This expression involves the time derivative of p with r fixed.We can also find an expression in terms of the material derivative by using the transport equation (A.67). Enforcing conservation of charge by setting that expression to zero,we have Dp(+()V.v(r.)-0. (1.12) Here Dp/Dt is the time rate of change of the charge density experienced by an observer moving with the current. We can state the large-scale form of the continuity equation in terms of a stationary volume.Integrating(1.11)over a stationary volume region V and using the divergence theorem.we find that fap(r.D dv-J(r.)-ds. Since V is not changing with time we have (1.13) Hence any increase of total charge within V must be produced by current entering V through S. As an example,suppose that in a bounded region of space p(r.t)=pore-8. We wish to find J and v,and to verify both versions of the continuity equation in point form.The spherical symmetry of p requires that J=.Application of(1.13)over a sphere of radius a gives Hence J= and therefore J=京2=prem The velocity is v==B好 and we have V.v =38/4.To verify the continuity equations,we compute the time derivatives 架=-ae =+ =-Bpore-+(EB月·(epoe-r) =-3Bpvre-#. 2001 by CRC Press LLC
This expression involves the time derivative of ρ with r fixed. We can also find an expression in terms of the material derivative by using the transport equation (A.67). Enforcing conservation of charge by setting that expression to zero, we have Dρ(r, t) Dt + ρ(r, t) ∇ · v(r, t) = 0. (1.12) Here Dρ/Dt is the time rate of change of the charge density experienced by an observer moving with the current. We can state the large-scale form of the continuity equation in terms of a stationary volume. Integrating (1.11) over a stationary volume region V and using the divergence theorem, we find that V ∂ρ(r, t) ∂t dV = − S J(r, t) · dS. Since V is not changing with time we have d Q(t) dt = d dt V ρ(r, t) dV = − S J(r, t) · dS. (1.13) Hence any increase of total charge within V must be produced by current entering V through S. Use of the continuity equation. As an example, suppose that in a bounded region of space we have ρ(r, t) = ρ0re−βt . We wish to find J and v, and to verify both versions of the continuity equation in point form. The spherical symmetry of ρ requires that J = rˆ Jr. Application of (1.13) over a sphere of radius a gives 4π d dt a 0 ρ0re−βt r 2 dr = −4π Jr(a)a2 . Hence J = rˆβρ0 r 2 4 e−βt and therefore ∇ · J = 1 r 2 ∂ ∂r (r 2 Jr) = βρ0re−βt . The velocity is v = J ρ = rˆβ r 4 , and we have ∇ · v = 3β/4. To verify the continuity equations, we compute the time derivatives ∂ρ ∂t = −βρ0re−βt , Dρ Dt = ∂ρ ∂t + v · ∇ρ = −βρ0re−βt + rˆβ r 4 · rˆρ0e−βt = −3 4 βρ0re−βt .
Note that the charge density decreases with time less rapidly for a moving observer than for a stationary one(3/4 as fast):the moving observer is following the charge outward, and p o r.Now we can check the continuity equations.First we see 器+p=-mre+e(）-0 as required for a moving observer;second we see 器+p小-+c=0 as required for a stationary observer. The continuity equation in fewer dimensions.The continuity equation can also be used to relate current and charge on a surface or along a line.By conservation of charge we can write 而，nc0ds=-fJ,c0m d (1.14) where ri is the vector normal to the curve r and tangential to the surface S.By the surface divergence theorem(B.20),the corresponding point form is 8pc.0+,J,c,0=0 (1.15) nce of the vector field J,.For instance,in rectangular coordinate have dx In cylindrical coordinates on the cylinder p=a,we would have g山-治+的 surface may be found in Tai [190],while many identities mav ily established by reference to Figure 1.2. exiting the surface during time At is given by △[1(w2,t)-1(41,1)1. Thus,the rate of net increase of charge within the system is -品∫.dl=-e-1a.1 (1.16) dt The corresponding point form is found by letting the length of the curve approach zero: a1.0+ap.2=0. (1.17) 1 where is arc length along the curve.As an example,suppose the line current on a circular loop antenna is approximately 10.0=6cos(g◆）osa
