Figure 1.1:Intersection of the averaging function of a point charge with a surface S,as the charge crosses S with velocity v:(a)at some time t=f,and (b)at t=12>n.The averaging function is represented by a sphere of radius a. Spatial averaging at timet eliminates currents associated with microscopic motions that are uncorrelated at the scale of he averaging radius (again,we do not consider the magnetic moments of particles).The assumption of a sufficiently large averaging radius leads to J(r,t)=p(r.t)v(r,t). (1.5) The total flux I(t)of current through a surface S is given by 1(t)=J(r.1).nds where f is the unit normal to S.Hence,using(4),we have d I0=∑9c.0:a)fr-ro》ds if fi stays approximately constant over the extent of the averaging function and S is not in motion.We see that the integral effectively intersects S with the averaging function sur- rounding each moving point charge (Figure 1.1).The time derivative of rif represents the velocity at which the averaging function is "carried across"the surface. Electric current takes a variety of forms,each described by the relation J=pv.Isolated charged particles(positive and negative)and charged insulated bodies moving through space comprise convection currents.Negatively-charged electrons moving through the positive background lattice within a conductor comprise a conduction current.Empirical sts that conduction currents are also described by the relation J=oE known as Ohm's law.A third type of current,called electrolytic current,results from the flow of positive or negative ions through a fluid. 1.3.2 Impressed vs.secondary sources In addition to the simple classification given above we may classify currents as primary or secondary,depending on the action that sets the charge in motion
Figure 1.1:Intersection of the averaging function of a point charge with a surface S, as the charge crosses S with velocity v:(a) at some time t = t1, and (b) at t = t2 > t1. The averaging function is represented by a sphere of radius a. Spatial averaging at time t eliminates currents associated with microscopic motions that are uncorrelated at the scale of the averaging radius (again, we do not consider the magnetic moments of particles). The assumption of a sufficiently large averaging radius leads to J(r, t) = ρ(r, t) v(r, t). (1.5) The total flux I(t) of current through a surface S is given by I(t) = S J(r, t) · nˆ d S where nˆ is the unit normal to S. Hence, using (4), we have I(t) = i qi d dt (ri(t) · nˆ) S f (r − ri(t)) d S if nˆ stays approximately constant over the extent of the averaging function and S is not in motion. We see that the integral effectively intersects S with the averaging function surrounding each moving point charge (Figure 1.1). The time derivative of r i ·nˆ represents the velocity at which the averaging function is “carried across” the surface. Electric current takes a variety of forms, each described by the relation J = ρv. Isolated charged particles (positive and negative) and charged insulated bodies moving through space comprise convection currents. Negatively-charged electrons moving through the positive background lattice within a conductor comprise a conduction current. Empirical evidence suggests that conduction currents are also described by the relation J = σE known as Ohm’s law. A third type of current, called electrolytic current, results from the flow of positive or negative ions through a fluid. 1.3.2 Impressed vs. secondary sources In addition to the simple classification given above we may classify currents as primary or secondary, depending on the action that sets the charge in motion.
It is helpful to separate primary or "impressed"sources,which are independent of the fields they source,from secondary sources which result from interactions between the sourced fields and the medium in which the fields exist.Most familiar is the conduc tion current set up in a conducting medium by an externally applied electric field.The impressed source concept is particularly important in circuit the ory,where independent voltage sources are modeled as providing primary voltage excitations that are indepen dent of applied load.In this way they differ from n the se that react to the effect produced by the application of primary source hmnmm In applied electro ay be so distant that return effects sed fields an be ignored.Othe at the i It of an antenna,the probe inser e and the ga power-line field in which a biological body is imme sed 1.3.3 Surface and line source densities Because they are spatially with so spatial dis raged effects it sour es and the fields the and lin e source de of actual,c describe scop sit The entity we is a continuous volume charge distributed f the e layer oratory ce. t withi ch point ea conta ds.w h)tthee d this quantity the surface p(r,w,t)=p(r,t)f(0.△）. where w is distance from S in the normal direction and A in some way parameterizes the "thickness"of the charge layer at r.The continuous density function f(x,A)satishies Cfx,△)dk=l and imf,△)=dx). For instance,we might have fa.A)= (1.6) △√元 With this definition the total charge contained in a cylinder normal to the surface at r and having cross-sectional area ds is dQ-p.(r.d5](w.8dw-p.ds. and the total charge contained within any cylinder oriented normal to S is 2()=p.(r.n)ds. (1.7) 2001 by CRC Press LLC
