NOTES AND DISCUSSIONS Note on group velocity and energy propagation Abraham Bers Department of Electrical Engineering Computer Science and Plasma Science Fusion Center, (Received 12 January 1999; accepted 10 June 1999) A general proof is given for the equality between group velocity and energy velocity for linear wave propagation in a homogeneous medium with arbitrary spatial and temporal dispersion. C 2000 American Association of Physics Teachers. t In the series of physics"Questions"in the American function. The Fourier-Laplace transform of this convolution urnal of Physics, K. M. Awati and T. Howes ask for a relationship is expressed by the conductivity tensor function general proof of the fact that in linear dispersive wave propa- of wave vector k and frequency w as gation, energy propagates at the group velocity. Several an- swers to this question have appeared recently, but a general J(k, o)=o(k, o).E(k,o) nondissipative medium, such a general proof is indeed The other linear response functions have similar interpreta- media. It can be readily argued that this type of proof can be =o(k, o)(-iwEo): K(k,)=I+X(k, o); E(k, w) carried out for any other(than electrodynamic)description of Eok(k, o). The Fourier-Laplace transform of Maxwells dynamics of a)medium. In an electrodynamic formulation, equations for the self-consistent electromagnetic fields are a stable, nondissipative, linear (or, more generally, linearized favor of the electromagnetic fields; one can easily visualize kXE=OLoH (2) doing the opposite. For exposing the simplest electrod- namic proof, I will focus on a nonmagnetic medium uoH) which is arbitrarily (spatially and temporally k×=-0e0k(k,ω)·E spersive-a plasma. The generalization to gnetic media is straightforward Taking kx(2)and using(3) to eliminate H, one find he continuum electrodynamics of a plasma-like mediu omogeneous set of equations for E Is described by Maxwells equations for the electromagnetic fields e andh. wherein the collective " mechanic D(ko).E=0 namics of the medium as a function of E and H are ex- where the dispersion tensor D is pressed by electric current and electric charge densities (, P). The latter, expressed as functions of E and H, are D(k, o)=kk-k21+-K(k,o) what one can call the electrodynamic response functions of the medium. For a plasma-like medium, since B=HoH, For nontrivial solutions of (4), Faradays equation provides a way of eliminating H in favor det[ d(k,o)]=D(k, w)=0 (6) of E, so that the response functions are only functions of E In addition, since J and p are related by the continuity equa- which is the dispersion relation giving, e.g., o(k). These are tion, the mechanical dynamics, regardless of the particular the natural modes of the system, with fields whose space model chosen for describing the dynamics, can be expressed time dependence exp[i(k-r-of)] is constrained by the disper by a single"electrical"response(or"influence")function, sion relation(6). Natural modes that are purely propagating e.g., the conductivity tensor o, or the susceptibility tensor x, waves are those for which solutions of (6)entail real k=k or the permittivity tensor K, or the dielectric tensor E-all of and real o=@r, i.e., o(k). The group velocity for such these being related to each other waves is given by In a homogeneous medium of the type we are considering the most general linear response function is one which ex presses both spatial and temporal dispersion through a con- volution integral in both space and time. Thus, for example, the current density J(, 1), at a point location F and time t, In a dissipation-free medium, the permittivity tensor is Her- depends upon the time-history and location-neighborhood mitian for real k and real a, K(kr, w)=Kh. In a linearly (consistent with causality and relativity)of the electric field stable and dissipation-free medium, the direction of signal E(r, n through the space-time conductivity tensor influence propagation is given by the direction of u Am J Phys. 68(5), May 2000 c 2000 American Association of Physics Teachers
