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 Teachersgiven 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 en￾ergy velocity follows simply from a variational form of Maxwell’s equa￾tions 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 inde￾pendently 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, ‘‘Geo￾metric 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 edi￾tors, 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 deter￾mine the effective resistance between the center and a vertex of an n-sided polygon made of resistors. The resistance con￾nected between the center and a vertex of the polygon is 1 V, whereas the resistance connected between any two succes￾sive 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 con￾dition, 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 in￾puts 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 recur￾sive 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 resis￾tances 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