MIT OpenCourseWare http://ocw.mit.edu Electromagnetic Field Theory: A Problem Solving Approach For any use or distribution of this textbook, please cite as follows: Markus Zahn, Electromagnetic Field Theory: A Problem Solving Approach. (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.ProMems 465(11) can be rewritten asF.= (A -,) O(H +H ) (14)2 axThe total force is thenf =sD Fdx= ( -l>o)sD( H2+H ) =o(IL-Io) N2,2D= )(15)2 swhere the fields at x = -0o are zero and the field at x = xo isgiven by (12). High permeability material is attracted toregions of stronger magnetic field. It is this force that causesiron materials to be attracted towards a magnet. Diamagneticmaterials (A <p0o) will be repelled.This same result can more easily be obtained using (6)where the flux through the gap isNIDQ = HD[jix +po(a-x)]= -NID[(L -po)x+alo] (16)so that the inductance isNQ N2DL = [(p -o)x + aizo] (17)I sThen the force obtained using (6) agrees with (15)f 1 21dL(x)dx-(-Ao) N212D (18)2sPROBLEMSSection 6-11. A circular loop of radius a with Ohmic conductivity oa andcross-sectional area A has its center a small distance D awayfrom an infinitely long time varying current.Cross-sectional area A-- --I,(a) Find the mutual inductance M and resistance R of theloop. Hint:dx 2 tan-' [JL-' tan (x/2)]a+b cosx = ' a+bJ rdr _(b) This loop is stationary. and has a self-inductance L.What is the time dependence of the induced short circuitcurrent when the line current is instantaneously stepped onto a dc level I at t = 0?(c) Repeat (b) when the line current has been on a longtime and is suddenly turned off at t = T.(d) If the loop has no resistance and is moving with radialvelocity v, = dr/dt, what is the short circuit current and opencircuit voltage for a dc line current?(e) What is the force on the loop when it carries a currenti ? Hint:+aCos d = -- sin [cos ;]D+a cos •b aD ._,a +D cos \)a/D/ a D+a cos /2. A rectangular loop at the far left travels with constantvelocity Ui. towards and through a dc surface current sheetKoi, at x = 0. The right-hand edge of the loop first reachesthe current sheet at t = 0.(a) What is the loop's open circuit voltage as a function oftime?(b) What is the short circuit current if the loop has self-inductance L and resistance R?(c) Find the open circuit voltage if the surface current isreplaced by a fluid with uniformly distributed volume cur-rent. The current is undisturbed as the loop passes through.466 Electromagnetic InductionI(t)I_·,-·Problems 467Koiy'1Specifically consider the case when d > b and then sketch theresults when d = b and d < b. The right edge of the currentloop reaches the volume current at t = 0.3. A short circuited rectangular loop of mass m and self-inductance L is dropped with initial velocity voi. between thepole faces of a magnet that has a concentrated uniformmagnetic field Boil. Neglect gravity.x v0(a) What is the imposed flux through the loop as a functionof the loop's position x (0 < x <s) within the magnet?(b) If the wire has conductivity ao and cross-sectional areaA, what equation relates the induced current i in the loop andthe loop's velocity?(c) What is the force on the loop in terms of i? Obtain asingle equation for the loop's velocity. (Hint: Let w0 =Bob2/mL, a = RIL.)(d) How does the loop's velocity and induced current varywith time?(e) If r-+oo, what minimum initial velocity is necessary forthe loop to pass through the magnetic field?4. Find the mutual inductance between the following cur-rents:(a) Toroidal coil of rectangular or circular cross sectionI -•Ut -b --Ut:--....(c)I Llia-bforoidross-section-a--D(b)coaxially centered about an infinitely long line current. Hint:Sdx 2fa+bcosx =tana b cos x Ja-6-1 I ?r tan(x/2)}a+b ,J r dr(b) A very long rectangular current loop, considered as twoinfinitely long parallel line currents, a distance D apart, car-rying the same current I in opposite directions near a smallrectangular loop of width a, which is a distance d away fromthe left line current. Consider the cases d +a <D, d <D <d +a, and d>D.5. A circular loop of radius a is a distance D above a pointmagnetic dipole of area dS carrying a current II.2 DI1dS468Electromagnetic InductionectlonII_Problems 469(a) What is the vector potential due to the dipole at allpoints on the circular loop? (Hint: See Section 5-5-1.)(b) How much flux of the dipole passes through the circu-lar loop?(c) What is the mutual inductance between the dipole andthe loop?(d) If the loop carries a current 12, what is the magneticfield due to 12 at the position of the point dipole? (Hint: SeeSection 5-2-4a.)(e) How much flux due to 12 passes through the magneticdipole?(f) What is the mutual inductance? Does your result agreewith (c)?6. A small rectangular loop with self-inductance L, Ohmicconductivity a, and cross-sectional area A straddles a currentsheet.,ýK(t)IItS(a) The current sheet is instantaneously turned on to a dclevel Koi, at t = 0.What is the induced loop current?(b) After a long time T the sheet current is instantaneouslyset to zero. What is the induced loop current?(c) What is the induced loop current if the current sheetvaries sinusoidally with time as Ko cos ot i,.7. A point magnetic dipole with area dS lies a distance dbelow a perfectly conducting plane of infinite extent. Thedipole current I is instantaneously turned on at t= 0.(a) Using the method of images, find the magnetic fieldeverywhere along the conducting plane. (Hint: i, •i, =sin 0,d=dS= wa2I470 Electromagnetic Inductionis ir = Cos 0.)(b) What is the surface current distribution?(c) What is the force on the plane? Hint:Sr3 dr (r2 + d'/4)(r2 +d2)5 6(r 2+d 2)4(d) If the plane has a mass M in the gravity field g, whatcurrent I is necessary to just lift the conductor? Evaluate forM= 10-s kg, d = 10- m, and a = 10-3 m.8. A thin block with Ohmic conductivity o and thickness 8moves with constant velocity Vi, between short circuitedperfectly conducting parallel plates. An initial surface currentKo is imposed at t = 0 when x = xo, but the source is thenremoved.. xix Depth D(a) The surface current on the plates K(t) will vary withtime. What is the magnetic field in terms of K(t)? Neglectfringing effects.(b) Because the moving block is so thin, the current
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