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Problems 475 (a) What is the open circuit Hall voltage? (Hint: The transverse current of each carrier must be zero.) (b) What is the short circuit Hall current? 17. A highly conducting hollow iron cylinder with permeability A and inner and outer radii R, and R 2 is concentric to an infinitely long dc line current (adapted from L. V. Bewley, Flux Linkages and Electromagnetic Induction. Macmillan, New York, 1952, pp. 71-77). d- -_-/_ Po (b) (d) (a) What is the magnetic flux density everywhere? Find the electromotive force (EMF) of the loop for each of the following cases. 476 ElecromagneticInduction (b) A highly conducting circuit abcd is moving downward with constant velocity Vo making contact with the surfaces of the cylinders via sliding brushes. The circuit is completed from c to d via the iron cylinder. (c) Now the circuit remains stationary and the iron cylinder moves upwards at velocity Vo. (d) Now a thin axial slot is cut in the cylinder so that it can slip by the complete circuit abcd, which remains stationary as the cylinder moves upwards at speed Vo0 . The brushes are removed and a highly conducting wire completes the c-d path. 18. A very long permanently magnetized cylinder Moi• rotates on a shaft at constant angular speed w. The inner and outer surfaces at r = a and r = b are perfectly conducting, so that brushes can make electrical contact. 4-V -- (a) If the cylinder is assumed very long compared to its radius, what are the approximate values of B and H in the magnet? (b) What is the open circuit voltage? (c) If the magnet has an Ohmic conductivity o, what is the equivalent circuit of this generator? (d) What torque is required to turn the magnet if the terminals are short circuited? 19. A single spoke wheel has a perfectly conducting cut The spoke has Ohmic conductivity ar and cross-sectional A. The wheel rotates at constant angular speed wo sinusoidally varying magnetic field B, = Bo cos at. (a) What is the open circuit voltage and short circuit rent? (b) What is the equivalent circuit? rim. area in a cur- Problems SB, 477 =Bocoswt eB t 20. An MHD machine is placed within a magnetic circuit. (a) A constant dc current if = Io is applied to the N turn coil. How much power is delivered to the load resistor RL? (b) The MHD machine and load resistor RL are now connected in series with the N turn coil that has a resistance Rf. No current is applied. For what minimum flow speed can the MHD machine be self-excited? 21. The field winding of a homopolar generator is connected in series with the rotor terminals through a capacitor C. The rotor is turned at constant speed w. (a) For what minimum value of rotor speed is the system self-excited? (b) For the self-excited condition of (a) what range of values of C will result in dc self-excitation or in ac selfexcitation? (c) What is the frequency for ac self-excitation? 478 Electromagnetic Induction C Section 6-4 22. An Ohmic block separates two perfectly conducting parallel plates. A dc current that has been applied for a long time is instantaneously turned off at t = 0. ' if) I1o 1-4 Depth D Ii D I " I 0 d (a) What are the initial and final magnetic field distributions? What are the boundary conditions? (b) What are the transient magnetic field and current distributions? (c) What is the force on the block as a function of time? 23. A block of Ohmic material is placed within a magnetic circuit. A step current Io is applied at t = 0. (a) What is the dc steady-state solution for the magnetic field distribution? (b) What are the boundary and initial conditions for the magnetic field in the conducting block? *(c) What are the transient field and current distributions? (d) What is the time dependence of the force on the conductor? (e) The current has been on a long time so that the system is in the dc steady state found in (a) when at t = T the current _II _ _ Problems 479 A(t) I -T oT7 is instantaneously turned off. What are the transient field and current distributions in the conductor? (f) If the applied coil current varies sinusoidally with time as i(t)=Iocos ot, what are the sinusoidal steady-state field and current distributions? (Hint: Leave your answer in terms of complex amplitudes.) (g) What is the force on the block? 24. A semi-infinite conducting block is placed between parallel perfect conductors. A sinusoidal current source is applied. locoscwt Depth D Depth D Y y (a) What are the magnetic field and current distributions within the conducting block? (b) What is the total force on the block? (c) Repeat (a) and (b) if the block has length d. 25. A current sheet that is turned on at t = 0 lies a distance d above a conductor of thickness D and conductivity or. The conductor lies on top of a perfectly conducting plane. (a) What are the initial and steady-state solutions? What are the boundary conditions? (b) What are the transient magnetic field and current distributions? (c) After a long time T, so that the system has reached the dc steady state, the surface current is set to zero. What are the subsequent field and current distributions? 