Chapter 3 The static electromagnetic field 3.1 Static fields and steady currents Perhaps the most carefully studied area of electromagnetics is that in which the fields are time-invariant. This area, known generally as statics, offers(1)the most direct op- portunities for solution of the governing equations, and(2) the clearest physical pictures of the electromagnetic field. We therefore devote the present chapter to a treatment of static fields. We begin to seek and examine specific solutions to the field equations however, our selection of examples is shaped by a search for insight into the behavior of the field itself, rather than a desire to catalog the solutions of numerous statics problems We note at the outset that a static field is physically sensible only as a limiting case of a time-varying field as the latter approaches a time-invariant equilibrium, and then only in local regions. The static field equations we shall study thus represent an idealized model of the physical fields If we examine the Maxwell-Minkowski equations(2. 1)-(2. 4)and set the time deriva- tives to zero, we obtain the static field marwell equation V×E(r)=0, v·D(r)=p(r) V×H(r)=J(r), V·B(r)=0. We note that if the fields are to be everywhere time-invariant, then the sources j and p must also be everywhere time-invariant. Under this condition the dynamic coupling between the fields described by Maxwell's equations disappears; any connection between E, D, B, and H imposed by the time-varying nature of the field is gone. For static fields ve also require that any dynamic coupling between fields in the constitutive relations vanish. In this static field limit we cannot derive the divergence equations from the curl equations, since we can no longer use the initial condition argument that the fields were identically zero prior to some time The static field equations are useful for approximating many physical situations in which the fields rapidly settle to a local, macroscopically-static state. This may occur o rapidly and so completely that, in a practical sense, the static equations describe the fields within our ability to measure and to compute. Such is the case when a capacitor is rapidly charged using a battery in series with a resistor; for example, a 1 pF capacitor charging through a 1 22 resistor reaches 99. 99% of its total charge static limit within ②2001 by CRC Press LLC
Chapter 3 The static electromagnetic field 3.1 Static fields and steady currents Perhaps the most carefully studied area of electromagnetics is that in which the fields are time-invariant. This area, known generally as statics, offers (1)the most direct opportunities for solution of the governing equations, and (2)the clearest physical pictures of the electromagnetic field. We therefore devote the present chapter to a treatment of static fields. We begin to seek and examine specific solutions to the field equations; however, our selection of examples is shaped by a search for insight into the behavior of the field itself, rather than a desire to catalog the solutions of numerous statics problems. We note at the outset that a static field is physically sensible only as a limiting case of a time-varying field as the latter approaches a time-invariant equilibrium, and then only in local regions. The static field equations we shall study thus represent an idealized model of the physical fields. If we examine the Maxwell–Minkowski equations (2.1)–(2.4) and set the time derivatives to zero, we obtain the static field Maxwell equations ∇ × E(r) = 0, (3.1) ∇ · D(r) = ρ(r), (3.2) ∇ × H(r) = J(r), (3.3) ∇ · B(r) = 0. (3.4) We note that if the fields are to be everywhere time-invariant, then the sources J and ρ must also be everywhere time-invariant. Under this condition the dynamic coupling between the fields described by Maxwell’s equations disappears; any connection between E, D, B, and H imposed by the time-varying nature of the field is gone. For static fields we also require that any dynamic coupling between fields in the constitutive relations vanish. In this static field limit we cannot derive the divergence equations from the curl equations, since we can no longer use the initial condition argument that the fields were identically zero prior to some time. The static field equations are useful for approximating many physical situations in which the fields rapidly settle to a local, macroscopically-static state. This may occur so rapidly and so completely that, in a practical sense, the static equations describe the fields within our ability to measure and to compute. Such is the case when a capacitor is rapidly charged using a battery in series with a resistor; for example, a 1 pF capacitor charging through a 1 resistor reaches 99.99% of its total charge static limit within 10 ps