It is helpful to separate primary or “impressed” sources, which are independent of the fields they source, from secondary sources which result from interactions between the sourced fields and the medium in which the fields exist. Most familiar is the conduction current set up in a conducting medium by an externally applied electric field. The impressed source concept is particularly important in circuit theory, where independent voltage sources are modeled as providing primary voltage excitations that are independent of applied load. In this way they differ from the secondary or “dependent” sources that react to the effect produced by the application of primary sources. In applied electromagnetics the primary source may be so distant that return effects resulting from local interaction of its impressed fields can be ignored. Other examples of primary sources include the applied voltage at the input of an antenna, the current on a probe inserted into a waveguide, and the currents producing a power-line field in which a biological body is immersed. 1.3.3 Surface and line source densities Because they are spatially averaged effects, macroscopic sources and the fields they source cannot have true spatial discontinuities. However, it is often convenient to work with sources in one or two dimensions. Surface and line source densities are idealizations of actual, continuous macroscopic densities. The entity we describe as a surface charge is a continuous volume charge distributed in a thin layer across some surface S. If the thickness of the layer is small compared to laboratory dimensions, it is useful to assign to each point r on the surface a quantity describing the amount of charge contained within a cylinder oriented normal to the surface and having infinitesimal cross section d S. We call this quantity the surface charge density ρs(r, t), and write the volume charge density as ρ(r,w, t) = ρs(r, t) f (w, ), where w is distance from S in the normal direction and in some way parameterizes the “thickness” of the charge layer at r. The continuous density function f (x, ) satisfies ∞ −∞ f (x, ) dx = 1 and lim →0 f (x, ) = δ(x). For instance, we might have f (x, ) = e−x2/2 √π . (1.6) With this definition the total charge contained in a cylinder normal to the surface at r and having cross-sectional area d S is d Q(t) = ∞ −∞ [ρs(r, t) d S] f (w, ) dw = ρs(r, t) d S, and the total charge contained within any cylinder oriented normal to S is Q(t) = S ρs(r, t) d S. (1.7)
We may describe a line charge as a thin "tube"of volume charge distributed along some contour I.The amount of charge contained between two planes normal to the contour and separated by a distance dl is described by the line charge density pr(r.t). The volume charge density associated with the contour is then p(r.p.1)=p(r.1)fs(p.A). where p is the radial distance from the contour in the plane normal torand f(p.A)is a density function with the properties f,a2pd=l and im(p,△)=dp 2r0 For example,we might have ,4)=a n42 (1.8) Then the total charge contained between planes separated by a distance dl is and the total charge contained between planes placed at the ends of a contour r is 2(r)=pr(r.1)dl. (1.9) We may define srface and line currents similarly. A surface current is merely a tothe vicinity of aceS.density may be represented using a surface current density function J,(r.),defined at each point r on the surface so that J(r,w,t)=J(r,t)f(w.A). 山a、 Here f(w.△)is some at mal to S.The total current flowing through a strip dl)=U,c,)a(ed训f,△)do=J.c,)a)dl through a Jc,p,)=i(r)c,)f(p,△)
We may describe a line charge as a thin “tube” of volume charge distributed along some contour . The amount of charge contained between two planes normal to the contour and separated by a distance dl is described by the line charge density ρl(r, t). The volume charge density associated with the contour is then ρ(r,ρ, t) = ρl(r, t) fs(ρ, ), where ρ is the radial distance from the contour in the plane normal to and fs(ρ, ) is a density function with the properties ∞ 0 fs(ρ, )2πρ dρ = 1 and lim →0 fs(ρ, ) = δ(ρ) 2πρ . For example, we might have fs(ρ, ) = e−ρ2/2 π2 . (1.8) Then the total charge contained between planes separated by a distance dl is d Q(t) = ∞ 0 [ρl(r, t) dl] fs(ρ, )2πρ dρ = ρl(r, t) dl and the total charge contained between planes placed at the ends of a contour is Q(t) = ρl(r, t) dl. (1.9) We may define surface and line currents similarly. A surface current is merely a volume current confined to the vicinity of a surface S. The volume current density may be represented using a surface current density function Js(r, t), defined at each point r on the surface