NOTES AND DISCUSSIONS Note on group velocity and energy propagation Abraham Bers Department of Electrical Engineering & Computer Science and Plasma Science & Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ~Received 12 January 1999; accepted 10 June 1999! A general proof is given for the equality between group velocity and energy velocity for linear wave propagation in a homogeneous medium with arbitrary spatial and temporal dispersion. © 2000 American Association of Physics Teachers. In the series of physics ‘‘Questions’’ in the American Journal of Physics, K. M. Awati and T. Howes ask for a general proof of the fact that in linear dispersive wave propagation, energy propagates at the group velocity.1 Several answers to this question have appeared recently,2 but a general proof has not been provided by any of them. For a stable and nondissipative3 medium, such a general proof is indeed available from classical, continuum electrodynamics of media.4 It can be readily argued that this type of proof can be carried out for any other ~than electrodynamic! description of a stable, nondissipative, linear ~or, more generally, linearized dynamics of a! medium. In an electrodynamic formulation, one chooses to eliminate the ‘‘mechanical’’ field variables in favor of the electromagnetic fields; one can easily visualize doing the opposite. For exposing the simplest electrodynamic proof, I will focus on a nonmagnetic medium (BW 5m0HW ) which is arbitrarily ~spatially and temporally! dispersive—a plasma. The generalization to linear waves in magnetic media is straightforward. The continuum electrodynamics of a plasma-like medium is described by Maxwell’s equations for the electromagnetic fields EW and HW , wherein the collective ‘‘mechanical’’ dynamics of the medium as a function of EW and HW are expressed by electric current and electric charge densities (JW,r). The latter, expressed as functions of EW and HW , are what one can call the electrodynamic response functions of the medium. For a plasma-like medium, since BW 5m0HW , Faraday’s equation provides a way of eliminating HW in favor of EW , so that the response functions are only functions of EW . In addition, since JW and r are related by the continuity equation, the mechanical dynamics, regardless of the particular model chosen for describing the dynamics, can be expressed by a single ‘‘electrical’’ response ~or ‘‘influence’’! function, e.g., the conductivity tensor sJ, or the susceptibility tensor Jx, or the permittivity tensor KJ, or the dielectric tensor Je—all of these being related to each other. In a homogeneous medium of the type we are considering, the most general linear response function is one which expresses both spatial and temporal dispersion through a convolution integral in both space and time. Thus, for example, the current density JW(rW,t), at a point location rW and time t, depends upon the time-history and location-neighborhood ~consistent with causality and relativity! of the electric field EW (rW,t) through the space–time conductivity tensor influence function. The Fourier–Laplace transform of this convolution relationship is expressed by the conductivity tensor function of wave vector kW and frequency v as JW~kW,v!5sJ~kW,v!•EW ~kW,v!. ~1! The other linear response functions have similar interpretations, and are simply related to each other: Jx(kW,v) 5sJ(kW,v)/(2ive 0); KJ(kW,v)5JI 1Jx(kW,v); Je (kW,v) 5e 0KJ(kW,v). The Fourier–Laplace transform of Maxwell’s equations for the self-consistent electromagnetic fields are then kW 3EW 5vm0HW ~2! and kW 3HW 52ve 0KJ~kW,v!•EW . ~3! Taking kW 3 ~2! and using ~3! to eliminate HW , one finds the homogeneous set of equations for EW : DJ ~kW,v!•EW 50, ~4! where the dispersion tensor DJ is DJ ~kW,v!5kW kW 2k2JI 1 v2 c2 J K~kW,v!. ~5! For nontrivial solutions of ~4!, det@DJ ~kW,v!#[D~kW,v!50, ~6! which is the dispersion relation giving, e.g., v(kW). These are the natural modes of the system, with fields whose space– time dependence exp@i(kW•rW2vt)# is constrained by the dispersion relation ~6!. Natural modes that are purely propagating waves are those for which solutions of ~6! entail real kW 5kWr and real v5vr , i.e., vr(kWr). The group velocity for such waves is given by vW g5]vr ]kWr . ~7! In a dissipation-free medium, the permittivity tensor is Hermitian for real kW and real v, KJ(kWr ,vr)5KJh . In a linearly stable and dissipation-free medium, the direction of signal propagation is given by the direction of vW g . 5 482 Am. J. Phys. 68 ~5!, May 2000 © 2000 American Association of Physics Teachers 482