3.1.1 Decoupling of the electric and magnetic fields For the remainder of this chapter we shall assume that there is coupling between E and H or between D and B in the constitutive relations. Then the static equations decouple into two independent sets of equations in terms of two independent sets of fields The static electric field set(E, D)is described by the equations V×E(r)=0 v·D(r)=p(r) (36) Integrating these over a stationary contour and surface, respectively, we have the large- scale forms (37) D. ds The static magnetic field set(B, H)is described by V×H(r)=J(r) (3.9) V·B(r)=0, or, in large-scale form H·d=/J-ds, 0. We can also specialize the Maxwell-Boffi equations to static form. Assuming that the fields, sources, and equivalent sources are time-invariant, the electrostatic field e(r)is described by the point-form equations V×E=0, (313) V·E (314) or the equivalent large-scale equations (p-v. p)dv (3.16) Similarly, the magnetostatic field B is described by V×B=0(J+Ⅴ×M (317) (318) (320) ②2001 by CRC Press LLC
3.1.1 Decoupling of the electric and magnetic fields For the remainder of this chapter we shall assume that there is no coupling between E and H or between D and B in the constitutive relations. Then the static equations decouple into two independent sets of equations in terms of two independent sets of fields. The static electric field set (E,D)is described by the equations ∇ × E(r) = 0, (3.5) ∇ · D(r) = ρ(r). (3.6) Integrating these over a stationary contour and surface, respectively, we have the largescale forms E · dl = 0, (3.7) S D · dS = V ρ dV. (3.8) The static magnetic field set (B,H)is described by ∇ × H(r) = J(r), (3.9) ∇ · B(r) = 0, (3.10) or, in large-scale form, H · dl = S J · dS, (3.11) S B · dS = 0. (3.12) We can also specialize the Maxwell–Boffi equations to static form. Assuming that the fields, sources, and equivalent sources are time-invariant, the electrostatic field E(r) is described by the point-form equations ∇ × E = 0, (3.13) ∇ · E = 1 0 (ρ −∇· P), (3.14) or the equivalent large-scale equations E · dl = 0, (3.15) S E · dS = 1 0 V (ρ −∇· P) dV. (3.16) Similarly, the magnetostatic field B is described by ∇ × B = µ0 (J +∇× M), (3.17) ∇ · B = 0, (3.18) or B · dl = µ0 S (J +∇× M) · dS, (3.19) S B · dS = 0. (3.20)
E=0 Figure 3.1: Positive point charge in the vicinity of an insulated, uncharged conductor It is important to note that any separation of the electromagnetic field into independent static electric and magnetic portions is illusory. As we mentioned in$ 2.3.2, the electric and magnetic components of the EM field depend on the motion of the observer. An observer stationary with respect to a single charge measures only a static electric field, while an observer in uniform motion with respect to the charge measures both electric and magnetic fields 3.1.2 Static field equilibrium and conductors Suppose we could arrange a group of electric charges into a static configuration in free pace. The charges would produce an electric field, resulting in a force on the distribution via the lorentz force law, and hence would begin to move. Regardless of how we arrange e charges they cannot maintain their original static configuration without the help of some mechanical force to counterbalance the electrical force. This is a statement of Earnshaws theorem, discussed in detail in$ 3.4.2 The situation is similar for charges within and on electric conductors. A conductor is a material having many charges free to move under external influences, both electric and non-electric. In a metallic conductor, electrons move against a background lattice of positive charges. An uncharged conductor is neutral: the amount of negative charge carried by the electrons is equal to the positive charge in the background lattice. The distribution of charges in an uncharged conductor is such that the macroscopic electric field is zero inside and outside the conductor. When the conductor is exposed to an addi- tional electric field, the electrons move under the influence of the Lorentz force, creating a conduction current. Rather than accelerating indefinitely, conduction electrons experi- ence collisions with the lattice, thereby giving up their kinetic energy. Macroscopically, the charge motion can be described in terms of a time-average velocity, hence a macro- scopic current density can be assigned to the density of moving charge. The relationship between the applied, or " impressed, field and the resulting current density is given by Ohm's law; in a linear, isotropic, nondispersive material this is J(r,n)=σ(r)E(r,D). (321) The conductivity o describes the impediment to charge motion through the lattice: the ②2001 by CRC Press LLC
Figure 3.1: Positive point charge in the vicinity of an insulated, uncharged conductor. It is important to note that any separation of the electromagnetic field into independent static electric and magnetic portions is illusory. As we mentioned in § 2.3.2, the electric and magnetic components of the EM field depend on the motion of the observer. An observer stationary with respect to a single charge measures only a static electric field, while an observer in uniform motion with respect to the charge measures both electric and magnetic fields. 