Sources of the Magnetic Field. Moving charges currents Ampere s Law Gauss Law in magnetism Magnetic materials

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1 Souces of the Magnetic Field Moving chages cuents Ampee s Law Gauss Law in magnetism Magnetic mateials

2 Biot-Savat Law ˆ ˆ θ ds P db out I db db db db ds ˆ 1 I P db in db db ds sinθ db μ 4 π 0 Ids ˆ B μ0i ds ˆ 4π μ 4π T. m / A pemeability of fee space

3 Magnetic Field Suounding a Thin, Staight Conducto ds ˆ kˆ ds ˆ kˆ dx cosθ db db μ I dx cosθ 4π ( ) kˆ 0 db kˆ μ0i dx cosθ 4π a cosθ x a tanθ dx a sec adθ θdθ cos θ db μ I a cos 4π a θ cosθdθ cos θ 0 μ0i cosθdθ 4πa 0I θ μ0i cosθdθ 1 sin a 4π θ μ B 4π a 1 ( sinθ θ ) If the wie is vey long, θ1 90 θ 90 then B μ0i πa

4 Field Due to a Cicula Ac of Wie μ0 ids sin 90 db 4 π R ds Rdφ B B μ i 4πR φ 0 db d μ0iφ 4πR 0 φ μ B 0i Full Cicle (φ π) R

5 Magnetic Foce Between Two Paallel Conductos l a I I a I l I lb I F π μ π μ F B F F 1 a I I l F B π μ 1 0

6 Concept Question On a compute chip, two conducting stips cay chage fom P to Q and fom R to S. If the cuent diection is evesed in both wies, the net magnetic foce of stip 1 on stip 1. emains the same.. eveses. 3. changes in magnitude, but not in diection. 4. changes to some othe diection. 5. othe

7 Ampèe s Law. μ0i B ds B ds 0 π μ B. ds μ I 0 ( π) I A line integal of B.ds aound a closed path equals μ 0 I, whee I is the total continuous cuent passing though any suface bounded by the closed path.

8 Long Cuent Caying Wie Fo > R B. ds B ds B μ0 μ0i B π Fo < R I π I πr I I R B ( π ) I B. s B μ ( π ) μ I I R 0 0 d μ0i πr

9 The Tooid By symmety, B is constant ove loop 1 and tangent to it B. d s Bds ( π ) NI B. ds B ds B μ0 B μ0ni π Outside the tooid: B 0 (Conside loop whose plane is pependicula to the sceen)

10 The Solenoid

11 The Magnetic Field of a Solenoid Conside loop Along path and 4, B is pependicula to ds Along path 3, B0 B. ds B. ds B ds path1 B. ds μ0ni path1 Bl B μ ni 0 whee n N l

12 Magnetic Flux Φ B. B da (WebeWbT.m ) Fo a unifom field making an angle θ with the suface nomal: Φ B BAcosθ Φ B 0 Φ B Φ B, max AB

13 Gauss Law in Magnetism Unlike electical fields, magnetic field lines end on themselves, foming loops. (no magnetic monopoles) B. da 0 Electic Field Lines Magnetic Field Lines

14 Magnetic Moments of Atoms Cuent caying loop B μ 0 I R μ IA Electon in obit T π ω I e T ω v eω π ev μ IA π π e μ m L n L m v ev π e ev e e m e h Electon spin e 4 μspin J / T m h e

15 Magnetic Moments of Atoms The net obital magnetic moment in most substances is zeo o vey small since the moments of electons in obit cancel each othe out. Similaly, the electon spin does not contibute a lage magnetic moment in substances with an even numbe of electons because electons pai up with ones of opposite spin.

Feomagnetism In cetain cystals, neighboing goups of atoms called

A feomagnetic cystal can be given a pemanent magnetization by applying

Unmagnetized cystal has domains of aligned magnetic moments.

16 Feomagnetism In cetain cystals, neighboing goups of atoms called domains have thei magnetic moments aligned. A feomagnetic cystal can be given a pemanent magnetization by applying an extenal field. Examples include Fe, Co, Ni. Unmagnetized cystal has domains of aligned magnetic moments. An extenal field inceases the sizes of the domains aligned with it. As the extenal field gets lage, the unaligned domains become smalle.

17 Paamagnetism Paamagnetic substances have a small but positive magnetism that comes fom atoms with pemanent magnetic moments. An extenal field can align these moments fo a net magnetization but the effect is weak and not pemanent. Cetain oganic compounds such as myoglobin ae paamagnetic.

18 Diamagnetism A vey small effect caused by the induction of an opposing field in the atoms by an extenal field. Supeconductos exhibit pefect diamagnetism (Meissne effect).

19 Summay Biot-Savat Law gives the magnetic field due to a cuent caying wie. Ampee s Law can simplify these calculations fo cases of high symmety. The magnetic flux though a closed suface is zeo. Solenoids and tooids can confine magnetic fields. Obital and spin magnetic moments of electons give ise to magnetism in matte.

20 Fo Next Class Reading Assignment Chapte 31 Faaday s Law WebAssign: Assignment 8

PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101

PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101 PHY 114 A Geneal Physics II 11 AM-1:15 PM TR Olin 11 Plan fo Lectue 1 Chaptes 3): Souces of Magnetic fields 1. Pemanent magnets.biot-savat Law; magnetic fields fom a cuent-caying wie 3.Ampee Law 4.Magnetic