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 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 p0o will be repelled This same result can more easily be obtained using 6 where the flux through the gap is NID Q HD jix po a x NID L po x alo 16 so that the inductance is L NQ N 2D I s p o x aizo 17 Then the force obtained using 6 agrees with 15 f 11 2 dL x dx Ao N 2 1 2D 2s 18 PROBLEMS Section 6 1 1 A circular loop of radius a with Ohmic conductivity oa and cross sectional area A has its center a small distance D away from an infinitely long time varying current 466 Electromagnetic Induction I t Cross sectional area A I a Find the mutual inductance M and resistance R of the loop Hint dx a b cosx J rdr 2 tan JL tan x 2 a b b This loop is stationary and has a self inductance L What is the time dependence of the induced short circuit current when the line current is instantaneously stepped on to a dc level I at t 0 c Repeat b when the line current has been on a long time and is suddenly turned off at t T d If the loop has no resistance and is moving with radial velocity v dr dt what is the short circuit current and open circuit voltage for a dc line current e What is the force on the loop when it carries a current i Hint aCos d b a sin D a cos cos D a D a a D cos D a cos 2 A rectangular loop at the far left travels with constant velocity Ui towards and through a dc surface current sheet Koi at x 0 The right hand edge of the loop first reaches the current sheet at t 0 a What is the loop s open circuit voltage as a function of time b What is the short circuit current if the loop has selfinductance L and resistance R c Find the open circuit voltage if the surface current is replaced by a fluid with uniformly distributed volume current The current is undisturbed as the loop passes through I Problems 467 1Koiy I Lli I Ut b Ut c Specifically consider the case when d 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 s within the magnet b If the wire has conductivity ao and cross sectional area A what equation relates the induced current i in the loop and the loop s velocity c What is the force on the loop in terms of i Obtain a single equation for the loop s velocity Hint Let w0 Bob 2 mL a RIL d How does the loop s velocity and induced current vary with time e If r oo what minimum initial velocity is necessary for the loop to pass through the magnetic field 4 Find the mutual inductance between the following currents a Toroidal coil of rectangular or circular cross section 468 Electromagnetic Induction a b ectlon foroid ross section a D b coaxially centered about an infinitely long line current Hint Sdx tan 2 fa bcosx a b cos x Ja 6 J 1 I r tan x 2 a b r dr b A very long rectangular current loop considered as two infinitely long parallel line currents a distance D apart carrying the same current I in opposite directions near a small rectangular loop of width a which is a distance d away from the 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 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 K 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 …
View Full Document
Unlocking...