MIT OpenCourseWare http://ocw.mit.edu Continuum Electromechanics For any use or distribution of this textbook, please cite as follows: Melcher, James R. Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. Copyright Massachusetts Institute of Technology. ISBN: 9780262131650. Also available online from MIT OpenCourseWare at http://ocw.mit.edu (accessed MM DD, YYYY) under Creative Commons license Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Problems for Chapter 11For Section 11.2:Prob. 11.2.1 Starting with the Lagrangian description of the particle motions afforded by Eq. 2,show that if E = -V1 and 1 is independent of time, the sum of potential and kinetic energies,m(V.")+q0, is invariant.For Section 11.3:Prob. 11.3.1 A planar version of the magnetron configuration considered in this section makes useof planar electrodes that play the role of the coaxial ones in Fig. 11.3.1. The cathode, which isat zero potential, is in the plane x = 0 while the anode is at x = a and has the potential V. Thus,ignoring space-charge effects, the electric potential is Vx/a. A uniform magnetic flux density, Bo,is imposed in the z direction.(a)Write the equations of motion for an electron in terms of its position r = xix + yi + z.(b)Combine these expressions to obtain a single second-order differential equation for x(t).(c)Under the assumption that V is constant, integrate this expression once (in a way analogous tothat used in obtaining Eq. 11.3.2) to obtain a potential well that is determined by the combinedeffects of the applied electric and magnetic fields. Use this result to generate a potential-wellpicture analogous to Fig. 11.3.2 and qualitatively describe how the particle trajectories areinfluenced by the applied potential. (Assume that particles leave the cathode with no velocity.)(d) Sketch typical trajectories in the x-y plane.(e)What is the critical potential, V = V ?For Section 11.4:Prob. 11.4.1 An electron has an initial velocity that is purely axial and moves in a region of con-stant potential and uniform axial magnetic field. Show that Eqs. 11.2.3-11.2.5 are satisfied by asubsequent motion in which the electron continues to move in only the z direction. How is it thatEq. 11.4.8 can predict the focusing effect of the field even though it only involves the axialcomponent of i?Prob. 11.4.2 In a magnetic lens, the potential, 0, is constant. Use Eq. 11.4.8 to show that allmagnetic lenses are converging, in the sense that dr/dz will be less as a particle leaves the lensthan as it enters.Prob. 11.4.3 If the focal length is long compared to the magnetic length of the magnetic lens, itis said to be a weak lens. In this case, most of the deflection occurs outside the field region withthe lens serving to redirect the particles. Essentially, r remains constant through the lens, but itsspatial derivative is altered. Use Eq. 11.4.8 to show that in this case the focal length isz+ -1e zShow that this general result for a weak lens is consistent with Eq. 11.4.12.Prob. 11.4.4 An electron beam crosses the z = 0 plane from the region z<0 with the potential V .As it crosses, a given electron has radial position ro and moves parallel to the z axis. In the z =0plane, a potential t -VoJo(yr) is imposed so that for z >0, 4 = VoJo(yr)exp(-yr) (See Eq. 2.16.18with jk÷y and m = 0.) Use the paraxial ray equation, Eq. 11.4.8, to determine the electron trajec-tory. Assume that yro << 1.For Section 11.5:Prob. 11.5.1 Under what conditions does an axial magnetic field suppress transverse electron motions?To answer this question take the potential distribution as having been determined from Eq. 11.7.7 andwrite the transverse components of the force equation with the potential terms as "drives." Arguethat transverse motions are small compared to longitudinal ones provided that the frequency is lowcompared to the electron-cyclotron frequency.Problems for Chap. 1111.71Prob. 11.5.2 An electron beam of annular cross section with outer radius a and inner radius b streamsin the z direction with velocity U.(a) Show that the transfer relations are as summarized in Table 2.16.2 with k2 y k2[1-W/(w-kU)2in the coefficients.(b) As an application of these transfer relations, consider a beam of radius b (no free-space core)with the potential constrained to be $cat the radius a. The region b< r <a is free space. Find^Cr(c) Find the eigenfrequencies of the temporal modes with a perfect conductor at r = a.For Sectiofi 11.6:Prob. 11.6.1 A physical situation is represented by m dependent variables x.x = X1i X..... x..... xmwhich satisfy m first-order partial differential equationsmm x. 3x.E [F 1 + G = 0j=l ij t ij z(For example, in Sec. 11.6, m = 2, x1 = p and x2 = v.) The coefficients Fij and Gij are functionsof the x's as well as (z,t).(a) Use the method of undetermined multipliers to find a determinantal equation for the firstcharacteristic equations.(b) Show that the same determinantal equation results by requiring that the coefficient matrixvanish.For Section 11.7:Prob. 11.7.1 There is a complete analogy between shallow-water gravity-wave dynamics and the one-dimensional compressible motions of gases studied in this section. Use Eqs. 9.13.11 and 9.13.12with V = 0 and b = constant to show that if y = 2, analogous quantities are2aP 5, +v vv, _ + gPThe analogy is exploited in the film "Waves in Fluids," which deals with both types of wave systems.(See Reference 10, Appendix C.)Prob. 11.7.2 The quasi-one-dimensional equations of motion for free surface flow contained by anelectric field (Fig. 9.13.3) are Eqs. 9.13.4 and 9.13.9, with A and p given in Fig. 9.13.3.Find the associated first and second characteristic equations.Prob. 11.7.3 An inviscid, incompressible fluid rests on a rigid flat bottom, as shown in Fig. 9.13.1.Consider the motions with V = 0.(a) Show that the first characteristic equations aredz +dz--= v + R(E) on C-dtand that the second characteristic equations arev = +R() + c, on C-where R(A) = 2 g.(b) A simple wave propagates into a region of constant depth Ec and zero velocity. At z = 0, E = S(t).Find v and E for z > 0, t > 0. Show that only if dEs/dt > 0 will a shock form.(c) Consider the initial value problem: when t = 0, v = v (z,0) = 1 and E = 0(z,0), whereProblems for Chap. 1111.72Prob. 11.7.3 (continued)1 for z < -3Eo(z,0) = -0.31z1+ 1.9 for -3< z < 31 for 3 < zFor computational purposes, set g =
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