But the potentials are not unique:changing them by a gauge transformation a A→A+7Λ， 0→φ- (23) with A any function of space and time,leaves the fields unchanged.It is sometimes useful to fix this ambiguity by specifying a condition on the potentials,such as V.A=0(Coulomb gauge),reducing the number of independent fields to 3.The two remaining Maxwell equa- tions become 02A 一E0 +V -V2A+(.A-J (24) 8t 0 720-7 (25) ©2010 by Charles Thorn
did not need the term. Maxwell realized, however, that without such a term, one would have ∇ · J = 0 implying the absurd conclusion that charge could never build up in a localized region. He resolved the absurdity by adding the time derivative term. This is a stunning example of world-class theoretical physics. Not only did it resolve an inconsistency of the equations, but it also implied the necessity of electromagnetic waves, which were experimentally confirmed only later. 1.6 Potentials and Gauge Invariance Returning to the 4 Maxwell equations, we see that two of them (the first and third) do not involve sources. They are completely linear, and we should be able to solve them once and for all. We first note that if B = ∇ × A, then ∇ · B = 0 for any A. The converse is also true: if ∇ · B = 0, then one can always find a vector potential A such that B = ∇ × A. So by using A instead of B to describe the magnetic field, the third Maxwell equation will automatically be satisfied. Plugging this into the first Maxwell equation leads to ∇ × ∂A ∂t + E = 0 (20) But any vector function with a vanishing curl can be expressed as the gradient of a scalar function, so we can write E + ∂A ∂t = −∇φ, or E = − ∂A ∂t − ∇φ (21) In summary we can put B = ∇ × A, E = −∇φ − ∂A ∂t (22) and then forget about the first and third Maxwell equations. This reduces the number of independent fields from 6 to 4. But the potentials are not unique: changing them by a gauge transformation A → A + ∇Λ, φ → φ − ∂Λ ∂t (23) with Λ any function of space and time, leaves the fields unchanged. It is sometimes useful to fix this ambiguity by specifying a condition on the potentials, such as ∇ · A = 0 (Coulomb gauge), reducing the number of independent fields to 3. The two remaining Maxwell equations become −0 ∂ 2A ∂t 2 + ∇ ∂φ ∂t = −∇2A + ∇(∇ · A) µ0 − J (24) −0 ∇ 2φ − ∇ · ∂A ∂t = ρ (25) 7 c 2010 by Charles Thorn
In Coulomb gauge there is a dramatic simplification: 102 20 A= -HoJ (26) -0720=p (27) which reveals that o solves a constraint equation,and A solves a wave equation. 2 Electrostatics For a particle moving in a potential,a static solution just has the particle sitting at rest at a relative minimum of the potential.But a static solution of Maxwell's equations is not so trivial:the fields won't depend on time,but they can have interesting dependence on the three spatial coordinates.If E,B are time independent the 4 Maxwell equations reduce to two independent pairs: -V×E=0, eoV·E=p (28) B 7×=J, 7.B=0 (29) 0 It follows,of course,that p and J are also independent of time and that V.J=0.Note that the presence of a current implies that charges are moving,but such that the charge density does not change:steady currents give rise to static magnetic fields.An absolutely static solution would have zero currents. Electrostatic problems involve solving the first pair of equations in different physical situations.The curl equation can be immediately solved in terms of the scalar potential by writing E=-Vo,with o solving Poisson's equations: -2o=p/Eo (30) If p=0,this is just the Laplace equation-V2=0,which has a large number of solutions. The simplest is the linear function o=-a.r which means a homogeneous electric field E a.Quadratic functions like ry,yz,xz,x2-y2,y2-22 also solve the equation.In fact one can find polynomial solutions of any order.Such solutions can always be added to any particular solution with p 0.If the charge distribution is localized we can fix this ambiguity by requiring the fields to vanish at infinity. 2.1 Point charge and the Dirac delta function Everyone knows that the potential for a point charge g sitting at the point a is 9 =4Tcolr -al (31) 8 ©2010 by Charles Thorn
