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ProMems 465 (11) can be rewritten as F.= ( A -,) O(H +H ax 2 ) (14) The total force is then f =sD =( Fdx - l>o)sD( H2+H ) =o (IL- Io) N2,2D = 2 )(15) s where the fields at x = -0o are zero and the field at x = xo is given by (12). High permeability material is attracted to regions of stronger magnetic field. It is this force that causes iron materials to be attracted towards a magnet. Diamagnetic materials (A b and then sketch the results when d = b and d < b. The right edge of the current loop reaches the volume current at t = 0. 3. A short circuited rectangular loop of mass m and selfinductance L is dropped with initial velocity voi. between the pole faces of a magnet that has a concentrated uniform magnetic field Boil. Neglect gravity. x v0 (a) What is the imposed flux through the loop as a function of the loop's position x (0 < x D. 5. A circular loop of radius a is a distance D above a point magnetic dipole of area dS carrying a current II. 2 D I1 dS II_ Problems 469 (a) What is the vector potential due to the dipole at all points on the circular loop? (Hint: See Section 5-5-1.) (b) How much flux of the dipole passes through the circular loop? (c) What is the mutual inductance between the dipole and the loop? (d) If the loop carries a current 12, what is the magnetic field due to 12 at the position of the point dipole? (Hint: See Section 5-2-4a.) (e) How much flux due to 12 passes through the magnetic dipole? (f) What is the mutual inductance? Does your result agree with (c)? 6. A small rectangular loop with self-inductance L, Ohmic conductivity a, and cross-sectional area A straddles a current sheet. t) ,ýK( II t S (a) The current sheet is instantaneously turned on to a dc level Koi, at t = 0.What is the induced loop current? (b) After a long time T the sheet current is instantaneously set to zero. What is the induced loop current? (c) What is the induced loop current if the current sheet varies sinusoidally with time as Ko cos ot i,. 7. A point magnetic dipole with area dS lies a distance d below a perfectly conducting plane of infinite extent. The dipole current I is instantaneously turned on at t= 0. (a) Using the method of images, find the magnetic field everywhere along the conducting plane. (Hint: i, •i, = sin 0, d 2 = wa dS= I 470 Electromagnetic Induction is ir = Cos 0.) (b) What is the surface current distribution? (c) What is the force on the plane? Hint: Sr 3 dr (r2 +d 2 )5 (r2 + d'/4) 6(r2 + d 2)4 (d) If the plane has a mass M in the gravity field g, what current I is necessary to just lift the conductor? Evaluate for M= 10-s kg, d = 10- m, and a = 10- 3 m. 8. A thin block with Ohmic conductivity o and thickness 8 moves with constant velocity Vi, between short circuited perfectly conducting parallel plates. An initial surface current Ko is imposed at t = 0 when x = xo, but the source is then removed. x . ix Depth D (a) The surface current on the plates K(t) will vary with time. What is the magnetic field in terms of K(t)? Neglect fringing effects. (b) Because the moving block is so thin, the current is uniformly distributed over the thickness 8. Using Faraday's law, find K(t) as a function of time. (c) What value of velocity will just keep the magnetic field constant with time until the moving block reaches the end? (d) What happens to the magnetic field for larger and smaller velocities? 9. A thin circular disk of radius a, thickness d, and conductivity o is placed in a uniform time varying magnetic field B(t). (a) Neglecting the magnetic field of the eddy currents, what is the current induced in a thin circular filament at radius r of thickness dr. _ · ·___~____·_· __· Problems 471 -1.1 d (d) (b) What power is dissipated in this incremental current loop? (c) How much power is dissipated in the whole disk? (d) If the disk is instead cut up into N smaller circular disks with negligible wastage, what is the approximate radius of each smaller disk? (e) If these N smaller disks are laminated together to form a thin disk of closely packed cylindrical wires, what is the power dissipated? Section 6-2 10. Find the self-inductance of an N turn toroidal coil of circular cross-sectional radius a and mean radius b. Hint: dO 2 rdr 2 2 tan (0/2) b+r b + r cos 0 f tan - .- rd6`b= 11. A large solenoidal coil of long length 11, radius a,, and number of turns NI coaxially surrounds a smaller coil of long length 12, radius a 2 , and turns N 2 . 472 Electromagnetic Induction 1 turns (a) Neglecting fringing field effects find the selfinductances and mutual inductances of each coil. (Hint: Assume the magnetic field is essentially uniform within the cylinders.) (b) What is the voltage across each coil in terms of iI and i 2 ? (c) If the coils are connected in series so that il = i 2 with the fluxes of each coil in the same direction, what is the total self-inductance? (d) Repeat (c) if the series connection is reversed so that ii=-i 2 and the fluxes due to each coil are in opposite directions. (e) What is the total self-inductance if the coils are connected in parallel so that v1 = v 2 or v 1 = -v2? 12. The iron core shown with infinite permeability has three gaps filled with different permeable materials. (a) What is the equivalent magnetic circuit? (b) Find the magnetic flux everywhere in terms of the gap reluctances. _S1-_ V1 Depth D Problems 473 (c) What is the total magnetic flux through each winding? (d) What is the self-inductance and mutual inductance of each winding? 13. A cylindrical shell of infinite permeability, length I and inner radius b coaxially surrounds a solid cylinder also with infinite permeability and length I but with smaller radius a so that there is a small gap g = b - a. An N 1 turn coil carrying a current I, is placed within two slots on the inner surface of the outer cylinder. (a) What is the magnetic field everywhere? Neglect all radial variations in the narrow air gap. (Hint: Separately consider 0 < 0 < 7r and ir < ( < 27-.) (b) What is the self-inductance of the coil? (c) A second coil with N 2 turns carrying a current 12 is placed in slots on the inner cylinder that is free to rotate. When the rotor is at angle 0, what is the total magnetic field due to currents I, and 12? (Hint: Separately consider 0< f <0, 0

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