480 ElectromagneticInduction K(t) o0,a = 0 2--X (d) What are the field and current distributions if the current sheet varies as Ko cos cot? 26. Distributed dc currents at x = 0 and x = I flow through a conducting fluid moving with constant velocity voix. 1 x Depth D (a) What are the magnetic field and current distributions? (b) What is the force on the fluid? 27. A sinusoidal surface current at x = 0 flows along parallel electrodes and returns through a conducting fluid moving to the right with constant velocity voi.. The flow is not impeded by the current source. The system extends to x = co. cos), K0 coswt 1o -------- -------- --=---- Vol Depth D Sx (a) What are the magnetic field and current density distributions? (b) What is the time-average force on the fluid? Problems 481 28. The surface current for the linear induction machine treated in Section 6-4-6 is now put a distance s below the conductor. (a) What are the magnetic field and current distributions in each region of space? (Hint: Check your answer with Section 6-4-6 when s = 0.) (b) Repeat (a) if s is set to zero but the conductor has a finite thickness d. 29. A superconducting block with plasma frequency wp is placed within a magnetic circuit with exciting current Io cos ot. Depth D (a) What are the magnetic field and current distributions in the superconductor? (b) What is the force on the block? Section 6.5 30. Find the magnetic energy stored and the self-inductance for the geometry below where the current in each shell is uniformly distributed. 31. Find the external self-inductance of the two wire lines shown. (Hint: See Section 2-6-4c.) 482 Electromagnetic Induction Depth I Depth I IE S 310 i 32. A coaxial cable with solid inner conductor is excited by a sinusoidally varying current Io cos to at high enough frequency so that the skin depth is small compared to the radius a. The current is now nonuniformly distributed over the inner conductor. Wt Io Cos (a) Assuming that H= H,(r)i,, what is the governing equation for H,(r) within the inner cylinder. (Hint: V2H = 0 V(V, H) -V x (V x H).) (b) Solve (a) for solutions of the form H,(r) = Re [fH(r) "']I Hint: Bessel's equation is • dy 2 2d y x ~+x i +(x -p y=O with solutions y =A 1Jp(x)+A 2YO(x) where Y, is singular at x = 0. (c) What is the current distribution? Hint: d 1 1U(x)] , + -J:(x) = Jo(x) Section 6-6 33. A reluctance motor is made by placing a high permeability material, which is free to rotate, in the air gap of a magnetic circuit excited by a sinusoidal current Io cos Oot. _···1__1 ____ · __ Problems 483 The inductance of the circuit varies as L(O)= Lo+ L 1 cos 20 where the maximum inductance Lo+L, occurs when 0 = 0 or 0 = 1r and the minimum inductance Lo-L 1 occurs when 0 = +Er/2. (a) What is the torque on the slab as a function of the angle 0? (b) The rotor is rotating at constant speed w, where 0 = wt + 8 and 8 is the angle of the rotor at t = 0. At what value of w does the torque have a nonzero time average. The reluctance motor is then a synchronous machine. Hint: cos2 wot sin 20 = l[sin 20 +cos 2wot sin 20] = f{sin 20 + ½[sin2(wot + 0) + sin 2(0 - wot)]} (c) What is the maximum torque that can be delivered by the shaft and at what angle 5 does it occur? 34. A system of two coupled coils have the following fluxcurrent relations: 484 ElectromagneticInduction d1 = Li()i, + M(O)i 2 c 2 = M(O)ii +L 2 (0)i 2 (a) What is the power p delivered to the coils? (b) Write this power in the form dW P=-+Tdt dO dt What are W and T? (c) A small coil is free to rotate in the uniform magnetic field produced by another coil. The flux-current relation is 1 = Lli, + Moi 2 sin 0 02 = Moi I sin 0 + L 2 i2 The coils are excited by dc currents I, and I2. What is the torque on the small coil? (d) If the small coil has conductivity or, cross-sectional area A, total length 1, and is short circuited, *what differential equation must the current il obey if 0 is a function of time? A dc current 12 is imposed in coil 2. (e) The small coil has moment of inertia J. Consider only small motions around 0 = 0 so that cos 0 - 1. With the torque and current equations linearized, try exponential solutions of the form est and solve for the natural frequencies. (f) The coil is released from rest at 0 = 00. What is O(t) and il(t)? Under what conditions are the solutions oscillatory? Damped? 35. A coaxial cable has its short circuited end free to move. (a) What is the inductance of the cable as a function of x? (b) What is the force on the end? 36. For the following geometries, find: (a) The inductance; (b) The force on the moveable member.