In order to determine the velocity with which energy is transported, one needs to first determine the appropriate for (18) mulation of energy and energy flow in a space-time disper sive medium. For the purely propagating wave modes in a This proves the equality of the group velocity and energy plasma-like medium, one can show that the average (in space velocity, Ug=ve, for purely propagating waves in a linear time)energy density is given by generally dispersive, and nondissipative"electric" medium =B+5E.,人,E like a plasma. Note that, in the above proof, no specific (8) model of the linear, loss-free, dispersive dynamics had to be where the first term is clearly the average magnetic energy sive dynamics of a loss-free medium Two remarks are in order. First, in relation to the assump. the electric field and in all of the collective ""mechanical" tion of a nondissipative medium, the Kramers-Kronig rela- fields. One can also show that the average energy flow den- tions for a dispersive medium require that the permittivity sity is given by tensor, K(kr, ar), have both a Hermitian and an anti- Hermitian part. The relative magnitudes of these parts can (=ReEx日 ) o, E ak E however, vary from region to region in(k,, o)space (9) Weakly damped waves [lo (k,)0], the group velocity direction for Po.(-dDh/akr)o U (dDb1o)a、(k (14) purely propagating wave modes having w(k,)=wr(k,),for some other kr, may not be the same as the direction of signal In addition, consider the variation of(11) and(12)with re pect to kr and K M. Awati and T Howes, Am J Phys, 64(11), 1353(1996) (Sk,)XE+k,x(SE)=(So,)uoH+o, lo( SH),(15) K T. MeDonald, C S Helrich, R.J. Mathar, S. Wong, and D Styer, Am (ok,)×H+k×(6H=-E0Kb)·E-o∈0Kh Well-defined wave propagation in a dispersive medium is understood to take place in a regime of weak dissipation where the modes are character- (16) these regimes that the concepts of group velocity, time, or space-averaged energies and their flows, and thus energy velocity, are well-defined. In thus Dot-multiplying(15) by H,(16) by -E, the complex considering these concepts, one can idealize the situation by considering conjugate of (I )by-(8H), the complex conjugate of( 2) Coe medium to be"nondissiraiNe s a. 30 (len for tempore as we do in the following variant electrodynamic ge wave energy and by(SE), and adding these equations, one obtains mentum and their flows in linear med sive media by S. M. Rytov, JETP 17, 930(1947), and ger (17) spatially and temporally dispersive media by M. E. Gertsenshte 680(1954). Independently, a simpler formulation of average from which one immediately finds and energy flow in linear media with spatial and temporal dispersion was 483 Am J Phys., VoL 68, No 5, May 2000 Notes and discussions
In order to determine the velocity with which energy is transported, one needs to first determine the appropriate formulation of energy and energy flow in a space–time dispersive medium. For the purely propagating wave modes in a plasma-like medium, one can show that the average ~in space or time! energy density is given by6 ^w&5 m0 4 uHW u 21 e 0 4 EW *• ]~vrKJh! ]vr •EW , ~8! where the first term is clearly the average magnetic energy density and the second term is the average energy density in the electric field and in all of the collective ‘‘mechanical’’ fields. One can also show that the average energy flow density is given by6 ^sW&5ReS 1 2 EW 3HW *D 2 e 0 4 vrEW *• ]KJh ]kWr •EW , ~9! where the first term is the average electromagnetic ~Poynting! energy flow density and the second term is the average collective ‘‘mechanical’’ energy flow density. Using ~8! and ~9!, one can define an energy flow velocity for a natural wave vr(kWr): vW e5 sW k wk , ~10! where sW k5^sW&uvr(k W r) and wk5^w&uvr(k W r) are the average wave energy flow density and wave energy density, respectively. The proof that ~7! and ~10! are equal to each other proceeds as follows. Consider Maxwell’s equations ~2! and ~3!, for kW 5kWr , v5vr , and KJ(kW,v)5KJh(kWr ,vr), kWr3EW 5vrm0HW , ~11! kWr3HW 52vre 0KJh~kWr ,vr!