Physics 212 Lecture 15 Amperes Law Physics 212

Physics 212 Lecture 15 Ampere’s Law Physics 212 Lecture 15, Slide 1

Infinite current-carrying wire LHS: RHS: General Case : 05 Physics 212 Lecture 15, Slide 2

Practice on Enclosed Currents Checkpoint 1 a Ienclosed = I For which loop is B·dl the greatest? A. Case 1 B. Case 2 C. Same : 08 Checkpoint 1 b Ienclosed = I Ienclosed = 0 For which loop is B·dl the greatest? A. Case 1 B. Case 2 C. Same Checkpoint 1 c Ienclosed = 0 For which loop is B·dl the greatest? A. Case 1 B. Case 2 C. Same Physics 212 Lecture 15, Slide 3

An infinitely long hollow conducting tube carries current I in the direction shown. Checkpoint 2 a Cylindrical Symmetry X X Enclosed Current = 0 Check cancellations What is the direction of the magnetic field inside the tube? A. clockwise B. counterclockwise C. radially inward to the center D. radially outward from the center E. the magnetic field is zero : 22 Physics 212 Lecture 15, Slide 4

Line Integrals I into screen : 12 Physics 212 Lecture 15, Slide 5

Ampere’s Law dl B dl : 14 B B dl Physics 212 Lecture 15, Slide 6

Ampere’s Law dl B B dl : 16 Physics 212 Lecture 15, Slide 7

Ampere’s Law dl B B dl : 16 Physics 212 Lecture 15, Slide 8

Which of the following current distributions would give rise to the B. d. L distribution at the right? A : 18 B C Physics 212 Lecture 15, Slide 9

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Match the other two: A : 21 B Physics 212 Lecture 15, Slide 13

Checkpoint 2 b : 22 Physics 212 Lecture 15, Slide 14

Simulation : 23 Physics 212 Lecture 15, Slide 15

Solenoid Several loops packed tightly together form a uniform magnetic field inside, and nearly zero magnetic field outside. 1 2 4 3 Assume a constant field inside the solenoid and zero field outside the solenoid. Apply Ampere’s law to find the magnitude of the field. : 28 n = # turns/length Physics 212 Lecture 15, Slide 16

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. • 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! For circular path concentric w/ shell Calculate B for the three regions separately: 1) r < a 2) a < r < b 3) r > b : 31 Physics 212 Lecture 15, Slide 17

Example Problem y I r a What does |B| look like for r < a ? b x so 0 (A) : 33 (B) (C) Physics 212 Lecture 15, Slide 18

Example Problem y I r a What does |B| look like for r > b ? b x I (A) : 35 (B) (C) Physics 212 Lecture 15, Slide 19

Example Problem y What does |B| look like for r > b ? dl LHS: I r B a RHS: (A) : 36 (B) b x (C) Physics 212 Lecture 15, Slide 20

Example Problem y I What is the current density j (Amp/m 2) in the conductor? (A) (B) a b x (C) j = I / area : 40 Physics 212 Lecture 15, Slide 21

Example Problem y I r a What does |B| look like for a < r < b ? (A) : 43 (B) b x (C) Physics 212 Lecture 15, Slide 22

Example Problem y What does |B| look like for a < r < b ? I r a b x Starts at 0 and increases almost linearly : 45 (A) (B) (C) Physics 212 Lecture 15, Slide 23

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. y a Sketch |B| as a function of r. I x b : 48 Physics 212 Lecture 15, Slide 24

Follow-Up y Add an infinite wire along the z axis carrying current I 0. What must be true about I 0 such that there is some value of r, a < r < b, such that B(r) = 0 ? A) |I 0| > |I| AND I 0 into screen a I I 0 X x b B) |I 0| > |I| AND I 0 out of screen C) |I 0| < |I| AND I 0 into screen D) |I 0| < |I| AND I 0 out of screen E) There is no current I 0 that can produce B = 0 there B will be zero if total current enclosed = 0 : 48 Physics 212 Lecture 15, Slide 25