Lect15 - rr B ⋅ d l = μ0 I enclosed ∫ 40 30 20 10 0...

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Unformatted text preview: rr B ⋅ d l = μ0 I enclosed ∫ 40 30 20 10 0 Lecture 15 Ampere’s Law Physics 212 Confused Avg = 3.3 Confident Physics 212 Lecture 15, Slide 1 Physics Infinite current-carrying wire B 2πr = μ0 I μ0 I B= 2πr Physics 212 Lecture 15, Slide 2 Physics Preflight 2 80 60 40 Both cases have the same current flowing through 20 them. Similar to a point charge in Gauss's Law, the magnitude of the magnetic field of the loop is not 0 dependent on the place of the wire, since the enclosed Physics 212 Physics current is the same. Lecture 15, Slide 3 Preflight 4 80 60 40 The loop in case 2 encloses no current, so integral B*dl is 0. 20 0 Physics 212 Lecture 15, Slide 4 Physics Preflight 6 Enclosed current = 0 for both cases 70 60 50 40 30 20 10 0 Physics 212 Lecture 15, Slide 5 Physics Preflight 8 Cylindrical Symmetry X X X X BB Enclosed Current = 0 Check cancellations 50 40 30 20 10 0 Physics 212 Lecture 15, Slide 6 Physics Ampere’s Law I into screen rr ∫ B ⋅ dl = μ0 I enc Physics 212 Lecture 15, Slide 7 Physics Ampere’s Law Ampere rr ∫ B ⋅ dl = μ0 I enc dl B dl B dl B Physics 212 Lecture 15, Slide 8 Physics Ampere’s Law rr ∫ B ⋅ dl = μ0 I enc dl B B dl Physics 212 Lecture 15, Slide 9 Physics dl B Ampere’s Law rr ∫ B ⋅ dl = μ0 I enc dl B B dl B dl Physics 212 Lecture 15, Slide 10 Physics Which of the following current distributions would give rise to the B.dL distribution at the right? BB A B C Physics 212 Lecture 15, Slide 11 Physics Physics 212 Lecture 15, Slide 12 Physics Physics 212 Lecture 15, Slide 13 Physics Physics 212 Lecture 15, Slide 14 Physics Match the other two: A B BB Physics 212 Lecture 15, Slide 15 Physics Preflight 10 80 60 40 Curl fingers around the current to get the thumb to point at the magnetic field. 20 0 Physics 212 Lecture 15, Slide 16 Physics Simulation Physics 212 Lecture 15, Slide 17 Physics Solenoid Several loops packed tightly together form a uniform magnetic field inside, and nearly zero magnetic field outside. 1 2 rr ∫ B ⋅ dl = μ0 I enc 2 4 3 r r 3r r 4r r 1r r ∫ B ⋅d l + ∫ B ⋅d l + ∫ B ⋅d l + ∫ B ⋅d l = μ0 I enc 1 2 3 4 BL + 0 + 0 + 0 = μ 0 I enc BL = μ 0 nLI B = μ 0 nI Physics 212 Lecture 15, Slide 18 Physics An infinitely long cylindrical shell with inner radius a and outer radius b carries a uniformly distributed current I out of the screen. Sketch |B| as a function of r. Example Problem y a I x • Conceptual Analysis – – 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! rr ∫ B ⋅ dl = μ0 I enc rr B ∫ dl = μ 0 I 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 Physics 212 Lecture 15, Slide 19 Physics Example Problem r What does |B| look like for r < a ? y I a b BB x rr ∫ B ⋅ dl = μ0 I enc 0 r so B = 0 (A) (B) (C) Physics 212 Lecture 15, Slide 20 Physics Example Problem r What does |B| look like for r > b ? y I a b BB x rr ∫ B ⋅ dl = μ0 I enc I (A) (B) (C) Physics 212 Lecture 15, Slide 21 Physics Example Problem dl What does |B| look like for r > b ? r y I a b x B rr ∫ B ⋅ dl = μ0 I ∫ Bdl B ∫ dl B 2π r = μ 0 I B 2π r μ0 I B= 2π r Physics 212 Lecture 15, Slide 22 Physics Example Problem r What does |B| look like for r > b ? y I a b x B= μ0 I 2π r (A) (B) (C) Physics 212 Lecture 15, Slide 23 Physics Example Problem y I What is the current density j (Amp/m2) in the conductor? a b BB x (A) I j= 2 πb (B) I j= 2 2 πb +πa (C) I j= 2 2 πb −πa Physics 212 Lecture 15, Slide 24 Physics Example Problem y I What is the current density j (Amp/m2) in the conductor? a b x j = I / area area = π b 2 − π a 2 I j= 2 2 πb −πa Physics 212 Lecture 15, Slide 25 Physics Example Problem y I What is the current density j (Amp/m2) in the conductor? a b x (A) I j= 2 πb (B) I j= 2 2 πb +πa (C) I j= 2 2 πb −πa Physics 212 Lecture 15, Slide 26 Physics Example Problem r What does |B| look like for a < r < b ? y I a b BB x (A) (B) (C) Physics 212 Lecture 15, Slide 27 Physics Example Problem r What does |B| look like for a < r < b ? y I a b x rr ∫ B ⋅ dl = μ0 I enc 2π rB = μ0 j areaencloses 2 2 areaenc = π r 2 − π a 2 I 2π rB = μ0 (π r − π a ) × π b2 − π a 2 ) ( B = μ0 (π r 2 −πa ) I × π b2 − π a 2 ) 2π r ( 2 Starts at 0 and increases almost linearly Physics 212 Lecture 15, Slide 28 Physics Example Problem r What does |B| look like for a < r < b ? y I a b x (A) (B) (C) Physics 212 Lecture 15, Slide 29 Physics An infinitely long cylindrical shell with inner radius a and outer radius b carries a uniformly distributed current I out of the screen. Sketch |B| as a function of r. Example Problem a y I x b Physics 212 Lecture 15, Slide 30 Physics 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 Follow-Up y I I0 X BB x A) |I0| > |I| AND I0 into screen B) |I0| > |I| AND I0 out of screen C) |I0| < |I| AND I0 into screen D) |I0| < |I| AND I0 out of screen b E) There is no current I0 that can produce B = 0 there B will be zero if total current enclosed = 0 Physics 212 Lecture 15, Slide 31 Physics ...
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This note was uploaded on 02/21/2011 for the course PHYS 212 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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