•EW . ~12! As followed from ~2! and ~3!, these entail the dispersion relation detF kWrkWr2kr 2JI 1 vr 2 c2 J Kh~kWr ,vr!G [Dh~kWr ,vr!50, ~13! giving vr(kWr), and thence the group velocity, vW g5]vr ]kr 5 ~2]Dh /]kWr!vr~k W r! ~]Dh /]vr!vr~k W r! . ~14! In addition, consider the variation of ~11! and ~12! with respect to kWr and vr : ~dkWr!3EW 1kWr3~dEW !5~dvr!m0HW 1vrm0~dHW !, ~15! ~dkWr!3HW 1kWr3~dHW !52e 0d~vrKJh!•EW 2vre 0KJh •~dEW !. ~16! Dot-multiplying ~15! by HW *, ~16! by 2EW *, the complex conjugate of ~11! by 2(dHW ), the complex conjugate of ~12! by (dEW ), and adding these equations, one obtains ~dkWr!•^sW&5~dvr!^w& ~17! from which one immediately finds ]vr ]kWr 5 sW k wk . ~18! This proves the equality of the group velocity and energy velocity, vW g5vW e , for purely propagating waves in a linear, generally dispersive, and nondissipative ‘‘electric’’ medium, like a plasma. Note that, in the above proof, no specific model of the linear, loss-free, dispersive dynamics had to be specified; the result ~18! is thus valid for any linear, dispersive dynamics of a loss-free medium. Two remarks are in order. First, in relation to the assumption of a nondissipative medium, the Kramers–Kro¨nig relations for a dispersive medium require that the permittivity tensor, KJ(kWr ,vr), have both a Hermitian and an antiHermitian part. The relative magnitudes of these parts can, however, vary from region to region in (kWr ,vr) space. Weakly damped waves @uvi(kr)u!uvr(kr)u#, for which ~18! holds, exist in regions of (kWr ,vr) where the anti-Hermitian part of KJ is small compared to its Hermitian part so that vr(kWr) is essentially determined by ~13!. Second, group velocity ~as its name is intended to remind us! applies to the velocity of a group of waves—a wave packet—and by ~18! this must also be true for energy velocity in a loss-free, dispersive medium.7 Allowing for the wave fields @exp(ikWr•rW 2ivrt)# to have amplitudes that vary slowly in space and time ~slowly compared to the fast scales of, respectively, kWr and vr!, their velocity is also found to be given by ~14!. In addition, averaged ~on the fast scales of either kWr or vr! energy and energy flow densities are again found to be given by ~8! and ~9!, respectively, and one can show that ~17! and ~18! also hold for such wave fields with slow space–time amplitude modulations. A detailed proof of the above, including the account of weak dissipation, is given by Bers ~Ref. 8!. Several concluding remarks are also in order. The above derivation carries through for a weakly inhomogeneous and/or weakly time-varying medium as long as geometrical optics is applicable to describe the wave propagation.8 The proof of ~18! can also be carried out for a weakly dissipative or weakly unstable medium.8 However, in a linearly unstable medium @i.e., in which for some kWr , v(kWr)5vr(kWr) 1ivi(kWr) has vi(kWr).0#, the group velocity direction for purely propagating wave modes having v(kWr)5vr(kWr), for some other kWr , may not be the same as the direction of signal propagation.9 1 K. M. Awati and T. Howes, Am. J. Phys. 64 ~11!, 1353 ~1996!. 2 K. T. McDonald, C. S. Helrich, R. J. Mathar, S. Wong, and D. Styer, Am. J. Phys. 66 ~8!, 656–661 ~1998!. 3 Well-defined wave propagation in a dispersive medium is understood to take place in a regime of weak dissipation where the modes are characterized by frequencies and wave numbers that are essentially real. It is only in these regimes that the concepts of group velocity, time-, or space-averaged energies and their flows, and thus energy velocity, are well-defined. In thus considering these concepts, one can idealize the situation by considering the medium to be ‘‘nondissipative’’ as we do in the following. 4 Covariant electrodynamic formulations of average wave energy and momentum and their flows in linear media were given for temporally dispersive media by S. M. Rytov, JETP 17, 930 ~1947!, and generalized to spatially and temporally dispersive media by M. E. Gertsenshtein, ibid. 26, 680 ~1954!. Independently, a simpler formulation of average wave energy and energy flow in linear media with spatial and temporal dispersion was 483 Am. J. Phys., Vol. 68, No. 5, May 2000 Notes and Discussions 483