so that J(r,w, t) = Js(r, t) f (w, ). Here f (w, ) is some appropriate density function such as (1.6), and the surface current vector obeys nˆ · Js = 0 where nˆ is normal to S. The total current flowing through a strip of width dl arranged perpendicular to S at r is d I(t) = ∞ −∞ [Js(r, t) · nˆl(r) dl] f (w, ) dw = Js(r, t) · nˆl(r) dl where nˆl is normal to the strip at r (and thus also tangential to S at r). The total current passing through a strip intersecting with S along a contour is thus I(t) = Js(r, t) · nˆl(r) dl. We may describe a line current as a thin “tube” of volume current distributed about some contour and flowing parallel to it. The amount of current passing through a plane normal to the contour is described by the line current density Jl(r, t). The volume current density associated with the contour may be written as J(r,ρ, t) = uˆ(r)Jl(r, t) fs(ρ, ),
where a is a unit vector along r,p is the radial distance from the contour in the plane normal to r,and fr(p,A)is a density function such as (1.8).The total current passing through any plane normal to r at r is 1()=(r,)a(r)a(r)(p,A)2xpde=). It is often convenient to employ singular models for continuous source densities.For p nly in the th We zero,thu tnce,the p(x.y.z.t)=p(x.y.1)f(.A). As△→0 we have px,z,)=p,)mf,△)=p,c,)8. It issmpleiy parameterized in terms of constant values of coordinate vrb constant values of re mus e taken to represent the 8-function properly. For instance,the rge on the surface of a cone at =o may be described using the distance normal to this surface,which is given by r-r: p(,8,p,)=p,p,t)8(r[0-lD Using the property &(ax)=8(x)/a,we can also write this as p8,)=AG,4,i6- 1.3.4 Charge conservation of energy,mo harge e;they have In ase th are true laws of physics ever,in r ern pny und ave come represen more aw ciated wit. amental symm try o elv.ea etry Is assocl ate with a mple.energy cor can be s wn to ar e fom the ob with respect to time;the laws of physics do no depend rom the ob vation th ant under translation,while angular momentum ses from inv riance under rotation. The law of cons ervation of charge also arises from a symmetry principle.But instea of being spatial or temporal in character,it is related to the invariance of electrostatic potential.Experiments show that there is no absolute potential,only potential difference The laws of nature are invariant with respect to what we choose as the "reference 2001 by CRC Press LLC
where uˆ is a unit vector along , ρ is the radial distance from the contour in the plane normal to , and fs(ρ, ) is a density function such as (1.8). The total current passing through any plane normal to at r is I(t) = ∞ 0 [Jl(r, t)uˆ(r) · uˆ(r)] fs(ρ, )2πρ dρ = Jl(r, t). It is often convenient to employ singular models for continuous source densities. For instance, it is mathematically simpler to regard a surface charge as residing only in the surface S than to regard it as being distributed about the surface. Of course, the source is then discontinuous since it is zero everywhere outside the surface. We may obtain a representation of such a charge distribution by letting the thickness parameter in the density functions recede to zero, thus concentrating the source into a plane or a line. We describe the limit of the density function in terms of the δ-function. For instance, the volume charge distribution for a surface charge located about the xy-plane is ρ(x, y,z, t) = ρs(x, y, t) f (z, ). As → 0 we have ρ(x, y,z, t) = ρs(x, y, t) lim →0 f (z, ) = ρs(x, y, t)δ(z). It is a simple matter to represent singular source densities in this way as long as the surface or line is easily parameterized in terms of constant values of coordinate variables. However, care must be taken to represent the δ-function properly. For instance, the density of charge on the surface of a cone at θ = θ0 may be described using the distance normal to this surface, which is given by rθ − rθ0: ρ(r,θ,φ, t) = ρs(r,φ, t)δ (r[θ − θ0]). Using the property δ(ax) = δ(x)/a, we can also write this as ρ(r,θ,φ, t) = ρs(r,φ, t) δ(θ − θ0) r . 1.3.4 Charge conservation There are four fundamental conservation laws in physics:conservation of energy, momentum, angular momentum, and charge. These laws are said to be absolute; they have never been observed to fail. In that sense they are true empirical laws of physics. However, in modern physics the fundamental conservation laws have come to represent more than just observed facts. Each law is now associated with a fundamental symmetry of the universe; conversely, each known symmetry is associated with a conservation principle. For example, energy conservation can be shown to arise from the observation that the universe is symmetric with respect to time; the laws of physics do not depend on choice of time origin t = 0. Similarly, momentum conservation arises from the observation that the laws of physics are invariant under translation, while angular momentum conservation arises from invariance under rotation. The law of conservation of charge also arises from a symmetry principle. But instead of being spatial or temporal in character, it is related to the invariance of electrostatic potential. Experiments show that there is no absolute potential, only potential difference. The laws of nature are invariant with respect to what we choose as the “reference”
