? this lecture. ? next lecture. What we have learned so far. a Q E F = q E a. F = q v B a. a Q in motion B. db/dt E. de/dt B.

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1 PHY 249 Lectue Notes Chapte 32: Page 1 of 12 What we have leaned so fa a a F q a a in motion F q v a a d/ Ae thee othe "static" chages that can make -field? this lectue d/? next lectue da dl Cuve Cuve µ enc dl enclosed ε t da A. Koytov

2 PHY 249 Lectue Notes Chapte 32: Page 2 of 12 Ae thee magnetic chages, o monopoles? The dipole field of a magnet ba is identical to the electic dipole field. Can one beak off the Noth-pole piece fom the est of the magnet (simila to taking away a positive chage fom an electic dipole)? N No! t does not wok One gets two dipole magnet bas. N S N S N S Loop This can be undestood, if the magnet ba field is made of many little dipole fields. f one adds these dipoles togethe, the net field is again a dipole field. All attempts to find monopoles (on eath, deep in oceans, in cosmic ay paticles) o ceate them in little big bangs poduced at high enegy acceleatos/collides failed Φ da A. Koytov

3 PHY 249 Lectue Notes Chapte 32: Page 3 of 12 Magnetism in matte lectons obiting a nucleus make little Magnetic Dipoles: Loop The magnetic field on the z-axis of a cuent loop with aea AπR 2 and cuent is given by z (z) 2kµ/z 3, when z >> R, whee the magnetic dipole moment µ A. 2 2 Note that the obital magnetic moment µ ob πr e /(2πR / v) πr e / 2Rv can be witten in tems of the obital angula momentum, L ob p Rmv, as follows e µ ob ob 2 m L. e lectons themselves spin like a top with some constant angula moment (non-stoppable!). So electons themselves behave like little Magnetic Dipoles. Nuclei, being made of chaged potons and neutal neutons, also spin same way as electons. So nuclei also behave like little Magnetic Dipoles (potons and neutons ae 2 times heavie than electons so thei contibution is not as significant). Depending on inteplay of all these intenal magnetic dipoles, mateials can be subdivided in fou diffeent categoies: paamagnetics diamagneics feomagnetics supeconductos When exposed to an extenal magnetic field these mateials will modify it, in thee vey diffeent ways! Recall: mateials also change electic field: dielectics make it smalle conductos make it zeo A. Koytov

4 PHY 249 Lectue Notes Chapte 32: Page 4 of 12 Paamagnetics Al, oxygen, Paamagnetics tend to slightly incease magnetic field. Obital and spin magnetic moments do not cancel. Atoms of these mateials have pemanent magnetic dipole moment µ. A loop with a cuent, being placed in extenal -field will otate and its magnetic dipole moment (-filed) will be paallel to the extenal field: F loop F Once lined up (we shall call it magnetized), it will make the extenal field a bit stonge: induced K m, whee K m is of the ode of 1.1 at oom tempeatues Themal inteactions (collisions) will tend to destoy the alignment: the highe tempeatue, the less magnetization is. Paamagnetics ae attacted towads stong magnetic field fom weake field egions. Thei induced field is in the same diection as the extenal field, so the situation eminds two magnets facing each othe with opposite poles: S N A. Koytov

5 PHY 249 Lectue Notes Chapte 32: Page 5 of 12 Diamagnetics Cu, Si, C(dimond), Diamagnetics tend to slightly decease magnetic field. Obital and spin magnetic moments in these atoms cancel. ualitative classical (naïve) explanation: Conside electons obiting inside atoms. Half of them go clockwise, half counteclockwise. When one tuns a magnetic field on (into the page), the changing -filed will induce counteclockwise emf in the all loops, thus, acceleating electons going clockwise and slowing down those going counteclockwise. The magnetic dipole moments of the fome (pointing out of page) will incease; the magnetic dipole moments of the latte (pointing into the page) will decease. The net esult is a small field pointing out of the page, opposite to the extenal -field. v 1 2 v 1 2 Thus, the oveall magnetic filed becomes a bit smalle: induced K m, whee K m is of the ode of.9999 Since diamagnetics poduce field opposite to the extenal one, these mateials ae epelled fom stong magnetic field (just as two magnets facing each othe with the same poles). A. Koytov

6 PHY 249 Lectue Notes Chapte 32: Page 6 of 12 Feomagnetics Fe, Ni, Co, a few othe ae elements Feomagnetics incease magnetic field thousands-fold. Due to quantum physical inteactions between electon spins, magnetic moments of atoms spontaneously line up (as long as the tempeatue is kept below some citical limit, e.g. 143 K fo ion). The spontaneous alignment is diffeent in diffeent pats of ion. The smallest chunks with the same alignment ae called domains. Since all domains have thei diection of alignment the net field is zeo. The extenal field as week as ~1-3 T will help to e-oient almost all incoectly aligned domains and the net field poduced by feomagnetic eaches its maximum ~1 T. induced µ, whee µ~ , when is vey week Howeve, once the satuation is eached, the stengthening of the extenal magnetic field is not as spectacula, anymoe..g., if one has 1 T extenal field, the net field is just twice lage: 1 T extenal field itself plus 1 T fom magnetized feomagnetic induced ~1-2 T and pactically -independent, once satuation is eached Hysteesis: Once the field is emoved, domains may stay aligned (the level of memoy depends on the alloy) and the feomagnetic becomes pemanently magnetized A. Koytov