3.1.2 Static field equilibrium and conductors Suppose we could arrange a group of electric charges into a static configuration in free space. The charges would produce an electric field, resulting in a force on the distribution via the Lorentz force law, and hence would begin to move. Regardless of how we arrange the charges they cannot maintain their original static configuration without the help of some mechanical force to counterbalance the electrical force. This is a statement of Earnshaw’s theorem, discussed in detail in § 3.4.2. The situation is similar for charges within and on electric conductors. A conductor is a material having many charges free to move under external influences, both electric and non-electric. In a metallic conductor, electrons move against a background lattice of positive charges. An uncharged conductor is neutral: the amount of negative charge carried by the electrons is equal to the positive charge in the background lattice. The distribution of charges in an uncharged conductor is such that the macroscopic electric field is zero inside and outside the conductor. When the conductor is exposed to an additional electric field, the electrons move under the influence of the Lorentz force, creating a conduction current. Rather than accelerating indefinitely, conduction electrons experience collisions with the lattice, thereby giving up their kinetic energy. Macroscopically, the charge motion can be described in terms of a time-average velocity, hence a macroscopic current density can be assigned to the density of moving charge. The relationship between the applied, or “impressed,” field and the resulting current density is given by Ohm’s law; in a linear, isotropic, nondispersive material this is J(r, t) = σ(r)E(r, t). (3.21) The conductivity σ describes the impediment to charge motion through the lattice: the
Figure 3.2: Positive point charge near a grounded conductor higher the conductivity, the farther an electron may move on average before undergoing Let us examine how a state of equilibrium is established in a conductor. We shall con- sider several important situations. First, suppose we bring a positively charged particle into the vicinity of a neutral, insulated conductor(we say that a conductor is "insulated" if no means exists for depositing excess charge onto the conductor). The Lorentz force on the free electrons in the conductor results in their motion toward the particle(Figure 3.1). A reaction force F attracts the particle to the conductor. If the particle and the conductor are both held rigidly in space by an external mechanical force, the electrons within the conductor continue to move toward the surface. In a metal. when these elec- trons reach the surface and try to continue further they experience a rapid reversal in the direction of the Lorentz force, drawing them back toward the surface. A sufficiently large force(described by the work function of the metal) will be able to draw these charges from the surface, but anything less will permit the establishment of a stable equilibrium at the surface. If o is large then equilibrium is established quickly, and a nonuniform static charge distribution appears on the conductor surface. The electric field within the conductor must settle to zero at equilibrium, since a nonzero field would be associated with a current J= oE. In addition, the component of the field tangential to the surface must be zero or the charge would be forced to move along the surface. At equilibrium, the field within and tangential to a conductor must be zero. Note also that equilibriu cannot be established without external forces to hold the conductor and particle in place Next, suppose we bring a positively charged particle into the vicinity of a grounded (rather than insulated) conductor as in Figure 3. 2. Use of the term "grounded"means that the conductor is attached via a filamentary conductor to a remote reservoir of charge known as ground; in practical applications the earth acts as this charge reservoir. Charges are drawn from or returned to the reservoir, without requiring any work, in response to the Lorentz force on the charge within the conducting body. As the particle approaches negative charge is drawn to the body and then along the surface until a static equilibrium is re-established. Unlike the insulated body, the grounded conductor in equilibrium has excess negative charge, the amount of which depends on the proximity of the particle Again, both particle and conductor must be held in place by external mechanical forces and the total field produced by both the static charge on the conductor and the particle must be zero at points interior to the conductor. Finally, consider the process whereby excess charge placed inside a conducting body redistributes as equilibrium is established. We assume an isotropic, homogeneous con lucting body with permittivity e and conductivity o. An initially static charge with ②2001 by CRC Press LLC
Figure 3.2: Positive point charge near a grounded conductor. higher the conductivity, the farther an electron may move on average before undergoing a collision. Let us examine how a state of equilibrium is established in a conductor. We shall consider several important situations. First, suppose we bring a positively charged particle into the vicinity of a neutral, insulated conductor (we say that a conductor is “insulated” if no means exists for depositing excess charge onto the conductor). The Lorentz force on the free electrons in the conductor results in their motion toward the particle (Figure 3.1). A reaction force F attracts the particle to the conductor. If the particle and the conductor are both held rigidly in space by an external mechanical force, the electrons within the conductor continue to move toward the surface. In a metal, when these electrons reach the surface and try to