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 terms Problems are different for example the homopolar machine has essentially a one turn armature and is always a low voltage high current device Thus matching a wire wound coil to this low impedance is difficult and few homopolar machines can be self excited either with series field windings or with shunt field windings Consequently most homopolar machines have separately excited field windings Because of the similarity between homopolar and commutator machine characteristics we terminate the discussion here and treat homopolar machines further in the problems at the end of this chapter 6 5 DISCUSSION In this chapter we have made the necessary generalizations of electromagnetic theory that are needed for analyzing quasistatic systems with materials in relative motion This has involved transformations for source and field quantities between inertial reference frames boundary conditions for moving boundaries and constituent relations for moving materials In addition to some simple examples we have made an extensive analysis of dc rotating machines because they are devices that are particularly amenable to analysis by the generalized field theory Having completed the generalization of field theory with illustrative examples of lumped parameter systems we are now prepared to proceed to continuum electromechanical problems In Chapter 7 we consider systems with specified mechanical motion and in which electromagnetic phenomena must be described with a continuum viewpoint PROBLEMS 6 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 X2 X1 Fig 6P 1 Fields and Moving Media a Write the transformation laws like 6 1 6 that relate primed coordinates to the unprimed coordinates b Given that Vyis a function of r 0 z t find ap 8at the rate of change with respect to time of vy for an observer in the rotating frame in terms of derivatives with respect to r 0 z t 6 2 A magnetic field distribution B B0 sin kx1 i2 exists in the laboratory frame What is the 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 iBo cos wt kx is produced in the laboratory by two windings distributed in space such that the number of turns per unit length 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 linear version of Problem 4 10 which was cylindrical w radial frequency 27rf 2fr k 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 of external forces av 1 p p v V v Vp 0 op a V pv 0 2 3 p p p where p is the pressure p the mass density and v the velocity of the fluid Equation 1 is Newton s law for a fluid 2 is the law of conservation of mass and 3 is a constitutive relation relating the pressure and density Are these equations invariant to a Galilean transformation to a coordinate system given by r r vt If so find v p p as a function 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 in the stationary frame and velocity v voi z See Fig 6P 5 r EO go Fig 6P 5 Problems a Using only the transformation law for charge density find the electric and magnetic fields in a reference frame that is at rest with the electrons b Without using any transformation laws find the electric and magnetic fields in the stationary frame c Show that a and b are consistent with the electric field system transformation laws 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 respectively The cylinders are only very slightly conducting so that as they rotate with the 1 Fig 6P 6 angular velocity o they carry along the charges induced on their surfaces As viewed from a frame rotating with the cylinders the charges are stationary We wish to compute the resulting fields a Compute the electric field between the cylinders and the surface charge densities cylinders respectively a and a on the inner and outer b Use the transformation for the current density to compute the current density from the results of part a c In turn use the current density to compute the magnetic field intensity H between the cylinders d Now use the field transformation for the magnetic field intensity to check the result 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 a y y a sin wt cos too urface current A0z r no L Surface current 0 o Fig 6P 7 O Fields and Moving Media function of space and time Both boundaries can be considered perfect conductors so that n B 0 on their surfaces In what follows assume that a d and that the magnetic field intensity between the plates takes the form H 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 Compute the electric field intensity by using the magnetic field found in part a Now check to see that this boundary condition is satisfied to linear terms 6 8 The system shown in Fig 6P 8 consists of a coaxial line of inner radius a and outer radius b with an
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