Note that the charge density decreases with time less rapidly for a moving observer than for a stationary one (3/4 as fast):the moving observer is following the charge outward, and ρ ∝ r. Now we can check the continuity equations. First we see Dρ Dt + ρ∇ · v = −3 4 βρ0re−βt + (ρ0re−βt ) 3 4 β = 0, as required for a moving observer; second we see ∂ρ ∂t +∇· J = −βρ0re−βt + βρ0e−βt = 0, as required for a stationary observer. The continuity equation in fewer dimensions. The continuity equation can also be used to relate current and charge on a surface or along a line. By conservation of charge we can write d dt S ρs(r, t) d S = − Js(r, t) · mˆ dl (1.14) where mˆ is the vector normal to the curve and tangential to the surface S. By the surface divergence theorem (B.20), the corresponding point form is ∂ρs(r, t) ∂t + ∇s · Js(r, t) = 0. (1.15) Here ∇s · Js is the surface divergence of the vector field Js. For instance, in rectangular coordinates in the z = 0 plane we have ∇s · Js = ∂ Jsx ∂x + ∂ Jsy ∂y . In cylindrical coordinates on the cylinder ρ = a, we would have ∇s · Js = 1 a ∂ Jsφ ∂φ + ∂ Jsz ∂z . A detailed description of vector operations on a surface may be found in Tai [190], while many identities may be found in Van Bladel [202]. The equation of continuity for a line is easily established by reference to Figure 1.2. Here the net charge exiting the surface during time t is given by t[I(u2, t) − I(u1, t)]. Thus, the rate of net increase of charge within the system is d Q(t) dt = d dt ρl(r, t) dl = −[I(u2, t) − I(u1, t)]. (1.16) The corresponding point form is found by letting the length of the curve approach zero: ∂ I(l, t) ∂l + ∂ρl(l, t) ∂t = 0, (1.17) where l is arc length along the curve. As an example, suppose the line current on a circular loop antenna is approximately I(φ, t) = I0 cos ωa c φ cos ωt,
Figure 1.2:Linear form of the continuityequation. light.We wish to find the 1.=6m(g）mo Thus g”=-hn()mm=-n0 al a1 Integrating with respect to time and ignoring any constant(static)charge,we have pu,n-sn(g）na Or pp,)=sin(p）sina Note that we could have used the chain rule ,0_81,0 aL 路ma-[-日 ity continuity equation (1.11)directly to surface and line dis ons written in op of the previous example,we Jr,t)=b8(p-a)8(2)1(p,t). Substitution into (1.11)then gives [s(0-ae1,1=-p2 at The divergence formula for cylindrical coordinates gives 5(p((=3p(r.) p8中 2001 by CRC Press LLC
Figure 1.2:Linear form of the continuityequation. where a is the radius of the loop, ω is the frequency of operation, and c is the speed of light. We wish to find the line charge density on the loop. Since l = aφ, we can write I(l, t) = I0 cos ωl c cos ωt. Thus ∂ I(l, t) ∂l = −I0 ω c sin ωl c cos ωt = −∂ρl(l, t) ∂t . Integrating with respect to time and ignoring any constant (static) charge, we have ρ(l, t) = I0 c sin ωl c sin ωt or ρ(φ, t) = I0 c sin ωa c φ sin ωt. Note that we could have used the chain rule ∂ I(φ, t) ∂l = ∂ I(φ, t) ∂φ ∂φ ∂l and ∂φ ∂l = ∂l ∂φ −1 = 1 a to calculate the spatial derivative. We can apply the volume density continuity equation (1.11) directly to surface and line distributions written in singular notation. For the loop of the previous example, we write the volume current density corresponding to the line current as J(r, t) = φˆ δ(ρ − a)δ(z)I(φ, t). Substitution into (1.11) then gives ∇ · [φˆ δ(ρ − a)δ(z)I(φ, t)] = −∂ρ(r, t) ∂t . The divergence formula for cylindrical coordinates gives δ(ρ − a)δ(z) ∂ I(φ, t) ρ∂φ = −∂ρ(r, t) ∂t
Next we substitute for/(中，t)to get -6egs血('p）sp-a)()o at=_prl o C Finally,integrating with respect to time and ignoring any constant term,we have (.1)(p-a)6(2)sin()sinot. where we have set p=a because of the presence of the factor (p-a). 1.3.5 Magnetic charge o m he ofte r打tan a single discretely charged particle from which the electric field diverges.In contrast. experiments show that magnetic fields are created only by currents or by time changing electric fields:hence,magnetic fields have moving electric charge as their source.The elemental source of magnetic field is the agnetic dipole.r senting a tiny loop of electric current(or a spinning electric pa article).The observation made in 1269 by Pierre De Maricourt,that e n the smallest】 agnet has two poles,still holds today. In a world filled with symmetry at the fundamental level,we find it hard to understand er aa y would be exhibited by a magnetic monopole.In 1931 Paul Dirac invigorated the search for magnetic monopoles by making the first strong theoretical argument for their existence. Dirac showed that the existence of magnetic monopoles would imply the quantization of electric charge,and would thus provide an explanation for one of the great puzzles vers in the of the universe. If magnetic monopoles are ever found to exist.there will be both positive and negativelv charged particles whose motions will constitute currents.We can define a macroscopic magnetic charge density m and current densityJ exactly as we did with electric charge and use conservation of magnetic charge to provide a continuity equation: V.J(+3P=(r.)