continue further they experience a rapid reversal in the direction of the Lorentz force, drawing them back toward the surface. A sufficiently large force (described by the work function of the metal)will be able to draw these charges from the surface, but anything less will permit the establishment of a stable equilibrium at the surface. If σ is large then equilibrium is established quickly, and a nonuniform static charge distribution appears on the conductor surface. The electric field within the conductor must settle to zero at equilibrium, since a nonzero field would be associated with a current J = σE. In addition, the component of the field tangential to the surface must be zero or the charge would be forced to move along the surface. At equilibrium, the field within and tangential to a conductor must be zero. Note also that equilibrium cannot be established without external forces to hold the conductor and particle in place. Next, suppose we bring a positively charged particle into the vicinity of a grounded (rather than insulated)conductor as in Figure 3.2. Use of the term “grounded” means that the conductor is attached via a filamentary conductor to a remote reservoir of charge known as ground; in practical applications the earth acts as this charge reservoir. Charges are drawn from or returned to the reservoir, without requiring any work, in response to the Lorentz force on the charge within the conducting body. As the particle approaches, negative charge is drawn to the body and then along the surface until a static equilibrium is re-established. Unlike the insulated body, the grounded conductor in equilibrium has excess negative charge, the amount of which depends on the proximity of the particle. Again, both particle and conductor must be held in place by external mechanical forces, and the total field produced by both the static charge on the conductor and the particle must be zero at points interior to the conductor. Finally, consider the process whereby excess charge placed inside a conducting body redistributes as equilibrium is established. We assume an isotropic, homogeneous conducting body with permittivity and conductivity σ. An initially static charge with
density po(r) is introduced at time t=0. The charge density must obey the continuit V·J(r,t)= dp(r, t) since =gE. we have dp(r, t) aV·E(r,1)= By Gausss law, V.E can be eliminated dp(r, t) p(r, t) Solving this differential equation for the unknown p(r, t) we have p(r, t)=po(r)e-at/e (322) The charge density within a homogeneous, isotropic conducting body decreases exponen tially with time, regardless of the original charge distribution and shape of the body. o course, the total charge must be constant, and thus charge within the body travels to the surface where it distributes itself in such a way that the field internal to the body approaches zero at equilibrium. The rate at which the volume charge dissipates is deter mined by the relazation time e/o; for copper(a good conductor) this is an astonishingly small 10-19 s. Even distilled water, a relatively poor conductor, has E/o=10-6s.Thus we see how rapidly static equilibrium can be approached 3.1.3 Steady current Since time-invariant fields must arise from time-invariant sources, we have from the continuity equation V·J(r)=0. In large-scale form this is J·ds=0. (3.24) A current with the property(3.23)is said to be a steady current. By(3. 24), a steady current must be completely lineal (and infinite in extent)or must form closed loops. However, if a current forms loops then the individual moving charges must undergo acceleration(from the change in direction of velocity). Since a single accelerating particle radiates energy in the form of an electromagnetic wave, we might expect a large steady loop current to produce a great deal of radiation. In fact, if we superpose the fields produced by the many particles comprising a steady current, we find that a steady current produces no radiation 91. Remarkably, to obtain this result we must consider the exact relativistic fields, and thus our finding is precise within the limits of our macroscopic If we try to create a steady current in free space, the flowing charges will tend to disperse because of the Lorentz force from the field set up by the charges, and th resulting current will not form closed loops. a beam of electrons or ions will produce both an electric field(because of the nonzero net charge of the beam)and a magnetic field (because of the current). At nonrelativistic particle speeds, the electric field produces an outward force on the charges that is much greater than the inward (or pinch) force produced by the magnetic field. Application of an additional, external force will allow ②2001 by CRC Press LLC
