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 3For Section 3.3:Prob. 3.3.1 In writing Eq. 3.2.3, the inertia of the charge carriers is ignored. Add inertial termsto the equations, assume that the magnetic field is zero and consider an imposed electric field ý =Re 2 exp(jwt). Show that the effects of inertia are negligible if W << V+. For copper, the electronmobility is about 3 x 10- 3 m2/volt sec, while q /m = 1.76 x 1011 m2/sec2 volt. What must the frequencybe to make the electron inertia significant?For Section 3.5:Prob. 3.5.1 For the system of Probs. 2.11.1 and 2.13.1,(a) Show that the reciprocity condition requires that C21 = C12.(b)Find the electrical forces (fl,f2) in terms of(vl,v2,El,2) that tend to displace the movableplate in the directions (El'E2,) respectively.Prob. 3.5.2 In Fig. 3.6.1, a dielectric slab is pictured as being pulled upward between plane parallelelectrodes from a dielectric fluid having the same permittivity as the slab.(a)What is the total coenergy, w'(v,ý)? (Ignore fringing fields.)(b)Use the force-energy relation, Eq. 3.5.9,to find the polarization force tending to make the slabrise.Prob. 3.5.3 Determine the electrical force tending to increase the displacement E of the saturabledielectric material of Prob. 2.13.2.Prob. 3.5.4 For the MQS configuration described in Probs. 2.12.1 and 2.14.1,(a)Find the radial surface force density Tr by using the coenergy function to obtain Tr(il,i2',).(b) Compare the operations necessary to obtain Tr(X1,Nix) using the energy function w to thoseusing w'. Even though the coenergy formulation is more convenient for this problem, the energyfunction is more convenient if one or more flux linkages are constrained.(c)If the inner coil is shorted at a time when its flux linkage is X2 = 0, what is Tr(X• )?For Section 3.6:Prob. 3.6.1 In a fluid at rest, external force densities are held in equilibrium by the gradientof the fluid pressuie p. Hence, force equilibrium for each incremental volume of the fluid subjectto a force density F is represented by4.Vp = FSuppose that the bottom of the dielectric slab pictured in Fig. 3.6.1 is well above the lower edgesof the electrodes, so that the fringing field, and hence the VE2, is confined to the liquid dielectric.Then there is no Kelvin force density acting on the slab, and the force density of Eq. 3.6.7 prevails inthe liquid. Use Eq. 3.6.7 in Eq. 3.6.1 and integrate from the exterior free surface to the bottom of theslab to find the fluid pressure acting on the bottom of the slab. Show that this pressure, acting overthe bottom of the slab, gives a net upward force that is consistent with the result of Prob. 3.5.2.Prob. 3.6.2 Use arguments similar to those leading to Eq. 3.6.4 to show that the torque on an electricdipole isT=PxEBased on arguments similar to those used in deducing Eq. 3.6.12 from Eq. 3.6.5, argue that the torqueon a magnetic dipole isT = o0m x HFor Section 3.7:Prob. 3.7.1 Show that the last paragraph in Sec. 3.7 is correct.Problems for Chap. 33.23For Section 3.9:Prob. 3.9.1 One way to show that Eq. 3.9.17 can be used to compute T is to write Eq. 3.9.16 inCartesian coordinates and use the symmetry of the stress tensor to bring the components of r insidethe spatial derivatives. Carry out these steps and then use the tensor form of Gauss' theorem toobtain Eq. 3.9.17.For Section 3.10:Prob. 3.10.1 For certain purposes, the electric force density in an incompressible liquid with nofree charge density might be represented asF = 2V(EE)where E is a function of the spatial coordinates. Show that this differs from Eq. 3.7.22 by the grad-ient of a pressure and that the accompanying stress components areT = £E.E.ij EE13Prob. 3.10.2 A fluid has the electrical constitutive law+ + +++4-D = alE + a2(E'E)EIt is inhomogeneous, so that al and a2 are functions of the spatial coordinates. There is no freecharge density and the fluid can be assumed incompressible. Integrate the conservation of coenergyequations to show that the coenergy density is1 ++ ~2 +-+ 2' = 22lE'E + -(E.E).f Find the force density F in terms of E, al and a2. Find the stress tensor T.ij associated with thisforce density. Prove that F can be written in the form = -V~ + VW, where P is the polarizationdensity.Prob. 3.10.1 For certain purposes, the electrical force density in an incompressible liquid with noi4 dfree charge dens.LLy M ghILL Ube represente asiLF EV (E*E)F2where s is a function of the spatial coordinates. Show that this differs from Eq. 3.7.22 by thegradient of a pressure, and that the accompanying stress components areTj = SE.E.Prob. 3.10.2 A fluid has the electrical constitutive law_ +4. 4_+ + +D = (So+a1)E+ 2(E)EIt is inhomogeneous, so that al and a2 are functions of the spatial coordinates. There is no freecharge density and the fluid can be assumed incompressible. Integrate the conservation of coenergyequations to show that the coenergy density is1 4-+ a2 2W' = -(o a1)E2E +E (E*E)Find the force density F rn terms of E, al and a2.Find the stress tensor Tij associated withthis force density. Prove that F can be written in the formF = P .VE + Vrwhere P is the polarization density.Problems for Chap. 373.24Prob. 3.10.3 Fig. P3.10.3 shows a circular cylindrical tube of innerradius a into which a second tube of outer radius b projects half way.On top of this inner tube is a "blob" of liquid metal (shown inside thebroken-line box) having an arbitrary shape, but having a base radiusequal to that of the inner tube. The outer and inner tubes, as well asthe blob, are all essentially perfectly conducting on the time scale ofinterest. When t=0 , there are no magnetic fields. When t=O+, the outer-t.tube is used to produce a magnetic flux which has density Bo z a distance2 >> a above the end of the inner tube. What is the magnetic flux dens-ity over the cross section of the annulus between tubes a distance 2(2>> a) below the end of the inner tube? Sketch the distribution ofsurface
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