In Coulomb gauge there is a dramatic simplification: ∇2 − 1 c 2 ∂ 2 ∂t 2 A = 1 c 2 ∇ ∂φ ∂t − µ0J (26) −0∇2φ = ρ (27) which reveals that φ solves a constraint equation, and A solves a wave equation. 2 Electrostatics For a particle moving in a potential, a static solution just has the particle sitting at rest at a relative minimum of the potential. But a static solution of Maxwell’s equations is not so trivial: the fields won’t depend on time, but they can have interesting dependence on the three spatial coordinates. If E, B are time independent the 4 Maxwell equations reduce to two independent pairs: −∇ × E = 0, 0∇ · E = ρ (28) ∇ × B µ0 = J, ∇ · B = 0 (29) It follows, of course, that ρ and J are also independent of time and that ∇ · J = 0. Note that the presence of a current implies that charges are moving, but such that the charge density does not change: steady currents give rise to static magnetic fields. An absolutely static solution would have zero currents. Electrostatic problems involve solving the first pair of equations in different physical situations. The curl equation can be immediately solved in terms of the scalar potential by writing E = −∇φ, with φ solving Poisson’s equations: −∇2φ = ρ/0 (30) If ρ = 0, this is just the Laplace equation −∇2φ = 0, which has a large number of solutions. The simplest is the linear function φ = −a · r which means a homogeneous electric field E = a. Quadratic functions like xy, yz, xz, x 2 − y 2 , y 2 − z 2 also solve the equation. In fact one can find polynomial solutions of any order. Such solutions can always be added to any particular solution with ρ 6= 0. If the charge distribution is localized we can fix this ambiguity by requiring the fields to vanish at infinity. 2.1 Point charge and the Dirac delta function Everyone knows that the potential for a point charge q sitting at the point a is φ = q 4π0|r − a| . (31) 8 c 2010 by Charles Thorn
It is easy to show that the Laplacian -V2 applied to this function when r is away from a is 0.This is consistent with Poisson's equation since the charge density of a point charge is zero everywhere except its location.However,if we integrate the point charge density over any region that encloses a,we should get the total charge g.Since the volume of that region can be arbitrarily small,the only way this could be is if p(a)=oo.Dirac introduced an "improper function"(r-a)to describe this situation.It satisfies: 6(r-a)=0,for all r≠a, dxd(r-a)=l,fora∈R (32) With this concept we can write the charge density of a point charge sitting at a as p go(r-a).Then for a "unit"point charge (by which we mean g=eo) -72、1 =(r-a) (33) 4πr-a so that Poisson's equation is formally satisfied. The potential for an arbitrary charge distribution is given by superposing the solutions for point charges: (r）= dx'、pr) 4πolr-r' (34) which makes sense as long as r2p→0asr→o. 2.2 Interfaces between different materials In materials Maxwell's equations assume the form ∂B 7.D=p, 7·B=0, 7xE=- Ot' V×H= D +J (35) D=0E+P→eE, H=1B-M→B (36) 0 L The second forms for D,II assume the material is homogeneous and isotropic.Applying Gauss's law (the first two equations)with a "pillbox"volume about a small area on the interface,one learns that n.B2=iB1,n·D2-i·D1=0 (37) where n is the normal to the interface,directed from material 1 to material 2;and o is the surface charge density at the interface.The conditions on tangential components are obtained using Stokes theorem: asa.(vxv)=pa.v (38) 9 ©2010 by Charles Thorn
It is easy to show that the Laplacian −∇2 applied to this function when r is away from a is 0. This is consistent with Poisson’s equation since the charge density of a point charge is zero everywhere except its location. However, if we integrate the point charge density over any region that encloses a, we should get the total charge q. Since the volume of that region can be arbitrarily small, the only way this could be is if ρ(a) = ∞. Dirac introduced an “improper function” δ(r − a) to describe this situation. It satisfies: δ(r − a) = 0, for all r 6= a, Z R d 3 xδ(r − a) = 1, for a ∈ R (32) With this concept we can write the charge density of a point charge sitting at a as ρ = qδ(r − a). Then for a “unit” point charge (by which we mean q = 0) −∇2 1 4π|r − a| = δ(r − a) (33) so that Poisson’s equation is formally satisfied. The potential for an arbitrary charge distribution is given by superposing the solutions for point charges: φ(r) = Z d 3x 0 ρ(r 0 ) 4π0|r − r 0 | (34) which makes sense as long as r 2ρ → 0 as r → ∞. 2.2 Interfaces between different materials In materials Maxwell’s equations assume the form ∇ · D = ρ, ∇ · B = 0, ∇ × E = − ∂B ∂t , ∇ × H = ∂D ∂t + J (35) D = 0E + P → E, H = 1 µ0 B − M → 1 µ B (36) The second forms for D, H assume the material is homogeneous and isotropic. Applying Gauss’s law (the first two equations) with a “pillbox” volume about a small area on the interface, one learns that nˆ · B2 = nˆ · B1, nˆ · D2 − nˆ · D1 = σ (37) where nˆ is the normal to the interface, directed from material 1 to material 2; and σ is the surface charge density at the interface. The conditions on tangential components are obtained using Stokes theorem: Z dSnˆ · (∇ × V ) = I dl · V (38) 9 c 2010 by Charles Thorn