density ρ0(r) is introduced at time t = 0. The charge density must obey the continuity equation ∇ · J(r, t) = −∂ρ(r, t) ∂t ; since J = σE, we have σ∇ · E(r, t) = −∂ρ(r, t) ∂t . By Gauss’s law, ∇ · E can be eliminated: σ ρ(r, t) = −∂ρ(r, t) ∂t . Solving this differential equation for the unknown ρ(r, t) we have ρ(r, t) = ρ0(r)e−σt/ . (3.22) The charge density within a homogeneous, isotropic conducting body decreases exponentially with time, regardless of the original charge distribution and shape of the body. Of course, the total charge must be constant, and thus charge within the body travels to the surface where it distributes itself in such a way that the field internal to the body approaches zero at equilibrium. The rate at which the volume charge dissipates is determined by the relaxation time /σ; for copper (a good conductor)this is an astonishingly small 10−19 s. Even distilled water, a relatively poor conductor, has /σ = 10−6 s. Thus we see how rapidly static equilibrium can be approached. 3.1.3 Steady current Since time-invariant fields must arise from time-invariant sources, we have from the continuity equation ∇ · J(r) = 0. (3.23) In large-scale form this is S J · dS = 0. (3.24) A current with the property (3.23)is said to be a steady current. By (3.24), a steady current must be completely lineal (and infinite in extent)or must form closed loops. However, if a current forms loops then the individual moving charges must undergo acceleration (from the change in direction of velocity). Since a single accelerating particle radiates energy in the form of an electromagnetic wave, we might expect a large steady loop current to produce a great deal of radiation. In fact, if we superpose the fields produced by the many particles comprising a steady current, we find that a steady current produces no radiation [91]. Remarkably, to obtain this result we must consider the exact relativistic fields, and thus our finding is precise within the limits of our macroscopic assumptions. If we try to create a steady current in free space, the flowing charges will tend to disperse because of the Lorentz force from the field set up by the charges, and the resulting current will not form closed loops. A beam of electrons or ions will produce both an electric field (because of the nonzero net charge of the beam)and a magnetic field (because of the current). At nonrelativistic particle speeds, the electric field produces an outward force on the charges that is much greater than the inward (or pinch)force produced by the magnetic field. Application of an additional, external force will allow
the creation of a collimated beam of charge, as occurs in an electron tube where a series of permanent magnets can be used to create a beam of steady current More typically, steady currents are created using wire conductors to guide the moving charge. When an external force, such as the electric field created by a battery, is applie to an uncharged conductor, the free electrons will begin to move through the positive lattice, forming a current. Each electron moves only a short distance before colliding with the positive lattice, and if the wire is bent into a loop the resulting macroscopic current will be steady in the sense that the temporally and spatially averaged microscopic current will obey V.J=0. We note from the examples above that any charges attempting to leave the surface of the wire are drawn back by the electrostatic force produced by th esulting imbalance in electrical charge. For conductors, the drift "velocity associated with the moving electrons is proportional to the applied field ud=-HeE where ue is the electron mobility. The mobility of copper (3.2 x 10-'m2/V. s) is such that an applied field of 1 V/m results in a drift velocity of only a third of a centimeter per second Integral properties of a steady current. Steady currents obey several useful inte- gral properties. To develop these properties we need an integral identity. Let f(r) and g(r)be scalar functions, continuous and with continuous derivatives in a volume region V. Let J represent a steady current field of finite extent, completely contained within V. We begin by using(B 42) to expand v·(fgJ=fg(V·J)+J.V(fg) Noting that V.J=0 and using(B 41), we get V·(fgJ)=(fJ)·Vg+(gJ)·Vf Now let us integrate over V and employ the divergence theorem fg)J·d (fJ)·Vg+(gJ)·VfdV. Since J is contained entirely within S, we must have f J=0 everywhere on S. Hence (fJ)·Vg+(gJ·Vfdv=0. We can obtain a useful relation by letting f =l and g=x; in(3. 25), where(x, y, 2)= Ji(rdV=0, (326) where Ji=Jx and so on. Hence the volume integral of any rectangular component of J is zero. Similarly, letting f=g=x; we find that Ji(r)dV=0 (327) With f = xi and g=xj we obtain Lx J; (r)+xj Ji (r)dv=0 ②2001 by CRC Press LLC
the creation of a collimated beam of charge, as occurs in an electron tube where a series of permanent magnets can be used to create a beam of steady current. More typically, steady currents are created using wire conductors to guide the moving charge. When an external force, such as the electric field created by a battery, is applied to an uncharged conductor, the free electrons will begin to move through the positive lattice, forming a current. Each electron moves only a short distance before colliding with the positive lattice, and if the wire is bent into a loop the resulting macroscopic current will be steady in the sense that the temporally and spatially averaged microscopic current will obey ∇ · J = 0. We note from the examples above that any charges attempting to leave the surface of the wire are drawn back by the electrostatic force produced by the resulting imbalance in electrical charge. For conductors, the “drift” velocity associated with the moving electrons is proportional to the applied field: ud = −µeE where µe is the electron mobility. The