Lect15 - Physics 212 Lecture 15 Ampere’s Law B ⋅ d ℓ...

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Unformatted text preview: Physics 212 Lecture 15 Ampere’s Law B ⋅ d ℓ = µ0 I enclosed ∫ 40 30 20 10 0 Confused Avg = 3.2 Confident Physics 212 Lecture 15, Slide 1 Music Who is the Artist? BB A) B) C) D) E) Albert Collins Buddy Guy Coco Montoya John Mayall John Mayall Tommy Castro One of my favorite blues guitar players He does a great live show He Started out as drummer for Albert Collins Started Also played in Bluesbreakers with John Mayall Also Bluesbreakers with Mayall Physics 212 Lecture 15, Slide 2 Physics 212 Lecture 15 Ampere’s Law B ⋅ d ℓ = µ0 I enclosed ∫ 40 30 20 10 0 Confused Avg = 3.2 Confident Physics 212 Lecture 15, Slide 3 Infinite current-carrying wire B 2πr = µ0 I :05 µ0 I B= 2πr Physics 212 Lecture 15, Slide 4 Preflight 2 80 60 40 only depends on the current enclosed, only ∫ B ⋅ d ℓthe same in both cases. which is which is the same in both cases. :08 20 0 Physics 212 Lecture 15, Slide 5 Preflight 4 100 80 60 40 :09 In case 2, the loop does not enclose any current, so integral b.dl is zero. And in case 1, the loop does enclose current and therefore the value is non zero. 20 0 Physics 212 Lecture 15, Slide 6 Preflight 6 Enclosed current = 0 For both cases 70 60 50 40 30 20 10 0 :09 Physics 212 Lecture 15, Slide 7 Cylindrical Symmetry Preflight 8 X BB X X X Enclosed Current = 0 Check cancellations 50 40 30 20 10 0 :22 Physics 212 Lecture 15, Slide 8 Ampere’s Law I into screen ∫ B ⋅ dl = µ I 0 enc :12 Physics 212 Lecture 15, Slide 9 ∫ B ⋅ dl = µ I Ampere’s Law Ampere 0 enc dl B dl B dl :14 B Physics 212 Lecture 15, Slide 10 10 ∫ B ⋅ dl = µ I Ampere’s Law 0 enc dl B B dl B dl :16 Physics 212 Lecture 15, Slide 11 11 ∫ B ⋅ dl = µ I Ampere’s Law dl 0 enc B B dl B dl :16 Physics 212 Lecture 15, Slide 12 12 Which of the following current Which distributions would give rise to the B.dL distribution at the right? A :18 B BB C Physics 212 Lecture 15, Slide 13 13 :19 Physics 212 Lecture 15, Slide 14 14 :19 Physics 212 Lecture 15, Slide 15 15 :19 Physics 212 Lecture 15, Slide 16 16 Match the other two: A :21 B BB Physics 212 Lecture 15, Slide 17 17 Preflight 10 80 60 Use the right hand rule and curl your Use fingers along the direction of the current. current. :22 40 20 0 Physics 212 Lecture 15, Slide 18 18 Simulation :23 Physics 212 Lecture 15, Slide 19 19 Solenoid Several loops packed tightly together form a uniform magnetic field inside, and nearly zero magnetic field outside. 1 2 4 3 ∫ B ⋅ dl = µ I 0 enc 2 3 4 1 ∫ B ⋅d ℓ + ∫ B ⋅d ℓ + ∫ B ⋅d ℓ + ∫ B ⋅d ℓ = µ I 0 enc 1 2 3 BL + 0 + 0 + 0 = µ 0 I enc BL = µ 0 nLI :28 4 B = µ 0 nI Physics 212 Lecture 15, Slide 20 20 y Example Problem An infinitely long cylindrical shell with inner radius a and outer radius b carries a uniformly distributed current I into the screen. a Sketch |B| as a function of r. • Conceptual Analysis – – I x b Complete cylindrical symmetry (can only depend on r) ⇒ can use Ampere’s law to calculate B B field can only be clockwise, counterclockwise or zero! ∫ B ⋅ dl = µ I B ∫ dl = µ I 0 enc 0 enc For circular path concentric w/ shell • Strategic Analysis Calculate B for the three regions separately: 1) r < a 2) a < r < b 3) r > b :31 Physics 212 Lecture 15, Slide 21 21 y Example Problem I BB r a What does |B| look like for r < a ? ∫ B ⋅ dl = µ I 0 enc b x so B = 0 0 (A) (A) :33 (B) (B) (C) Physics 212 Lecture 15, Slide 22 22 y Example Problem I r a What does |B| look like for r > b ? BB b x ∫ B ⋅ dl = µ I 0 enc I (A) (A) :35 (B) (B) (C) Physics 212 Lecture 15, Slide 23 23 y Example Problem dl r I B a What does |B| look like for r > b ? b x ∫ B ⋅ dl = µ I 0 ∫ Bdl B ∫ dl B 2π r :36 B 2π r = µ0 I µ0 I B= 2π r Physics 212 Lecture 15, Slide 24 24 y Example Problem I r a What does |B| look like for r > b ? B= (A) (A) :37 b x µ0 I 2π r (B) (B) (C) Physics 212 Lecture 15, Slide 25 25 y Example Problem I a What is the current density j (Amp/m2) in the conductor? (A) (A) :40 I j= 2 πb (B) (B) I j= 2 2 πb +πa (C) BB b x I j= 2 2 πb −πa Physics 212 Lecture 15, Slide 26 26 y Example Problem I a What is the current density j (Amp/m2) in the conductor? j = I / area b x area = π b2 − π a 2 I j= 2 2 πb −πa :41 Physics 212 Lecture 15, Slide 27 27 y Example Problem I a What is the current density j (Amp/m2) in the conductor? (A) (A) :42 I j= 2 πb (B) (B) I j= 2 2 πb +πa (C) b x I j= 2 2 πb −πa Physics 212 Lecture 15, Slide 28 28 y Example Problem I r a What does |B| look like for a < r < b ? (A) (A) :43 (B) (B) BB b x (C) Physics 212 Lecture 15, Slide 29 29 y Example Problem I r a What does |B| look like for a < r < b ? b x ∫ B ⋅ dl = µ I 0 enc 2π rB = µ0 j areaencloses areaenc = π r − π a 2 2 I 2π rB = µ0 (π r − π a ) × π b2 − π a 2 ) ( 2 B = µ0 :45 (π r 2 −πa ) I × π b2 − π a 2 ) 2π r ( 2 2 Starts at 0 and increases almost linearly Physics 212 Lecture 15, Slide 30 30 y Example Problem I r a What does |B| look like for a < r < b ? (A) (A) :46 (B) (B) b x (C) Physics 212 Lecture 15, Slide 31 31 y Example Problem An infinitely long cylindrical shell with inner radius a and outer radius b carries a uniformly distributed current I out of the screen. a Sketch |B| as a function of r. I x b :48 Physics 212 Lecture 15, Slide 32 32 y Follow-Up Add an infinite wire along the z axis carrying current I0. What must be true about I0 such that there is some value of r, a < r < b, such that B(r) = 0 ? A) |I0| > |I| AND I0 into screen A) |I a I I0 X BB x b B) |I0| > |I| AND I0 out of screen B) |I C) |I0| < |I| AND I0 into screen C) |I D) |I0| < |I| AND I0 out of screen D) |I E) There is no current I0 that can produce B = 0 there E) There B will be zero if total current enclosed = 0 :48 Physics 212 Lecture 15, Slide 33 33 ...
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