7 PHY 249 Lectue Notes Chapte 32: Page 7 of 12 Supeconductos (Many metals at vey low tempeatues of a few K) (Some ceamics at modestly low tempeatues of ~1 K) Supeconductos expel magnetic field completely -field inside supeconductos is always zeo. Conside fee electons inside. When one tuns a magnetic field on (into the page), the changing -filed will induce counteclockwise emf that would cause cuents: emf / R Since R is exactly zeo, this would cause infinitely lage cuents What actually happens is that the fee electons instantaneously acceleate and come in motion to pevent the change of the -field flux. A. Koytov

8 PHY 249 Lectue Notes Chapte 32: Page 8 of 12 Maxwell's quations as we know them so fa. Gauss' Law ( ): Φ da enclosed ε Chage. Gauss' Law fo Magnetism (No Magnetic Chages!): Φ da. Faaday's Law of nduction (d/ ): Cuve dl Changing Magnetic Field V. Ampee's Law ( in motion ): dl Cuve µ enclosed Cuent A. Koytov

9 PHY 249 Lectue Notes Chapte 32: Page 9 of 12 Finding the Missing Tem (d/?) ε + - C S 2 R S 1 C 1 We ae looking fo a new tem in Ampee's Law of the fom, C1 dl µ + δ whee δ is an unknown constant and Φ S d A S is any suface bounded by the cuve C 1.,, Case (use suface S 1 ): f we use the suface S 1 which is bounded by the cuve C 1 then C 1 dl since though the suface S 1. µ + δ Must be equal! Case (use suface S 2 ): f we use the suface S 2 which is bounded by the cuve C 1 then dl C 1 since though the suface S 2. µ δ δ + da t S1 µ, δ A t, Capacito: Thus, δ µ ε σ ε ε A and t 1 ε A d ε A. And finally, Ampee's Law (complete): dl µ + µ ε Cuve. A. Koytov

10 PHY 249 Lectue Notes Chapte 32: Page 1 of 12 Complete Maxwell's quations. Gauss' Law ( ): Φ da enclosed ε Chage. Gauss' Law fo Magnetism (No Magnetic Chages!): Φ da. Faaday's Law of nduction (d/ ): Cuve dl Changing Magnetic Field V. Ampee's Law ( in motion, d/ ): dl µ + µ ε Cuve Cuent Changing lectic Field A. Koytov

11 PHY 249 Lectue Notes Chapte 32: Page 11 of 12 What s the deal with monopoles? f we had monopoles, Maxwell s equations would be vey symmetical: Φ Φ da da δ 1 enclosed m ε Cuve dl δ 2 & m dl µ + µ ε Cuve d Φ Note that the thid equation povides a neat way of detecting monopoles no matte how slow they move. A supeconducting coil cannot have inside (), othewise, it would ceate infinitely lage cuents. Thus, the -filed integal is zeo: δ & 2 y taking a time-integal fom this equation, one can see that the change of magnetic field flux would indicate a passage of a monopole: so m t final t final δ & 2 m t initial t initial that Φ 2 m Afte passage of a monopole, the flux changes and stays changed foeve! δ A. Koytov

12 PHY 249 Lectue Notes Chapte 32: Page 12 of 12 lectic & Magnetic Fields that Change with Time Changing Magnetic Field Poduces an lectic Field: -out inceasing with time A unifom magnetic field is confined to a cicula egion of adius,, and is inceasing with time. What is the diection and magnitude of the induced electic field at the adius? Answe: f choose the loop oientation to be counteclockwise then my thumb would give positive diection out of page ( is positive and d/ also is positive). Φ (t)a with A π 2. Faaday's Law of nduction tells us that d Φ dl Cicle 2 d, 2π ( ) π and hence () -(/2) d/. Since d/ > (inceasing with time), is negative which means that it points opposite to my chosen oientation. Changing lectic Field Poduces a Magnetic Field: -out inceasing with time A unifom electic field is confined to a cicula egion of adius,, and is inceasing with time. What is the diection and magnitude of the induced magnetic field at the adius? Answe: f choose the loop oientation to be counteclockwise then Φ (t)a with A π 2. Ampee's Law (with J ): 2 π d dl 2π( ) ε µ 2 c, Cicle and hence () (/2c 2 ) d/. Since d/ > (inceasing with time), is positive which means that it points in the diection of my chosen oientation. Note: Choice of the loop oientation (counteclockwise o clockwise) is abitay the answe should not depend on you choice. Fo example: if you had chosen the loop oientation in the fist example to be clockwise, you thumb would indicate positive diection (fo -field flux and its deivative) to be into the page. Then, the flux would be negative, d/ would also be negative. Theefoe, the integal dl must be positive. Thus, -filed must be along the positive loop diection, i.e. clockwise same answe! A. Koytov