mobility of copper (3.2 × 10−3m2/V · s)is such that an applied field of 1 V/m results in a drift velocity of only a third of a centimeter per second. Integral properties of a steady current. Steady currents obey several useful integral properties. To develop these properties we need an integral identity. Let f (r) and g(r) be scalar functions, continuous and with continuous derivatives in a volume region V. Let J represent a steady current field of finite extent, completely contained within V. We begin by using (B.42)to expand ∇ · ( f gJ) = f g(∇ · J) + J · ∇( f g). Noting that ∇ · J = 0 and using (B.41), we get ∇ · ( f gJ) = ( f J) · ∇g + (gJ) · ∇ f. Now let us integrate over V and employ the divergence theorem: S ( f g)J · dS = V [( f J) · ∇g + (gJ) · ∇ f ] dV. Since J is contained entirely within S, we must have nˆ · J = 0 everywhere on S. Hence V [( f J) · ∇g + (gJ) · ∇ f ] dV = 0. (3.25) We can obtain a useful relation by letting f = 1 and g = xi in (3.25), where (x, y,z) = (x1, x2, x3). This gives V Ji(r) dV = 0, (3.26) where J1 = Jx and so on. Hence the volume integral of any rectangular component of J is zero. Similarly, letting f = g = xi we find that V xi Ji(r) dV = 0. (3.27) With f = xi and g = x j we obtain V xiJj(r) + x jJi(r) dV = 0. (3.28)
3.2 Electrostatics 3.2.1 The electrostatic potential and work The equation Ed=0 satisfied by the electrostatic field E(r)is particularly interesting. A field with zero circulation is said to be conservative. To see why, let us examine the work required to move a particle of charge Q around a closed path in the presence of e(r). Since work is the line integral of force and b=0, the work expended by the external system moving the charge against the lorentz for rce Is W=-∮(QE+QvxB)·d=-Q∮E·d=0 This property is analogous to the conservation property for a classical gravitational field: ly potential energy gained by raising a point mass is lost when the mass is lowered Direct experimental verification of the electrostatic conservative property is difficult aside from the fact that the motion of Q may alter e by interacting with the sources of E. By moving Q with nonuniform velocity (i.e, with acceleration at the beginning of the loop, direction changes in transit, and deceleration at the end) we observe a radiative loss of energy, and this energy cannot be regained by the mechanical system providing the motion. To avoid this problem we may assume that the charge is moved so slowly, or in such small increments, that it does not radiate. We shall use this concept later te determine the"assembly energy"in a charge distribution The electrostatic potential. By the point form of (3.29) V×E(r)=0 we can introduce a scalar field p= (r) such that E(r)=-VΦ(r) (330) The function p carries units of volts and is known as the electrostatic potential. Let us consider the work expended by an external agent in moving a charge between points Pl at ri and P, at r Q r):d=Q/d(r)=Q(r)-(r) The work W2l is clearly independent of the path taken between Pi and P2; the quantit d(r2) m)= (331) called the potential difference, has an obvious physical meaning as work per unit charge required to move a particle against an electric field between two point ②2001 by CRC Press LLC
3.2 Electrostatics 3.2.1 The electrostatic potential and work The equation E · dl = 0 (3.29) satisfied by the electrostatic field E(r) is particularly interesting. A field with zero circulation is said to be conservative. To see why, let us examine the work required to move a particle of charge Q around a closed path in the presence of E(r). Since work is the line integral of force and B = 0, the work expended by the external system moving the charge against the Lorentz force is W = − (QE + Qv × B) · dl = −Q E · dl = 0. This property is analogous to the conservation property for a classical gravitational field: any potential energy gained by raising a point mass is lost when the mass is lowered. Direct experimental verification of the electrostatic conservative property is difficult, aside from the fact that the motion of Q may alter E by interacting with the sources of E. By moving Q with nonuniform velocity (i.e., with acceleration at the beginning of the loop, direction changes in transit, and deceleration at the end)we observe a radiative loss of energy, and this energy cannot be regained by the mechanical system providing the motion. To avoid this problem we may assume that the charge is moved so slowly, or in such small increments, that it does not radiate. We shall use this concept later to determine the “assembly energy” in a charge distribution. The electrostatic potential. By the point form of (3.29), ∇ × E(r) = 0, we can introduce a scalar field = (r) such that E(r) = −∇(r). (3.30) The function carries units of volts and is known as the electrostatic potential. Let us consider the work expended by an external agent in moving a charge between points P1 at r1 and P2 at r2: W21 = −Q P2 P1 −∇(r) · dl = Q P2 P1 d(r) = Q [(r2) − (r1)] . The work W21 is clearly independent of the path taken between P1 and P2; the quantity V21 = W21 Q = (r2) − (r1) = − P2 P1 E · dl, (3.31) called the potential difference, has an obvious physical meaning as work per unit charge required to move a particle against an electric field between two points.