given by A. Bers in 1962; see w.P. Allis, S.J. Buchsbaum, and A Bers regions w(k,)=o(kr)+iw (k,), and o (kr)I is not small compared to Waves in Anisotropic Plasmas(MIT, Cambridge, MA, 1963), Sec. 8.5. A nown in this last reference, the equality between group velocity and en G).(kr)I. Then, calculating x from w,(k,) can give values for Uxthat ergy velocity follows simply from a variational form of Maxwell's equa exceed the speed of light and hence, in such regions, the group velocity no tions for an arbitrarily dispersive medium; this general proof is the basis of longer represents the velocity of energy flow, energy cannot travel at such speeds. (Academic, New York, 1960), Chap /opagation and Group Velocity A. Bers, "Linear Waves and Instabilities, "in Plasma Physics-Les See, for example, L. Brillouin, Wane Pre Houches 1972, edited by C. DeWitt and J. Peyraud( Gordon and Breach, 6For a loss-free, isotropic, and only temporally dispersive medium [Kh New York, London, Paris, 1975), Secs. Il and Vll; I B. Bernstein, "Geo- IK(or)and real]. (8)was first identified by L. Brillouin in 1932; se metric Optics in Space- and Time-Varying Plasmas, Phys. Fluids 18, 320 Ref. 5 above, Chap. IV. For a loss-free, anisotropic medium with spat (1975) nd temporal dispersion(K, =kg(k,, or),(8)and(9)were derived inde- A. Bers, ""Space-Time Evolution of Plasma Instabilities-Absolute and Convective in Handbook of Plasma Physics, general editors, M. N in Ref. 8 below Rosenbluth and R. Z. Sagdeev, Vol. I Basic Plasma Physics, volume edi- Note that for propagation in regions of strong absorption, it is well-known tors, A. A. Galeev and R N. Sudan(North-Holland, Amsterdam, 1983), that this equality breaks down, see Chap. V in Ref. 5 above. In such Chap.3.2,Sec.3.2.3 Analysis of doubly excited symmetric ladder networks V.V. Bapeswara Rao Department of Electrical Engineering, North Dakota State University, Fargo, North Dakota 58105 (Received 29 March 1999; accepted 14 June 1999) A simple procedure to determine the effective resistance between the center and a vertex of an n-sided polygon made of resistors is presented. C 2000 American Association of Physies Teachers In a recent paper Sidhu presented a procedure to deter- The resistance R2N+1 for any N, may be computed by nine the effective resistance between the center and a vertex considering appropriate ladder network. It should have(2N of an n-sided polygon made of resistors. The resistance con- +1) resistances of value 2G 0, 2N resistances of value 1n nected between the center and a vertex of the polygon is I n, and two 2 n resistances (one at each end). Note that the computation of successive voltages and currents is a recur sive vertices is 2G n2. In this note, we present a simple sive procedure. We now consider the determination of r2N alternative procedure that is easier to comprehend. The units The network used for the determination of R2N is derived for current(ampere)and voltage(volt)are not specified with from the network used for the determination of R2N+1.The respect to each variable in the text below. "'mid-section'' is modified as follows: Each of the two resis- Consider the ladder network- shown in Fig. 1. We let El tances from the vertex to the center is changed to 2 n2.The =E2=. From considerations of symmetry, 10=0. Further, resistance between the vertices is set to zero. The network 1=12,13=14,ls=l6,l7=l8,l9=1 and 111=11 for R4 is obtained from the circuit shown in Fig. 1. The propose to determine E for which 11=1. Under such a con- element values are as shown in Fig3 dition, VcN= I and 13=10+1=1 We analyze the circuit in Fig. 3 as earlier. We note that lo Further is zero. We propose to determine E for which 11=1. Under VBN=VBC+ VCN=(2G+1) such a condition, VCN=2. Then 13=I0 VBN=VBC+VCN=(2G+2) 17=l3+ls=(2G+2) AN=E=1(2G)+VBN=(4G2+6G+1) l9=VAM2=(2G2+3G+0.5) 1=17+l=(2G2+5G+2.5) If we use the same source to provide the two voltage in- P puts of the network as shown in Fig. 2, the current delivered Er by that source would be 111+112. Thus 1,=(4G-+10G +5). It may be seen that the network shown in Fig. 2 is a five-sided polygon of resistances defined in Ref. 1. Hence R5=E/1=VAN/12=(4G2+6G+1)(4G2+10G+5)9 Fig. 1. Ladder network. Am J Phys. 68(5), May 2000 o 2000 American Association of Physics Teachers