potential.This in turn is related to the invariance of Maxwell's equations under gauge transforms;the values of the electric and magnetic fields do not depend on which gauge transformation we use to relate the scalar potential to the vector potential A. We may state the conservation of charge as follows: The net charge in any closed system remains constant with time This does not mean that individual charges cannot be created or destroved.only that the total cha ge in any isolated system m t remain constant.Thus it is possible for a positron with charge e to annihilate an electron with charge -e without changing the net charge of the system.Only if a system is not closed c n its net charge be alt e moving charge constitutes e of the tion.To deriv this impc tant nsider a cl olds t ranspor (sce A.2). ains co onstant,and apply the Reyne The continuity equation.Consider a region of space occupied by a distribution of charge whose velocity is given by the vector field v.We surround a portion of charge by a surface S and let S deform as necessary to "follow"the charge as it moves.Since S always contains precisely the same charged particles,we have an isolated system for which the time rate of change of total charge must vanish.An expression for the time rate of change is given by the Reynolds transport theorem(A.66);we have2 The "D/D"notation indicates that the volume region V(t)moves with its enclosed particles.Since oy represents current density,we can write ap(r.dv+J().ds=0. (1.10) In this large- contin ange f of the charge den iyequatio ity for a xed spat At any time r rge de ty integrat over a volume is exactly compensated t exiting through the surrounding surface. We can obtain the continuity equation in point form by applying the divergence the- orem to the second term of (1.10)to get f「c+v.J,)w=0 v it Since V(r)is arbitrary we can set the integrand to zero to obtain ap(r.t) +VJc,)=0. (1.11) Note that Appendix A weuse the symbolutoreprese the velocity of a material and vto represe the velocity of a artificial surface 2001y CRC Press LLC
potential. This in turn is related to the invariance of Maxwell’s equations under gauge transforms; the values of the electric and magnetic fields do not depend on which gauge transformation we use to relate the scalar potential to the vector potential A. We may state the conservation of charge as follows: The net charge in any closed system remains constant with time. This does not mean that individual charges cannot be created or destroyed, only that the total charge in any isolated system must remain constant. Thus it is possible for a positron with charge e to annihilate an electron with charge −e without changing the net charge of the system. Only if a system is not closed can its net charge be altered; since moving charge constitutes current, we can say that the total charge within a system depends on the current passing through the surface enclosing the system. This is the essence of the continuity equation. To derive this important result we consider a closed system within which the charge remains constant, and apply the Reynolds transport theorem (see § A.2). The continuity equation. Consider a region of space occupied by a distribution of charge whose velocity is given by the vector field v. We surround a portion of charge by a surface S and let S deform as necessary to “follow” the charge as it moves. Since S always contains precisely the same charged particles, we have an isolated system for which the time rate of change of total charge must vanish. An expression for the time rate of change is given by the Reynolds transport theorem (A.66); we have2 DQ Dt = D Dt V(t) ρ dV = V(t) ∂ρ ∂t dV + S(t) ρv · dS = 0. The “D/Dt” notation indicates that the volume region V(t) moves with its enclosed particles. Since ρv represents current density, we can write V(t) ∂ρ(r, t) ∂t dV + S(t) J(r, t) · dS = 0. (1.10) In this large-scale form of the continuity equation, the partial derivative term describes the time rate of change of the charge density for a fixed spatial position r. At any time t, the time rate of change of charge density integrated over a volume is exactly compensated by the total current exiting through the surrounding surface. We can obtain the continuity equation in point form by applying the divergence theorem to the second term of (1.10) to get V(t) ∂ρ(r, t) ∂t +∇· J(r, t) dV = 0. Since V(t) is arbitrary we can set the integrand to zero to obtain ∂ρ(r, t) ∂t +∇· J(r, t) = 0. (1.11) 2Note that in Appendix A we use the symbol u to represent the velocity of a material and v to represent the velocity of an artificial surface.