T1 Figure 3.3: Demonstration of path independence of the electric field line integral Of course, the large-scale form(3.29 )also implies the path-independence of work in the electrostatic field. Indeed, we may pass an arbitrary closed contour r through Pr and P2 and then split it into two pieces TI and T2 as shown in Figure 3.3. Since EdI=-Q/EdI+0/EdI=0. 1-r2 we hav eE.dI=-Q/E We sometimes refer to p(r) as the absolute electrostatic potential. Choosing a suitable reference point Po at location ro and writing the potential difference as V21=[(r2)-Φ(ro)-[Φ(ri)-中(ro)] we can justify calling p(r) the absolute potential referred to Po. Note that Po might describe a locus of points, rather than a single point, since many points can be at the same value of E found from( 3.30), for simplicity we often choose ro such that (ro)=0. potential. Although we can choose any reference point without changing the result Several properties of the electrostatic potential make lectric fields. We know that, at equilibrium, the electrostatic field within a conducting body must vanish. By(3. 30) the potential at all points within the body must therefore have the same constant value. It follows that the surface of a conductor is an equipotential surface: a surface for which p(r) is constant As an infinite reservoir of charge that can be tapped through a filamentary conductor the entity we call"ground"must also be an equipotential object. If we connect a con- ductor to ground, we have seen that charge may flow freely onto the conductor. Since no work is expended, "grounding a conductor obviously places the conductor at the same absolute potential as ground. For this reason, ground is often assigned the role as the potential reference with an absolute potential of zero volts. Later we shall see that for sources of finite extent ground must be located at infinit ②2001 by CRC Press LLC
Figure 3.3: Demonstration of path independence of the electric field line integral. Of course, the large-scale form (3.29)also implies the path-independence of work in the electrostatic field. Indeed, we may pass an arbitrary closed contour through P1 and P2 and then split it into two pieces 1 and 2 as shown in Figure 3.3. Since −Q 1−2 E · dl = −Q 1 E · dl + Q 2 E · dl = 0, we have −Q 1 E · dl = −Q 2 E · dl as desired. We sometimes refer to (r) as the absolute electrostatic potential. Choosing a suitable reference point P0 at location r0 and writing the potential difference as V21 = [(r2) − (r0)] − [(r1) − (r0)], we can justify calling (r) the absolute potential referred to P0. Note that P0 might describe a locus of points, rather than a single point, since many points can be at the same potential. Although we can choose any reference point without changing the resulting value of E found from (3.30), for simplicity we often choose r0 such that (r0) = 0. Several properties of the electrostatic potential make it convenient for describing static electric fields. We know that, at equilibrium, the electrostatic field within a conducting body must vanish. By (3.30)the potential at all points within the body must therefore have the same constant value. It follows that the surface of a conductor is an equipotential surface: a surface for which (r) is constant. As an infinite reservoir of charge that can be tapped through a filamentary conductor, the entity we call “ground” must also be an equipotential object. If we connect a conductor to ground, we have seen that charge may flow freely onto the conductor. Since no work is expended, “grounding” a conductor obviously places the conductor at the same absolute potential as ground. For this reason, ground is often assigned the role as the potential reference with an absolute potential of zero volts. Later we shall see that for sources of finite extent ground must be located at infinity.
3.2.2 Boundary conditions Boundary conditions for the electrostatic field. The boundary conditions found for the dynamic electric field remain valid in the electrostatic case. Thus n12×(E1-E2) (3.32) Here f12 points into region 1 from region 2. Because the static curl and divergence equations are independent, so are the boundary conditions(3. 32)and(3. 33) For a linear and isotropic dielectric where D=EE, equation(3. 33)becomes (∈E1-∈2E2)=p (334) Alternatively, using D=EoE+P we can write (3. 33)as (Ps pps +p (335) n. P is the polarization surface charge with f pointing outward from the material body We can also write the boundary conditions in terms of the electrostatic potential. with E=-Vq, equation(3. 32) becomes d1(r)=Φ2(r) for all points r on the surface. Actually p and p2 may differ by a constant; because this constant is eliminated when the gradient is taken to find E, it is generally ignored We can write (3. 35)as dd1a中2 =-Ps-pPsl-PPs2 where the normal derivative is taken in the f12 direction. For a linear, isotropic dielectric (337) Again, we note that (3.36)and (3. 37) are independent Boundary conditions for steady electric current. The boundary condition on the normal component of current found in 8 2.8.2 remains valid in the steady current case. Assume that the boundary exists between two linear, isotropic conducting regions having onstitutive parameters(E1, 01) and(E2, 2), respectively. By(2.198)we have n12·(J1-J2)=-Vx·J (338) where f12 points into region 1 from region 2. A surface current will not appear on the boundary between two regions having finite conductivity, although a surface charge may cumulate there during the transient period when the currents are established 31]. If is influenced to move from the surface, it will move into the adjacent regions, ②2001 by CRC Press LLC