given by A. Bers in 1962; see W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas ~MIT, Cambridge, MA, 1963!, Sec. 8.5. As shown in this last reference, the equality between group velocity and energy velocity follows simply from a variational form of Maxwell’s equations for an arbitrarily dispersive medium; this general proof is the basis of this note. 5 See, for example, L. Brillouin, Wave Propagation and Group Velocity ~Academic, New York, 1960!, Chap. I. 6 For a loss-free, isotropic, and only temporally dispersive medium @KJh 5JI K(vr) and real#, ~8! was first identified by L. Brillouin in 1932; see Ref. 5 above, Chap. IV. For a loss-free, anisotropic medium with spatial and temporal dispersion @KJh5KJh(kWr ,vr)#, ~8! and ~9! were derived independently by several authors; see Ref. 4 above, and the treatment by Bers in Ref. 8 below. 7 Note that for propagation in regions of strong absorption, it is well-known that this equality breaks down; see Chap. V in Ref. 5 above. In such regions v(kWr)5vr(kWr)1ivi(kWr), and uvi(kWr)u is not small compared to uvr(kWr)u. Then, calculating vW g from vr(kWr) can give values for vg that exceed the speed of light and hence, in such regions, the group velocity no longer represents the velocity of energy flow; energy cannot travel at such speeds. 8 A. Bers, ‘‘Linear Waves and Instabilities,’’ in Plasma Physics—Les Houches 1972, edited by C. DeWitt and J. Peyraud ~Gordon and Breach, New York, London, Paris, 1975!, Secs. II and VII; I. B. Bernstein, ‘‘Geometric Optics in Space- and Time-Varying Plasmas,’’ Phys. Fluids 18, 320 ~1975!. 9 A. Bers, ‘‘Space-Time Evolution of Plasma Instabilities—Absolute and Convective’’ in Handbook of Plasma Physics, general editors, M. N. Rosenbluth and R. Z. Sagdeev, Vol. I Basic Plasma Physics, volume editors, A. A. Galeev and R. N. Sudan ~North-Holland, Amsterdam, 1983!, Chap. 3.2, Sec. 3.2.3. Analysis of doubly excited symmetric ladder networks V. V. Bapeswara Rao Department of Electrical Engineering, North Dakota State University, Fargo, North Dakota 58105 ~Received 29 March 1999; accepted 14 June 1999! A simple procedure to determine the effective resistance between the center and a vertex of an n-sided polygon made of resistors is presented. © 2000 American Association of Physics Teachers. In a recent paper1 Sidhu presented a procedure to determine the effective resistance between the center and a vertex of an n-sided polygon made of resistors. The resistance connected between the center and a vertex of the polygon is 1 V, whereas the resistance connected between any two successive vertices is 2G V. In this note, we present a simple alternative procedure that is easier to comprehend. The units for current ~ampere! and voltage ~volt! are not specified with respect to each variable in the text below. Consider the ladder network2 shown in Fig. 1. We let E1 5E25E. From considerations of symmetry, I050. Further, I15I2 , I35I4 , I55I6 , I75I8 , I95I10 , and I115I12 . We propose to determine E for which I151. Under such a condition, VCN51 and I35I01I151. Further, VBN5VBC1VCN5~2G11!, I55~2G11!, I75I31I55~2G12!, VAN5E5I7~2G!1VBN5~4G216G11!, I95VAN/25~2G213G10.5!, I115I71I95~2G215G12.5!. If we use the same source to provide the two voltage inputs of the network as shown in Fig. 2, the current delivered by that source would be I111I12 . Thus Is5(4G2110G 15). It may be seen that the network shown in Fig. 2 is a five-sided polygon of resistances defined in Ref. 1. Hence, R55E/Is5VAN /Is5(4G216G11)/(4G2110G15)V. The