=0. (1.18) With these new sources Maxwell's equations become appealingly symmetric.Despite uncertainties about the existence and physical nature of magnetic monopoles,magnetic charge and current have become an integral part of electromagnetic theory.We often use the concept of fictitious magnetic sources to make Maxwell's equations symmetric,and then derive various equivalence theorems for use in the solution of important problems. Thus we can put the idea of magnetic sources to use regardless of whether these sources actually exist
Next we substitute for I(φ, t) to get − I0 ρ ωa c sin ωa c φ δ(ρ − a)δ(z) cos ωt = −∂ρ(r, t) ∂t . Finally, integrating with respect to time and ignoring any constant term, we have ρ(r, t) = I0 c δ(ρ − a)δ(z)sin ωa c φ sin ωt, where we have set ρ = a because of the presence of the factor δ(ρ − a). 1.3.5 Magnetic charge We take for granted that electric fields are produced by electric charges, whether stationary or in motion. The smallest element of electric charge is the electric monopole: a single discretely charged particle from which the electric field diverges. In contrast, experiments show that magnetic fields are created only by currents or by time changing electric fields; hence, magnetic fields have moving electric charge as their source. The elemental source of magnetic field is the magnetic dipole, representing a tiny loop of electric current (or a spinning electric particle). The observation made in 1269 by Pierre De Maricourt, that even the smallest magnet has two poles, still holds today. In a world filled with symmetry at the fundamental level, we find it hard to understand why there should not be a source from which the magnetic field diverges. We would call such a source magnetic charge, and the most fundamental quantity of magnetic charge would be exhibited by a magnetic monopole. In 1931 Paul Dirac invigorated the search for magnetic monopoles by making the first strong theoretical argument for their existence. Dirac showed that the existence of magnetic monopoles would imply the quantization of electric charge, and would thus provide an explanation for one of the great puzzles of science. Since that time magnetic monopoles have become important players in the “Grand Unified Theories” of modern physics, and in cosmological theories of the origin of the universe. If magnetic monopoles are ever found to exist, there will be both positive and negatively charged particles whose motions will constitute currents. We can define a macroscopic magnetic charge density ρm and current density Jm exactly as we did with electric charge, and use conservation of magnetic charge to provide a continuity equation: ∇ · Jm(r, t) + ∂ρm(r, t) ∂t = 0. (1.18) With these new sources Maxwell’s equations become appealingly symmetric. Despite uncertainties about the existence and physical nature of magnetic monopoles, magnetic charge and current have become an integral part of electromagnetic theory. We often use the concept of fictitious magnetic sources to make Maxwell’s equations symmetric, and then derive various equivalence theorems for use in the solution of important problems. Thus we can put the idea of magnetic sources to use regardless of whether these sources actually exist
1.4 Problems 1.2 Repeat Problem 1.1 for a charged half plane=0. 1.3 Write the volume charge density for a singular surface charge located on the cylin- der p=po,entirely in terms of cylindrical coordinates.Find the total charge on the cylinder. 1.4 Repeat Problem 1.3 for a charged half plane=. 2001 by CRC Press LLC
1.4 Problems 1.1 Write the volume charge density for a singular surface charge located on the sphere r = r0, entirely in terms of spherical coordinates. Find the total charge on the sphere. 1.2 Repeat Problem 1.1 for a charged half plane φ = φ0. 1.3 Write the volume charge density for a singular surface charge located on the cylinder ρ = ρ0, entirely in terms of cylindrical coordinates. Find the total charge on the cylinder. 1.4 Repeat Problem 1.3 for a charged half plane φ = φ0