MIT OpenCourseWare http://ocw.mit.edu Electromechanical Dynamics For any use or distribution of this textbook, please cite as follows: Woodson, Herbert H., and James R. Melcher. Electromechanical Dynamics. 3 vols. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsProblemsare different; for example, the homopolar machine has essentially a one-turnarmature and is always a low-voltage, high-current device. Thus matching awire-wound coil to this low impedance is difficult and few homopolar ma-chines can be self-excited, either with series field windings or with shunt fieldwindings. Consequently, most homopolar machines have separately excitedfield windings.Because of the similarity between homopolar and commutator machinecharacteristics we terminate the discussion here and treat homopolar ma-chines further in the problems at the end of this chapter.6.5 DISCUSSIONIn this chapter we have made the necessary generalizations of electro-magnetic theory that are needed for analyzing quasistatic systems withmaterials in relative motion. This has involved transformations for sourceand field quantities between inertial reference frames, boundary conditionsfor moving boundaries, and constituent relations for moving materials. Inaddition to some simple examples, we have made an extensive analysis of dcrotating machines because they are devices that are particularly amenable toanalysis by the generalized field theory.Having completed the generalization of field theory with illustrativeexamples of lumped-parameter systems, we are now prepared to proceed tocontinuum electromechanical problems. In Chapter 7 we consider systemswith specified mechanical motion and in which electromagnetic phenomenamust be described with a continuum viewpoint.PROBLEMS6.1. Two frames of reference have a relative angular velocity f, as shown in Fig. 6P.1.In the fixed frame a point in space is designated by the cylindrical coordinates (r, 0, z).In the rotating frame the same point is designated by (r', 0', z'). Assume that t = t'.X2X1Fig. 6P.1Fields and Moving Media(a) Write the transformation laws [like (6.1.6)] that relate primed coordinates to theunprimed coordinates.(b) Given that Vyis a function of (r, 0, z,t), find ap/8at' (the rate of change with respectto time of vy for an observer in the rotating frame) in terms of derivatives withrespect to (r, 0, z, t).6.2. A magnetic field distribution B = B0 sin kx1 i2 exists in the laboratory frame. What isthe time rate of change for the magnetic field as viewed from the following:(a) An inertial frame traveling parallel to the xz-axis with speed V?(b) An inertial frame traveling parallel to the X2-axis with speed V?6.3. A magnetic field traveling wave of the form B = iBocos (wt -kx) is produced inthe laboratory by two windings distributed in space such that the number of turns per unitlength varies sinusoidally in space. The windings are identical except for a 900 separation.They are excited by currents of equal amplitude but 900 out of time phase. This is a linearversion of Problem 4.10, which was cylindrical.w = radial frequency = 27rf,2frk = wavenumber = --= wavelength,v•=-= f = phase velocity of wave.(a) If an observer is in an inertial frame traveling with speed V in the z-direction,what is the apparent frequency of the magnetic wave?(b) For what velocity will the wave appear stationary?6.4. The following equations describe the motions of an inviscid fluid in the absence ofexternal forces:avp -+ p(v. V)v + Vp = 0, (1)opa + V .pv = 0, (2)p = p(p), (3)where p is the pressure, p, the mass density, and v the velocity of the fluid. Equation 1 isNewton's law for a fluid, (2) is the law of conservation of mass, and (3) is a constitutiverelation relating the pressure and density. Are these equations invariant to a Galileantransformation to a coordinate system given by r' = r -vt? If so, find v', p', p' as afunction of the unprimed quantities v, p, p.6.5. A cylindrical beam of electrons has radius a, a charge density Po(1 -r/a) (Po < 0) inthe stationary frame, and velocity v = voiz.(See Fig. 6P.5.)rEO,goFig. 6P.5Problems(a) Using only the transformation law for charge density, find the electric andmagnetic fields in a reference frame that is at rest with the electrons.(b) Without using any transformation laws, find the electric and magnetic fields inthe stationary frame.(c) Show that (a) and (b) are consistent with the electric field system transformationlaws for E, H, and J,.6.6. A pair of cylinders coaxial with the z-axis, as shown in Fig. 6P.6, forms a capacitor.The inner and outer surfaces have the potential difference V and radii a and b, re-spectively. The cylinders are only very slightly conducting, so that as they rotate with theFig. 6P.6angular velocity o they carry along the charges induced on their surfaces. As viewed froma frame rotating with the cylinders, the charges are stationary. We wish to compute theresulting fields.(a) Compute the electric field between the cylinders and the surface charge densitiesa. and a, on the inner and outer cylinders, respectively.(b) Use the transformation for the current density to compute the current densityfrom the results of part (a).(c) In turn, use the current density to compute the magnetic field intensity H betweenthe cylinders.(d) Now use the field transformation for the magnetic field intensity to check theresult of part (c).6.7. A pair of perfectly conducting electrodes traps a magnetic field, as shown in Fig. 6P.7.One electrode is planar and at y = 0, the other has a small sinusoidal variation given as ay-toor no~ ~L_y = a sin wt cos + urface current................OA0z0'-**o*Surface currentFig. 6P.7€)1Fields and Moving Mediafunction of space and time. Both boundaries can be considered perfect conductors, so thatn -B = 0 on their surfaces. In what follows, assume that a << d and that the magneticfield intensity between the plates takes the formH = + h(x, y, tA) i + h1,(, y, t)i,where A is the flux trapped between the plates (per unit length in the z-direction) and h,and h, are small compared with A/lod.(a) Find the perturbation components h. and h,.(b) The solutions in part (a) must satisfy the boundary conditions: n X E = (n. v)B[boundary condition (6.2.22) of Table 6.1].
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