3.2.2 Boundary conditions Boundary conditions for the electrostatic field. The boundary conditions found for the dynamic electric field remain valid in the electrostatic case. Thus nˆ 12 × (E1 − E2) = 0 (3.32) and nˆ 12 · (D1 − D2) = ρs. (3.33) Here nˆ 12 points into region 1 from region 2. Because the static curl and divergence equations are independent, so are the boundary conditions (3.32)and (3.33). For a linear and isotropic dielectric where D = E, equation (3.33)becomes nˆ 12 · ( 1E1 − 2E2) = ρs. (3.34) Alternatively, using D = 0E + P we can write (3.33)as nˆ 12 · (E1 − E2) = 1 0 (ρs + ρPs1 + ρPs2) (3.35) where ρPs = nˆ · P is the polarization surface charge with nˆ pointing outward from the material body. We can also write the boundary conditions in terms of the electrostatic potential. With E = −∇, equation (3.32)becomes 1(r) = 2(r) (3.36) for all points r on the surface. Actually 1 and 2 may differ by a constant; because this constant is eliminated when the gradient is taken to find E, it is generally ignored. We can write (3.35)as 0 ∂1 ∂n − ∂2 ∂n = −ρs − ρPs1 − ρPs2 where the normal derivative is taken in the nˆ 12 direction. For a linear, isotropic dielectric (3.33)becomes 1 ∂1 ∂n − 2 ∂2 ∂n = −ρs. (3.37) Again, we note that (3.36)and (3.37)are independent. Boundary conditions for steady electric current. The boundary condition on the normal component of current found in § 2.8.2 remains valid in the steady current case. Assume that the boundary exists between two linear, isotropic conducting regions having constitutive parameters ( 1,σ1) and ( 2,σ2), respectively. By (2.198) we have nˆ 12 · (J1 − J2) = −∇s · Js (3.38) where nˆ 12 points into region 1 from region 2. A surface current will not appear on the boundary between two regions having finite conductivity, although a surface charge may accumulate there during the transient period when the currents are established [31]. If charge is influenced to move from the surface, it will move into the adjacent regions
(02,E2) Figure 3.4: Refraction of steady current at a material interface ather than along the surface, and a new charge will replace it, supplied by the current for finite conducting regions (3. 38)becomes n12·(J1-J2)=0 (339) a boundary condition on the tangential component of current can also be found Substituting E=J/o into(3. 32) we have JI J2 = We can also write this as Ju J2e (3.40) he We may combine the boundary conditions for the normal components of current electric field to better understand the behavior of current at a material boundary stituting E=J/o into ( 3. 34)we hay EIJun-EJ2n=p (341) where Jin= f12 JI and J2n =112. J2. Combining (3.41)with(3. 39), we have E1n(∈1 where Unless E102-01E2=0, a surface charge will exist on the interface between dissimilar urrent-carrying conductor We may also combine the vector components of current on each side of the boundary to determine the effects of the boundary on current direction(Figure 3. 4). Let 81.2 denote en J1.2 and f12 so that Jin = Ji cos 01, Jir= Ju sin 81 J2= J2 sin g. ②2001 by CRC Press LLC
Figure 3.4: Refraction of steady current at a material interface. rather than along the surface, and a new charge will replace it, supplied by the current. Thus, for finite conducting regions (3.38)becomes nˆ 12 · (J1 − J2) = 0. (3.39) A boundary condition on the tangential component of current can also be found. Substituting E = J/σ into (3.32)we have nˆ 12 × J1 σ1 − J2 σ2 = 0. We can also write this as J1t σ1 = J2t σ2 (3.40) where J1t = nˆ 12 × J1, J2t = nˆ 12 × J2. We may combine the boundary conditions for the normal components of current and electric field to better understand the behavior of current at a material boundary. Substituting E = J/σ into (3.34)we have 1 σ1 J1n − 2 σ2 J2n = ρs (3.41) where J1n = nˆ 12 · J1 and J2n = nˆ 12 · J2. Combining (3.41)with (3.39), we have ρs = J1n 1 σ1 − 2 σ2 = E1n 1 − σ1 σ2 2 = J2n 1 σ1 − 2 σ2 = E2n 1 σ2 σ1 − 2 where E1n = nˆ 12 · E1, E2n = nˆ 12 · E2. Unless 1σ2 − σ1 2 = 0, a surface charge will exist on the interface between dissimilar current-carrying conductors. We may also combine the vector components of current on each side of the boundary to determine the effects of the boundary on current direction (Figure 3.4). Let θ1,2 denote the angle between J1,2 and nˆ 12 so that J1n = J1 cos θ1, J1t = J1 sin θ1 J2n = J2 cos θ2, J2t = J2 sin θ2