resistance R2N11 for any N, may be computed by considering appropriate ladder network. It should have (2N 11) resistances of value 2G V, 2N resistances of value 1 V, and two 2 V resistances ~one at each end!. Note that the computation of successive voltages and currents is a recursive procedure. We now consider the determination of R2N . The network used for the determination of R2N is derived from the network used for the determination of R2N11 . The ‘‘mid-section’’ is modified as follows: Each of the two resistances from the vertex to the center is changed to 2 V. The resistance between the vertices is set to zero. The network for R4 is obtained from the circuit shown in Fig. 1. The element values are as shown in Fig. 3. We analyze the circuit in Fig. 3 as earlier. We note that I0 is zero. We propose to determine E for which I151. Under such a condition, VCN52. Then I35I01I151. Further, VBN5VBC1VCN5~2G12!, I55~2G12!, I75I31I55~2G13!, Fig. 1. Ladder network. 484 Am. J. Phys. 68 ~5!, May 2000 © 2000 American Association of Physics Teachers 484
6↓ ↓g1511|l2161|10 Fig. 2. Doubly excited symmetrical ladder network. Fig 3. Modification of Fig. 1 R4=VAN1=(4G2+8G+2)/(4G2+12G+8)9 AN=E=VBN+17(2G)=(4G2+8G+2) (2G2+4G+1)(2G2+6G+4)9 Is=VAM2=(2G2+4G+1), The procedure outlined in the note is suitable for class- room use. For a given value of G(say G=1), the effective I1=1y+l9=(2G2+6G+4) resistance Ro or R can be computed in a short time If we use the same source to provide the two voltage in- by that source would he lili. Thus I,(4G-+12G ME6(94)u, "polygons of unequal resistors,"AmJPhys. 62, puts of the network as shown in Fig. 2, the current delivered Van Valkenberg, Newwork Analysis(Prentice- Englewood +8).Thus, Cliffs, NJ, 1964) THE METRIC SYSTEM The only case for the French metric system is that it has become sufficiently universal so tha there are real advantages in making it completely universal. It cannot claim the slightest scientific alidity as its units are not based on any natural units and are psychologically not even particularly convenient. The decimal system is one of the less convenient systems of counting, though by no means the worst. The only argument for it is that when it doesnt really matter what we do, it is convenient to have everybody do the same thing, so let's all join the party Kenneth Boulding, Numbers and Measurement on a Human Scale, in The Metric Debate, edited by David F. Bartlett ( Colorado Associated University Press, Boulder, 1980), P. 64. Am J Phys., Vol 68, No 5, May 2000 Notes and discussions
VAN5E5VBN1I7~2G!5~4G218G12!, I95VAN/25~2G214G11!, I115I71I95~2G216G14!. If we use the same source to provide the two voltage inputs of the network as shown in Fig. 2, the current delivered by that source would be I111I12 . Thus Is5(4G2112G 18). Thus, R45VAN /Is5~4G218G12!/~4G2112G18!V 5~2G214G11!/~2G216G14!V. The procedure outlined in the note is suitable for classroom use. For a given value of G ~say G51!, the effective resistance R9 or R8 can be computed in a short time. 1 Satinder S. Sidhu, ‘‘Polygons of unequal resistors,’’ Am. J. Phys. 62, 815–816 ~1994!. 2 M. E. Van Valkenberg, Network Analysis ~Prentice–Hall, Englewood Cliffs, NJ, 1964!. THE METRIC SYSTEM The only case for the French metric system is that it has become sufficiently universal so that there are real advantages in making it completely universal. It cannot claim the slightest scientific validity as its units are not based on any natural units and are psychologically not even particularly convenient. The decimal system is one of the less convenient systems of counting, though by no means the worst. The only argument for it is that when it doesn’t really matter what we do, it is convenient to have everybody do the same thing, so let’s all join the party. Kenneth Boulding, ‘‘Numbers and Measurement on a Human Scale,’’ in The Metric Debate, edited by David F. Bartlett ~Colorado Associated University Press, Boulder, 1980!, p. 64. Fig. 2. Doubly excited symmetrical ladder network. Fig. 3. Modification of Fig. 1. 485 Am. J. Phys., Vol. 68, No. 5, May 2000 Notes and Discussions 485
