Physics 212 Lecture 15 Ampere’s Law

Infinite current-carrying wire LHS: RHS: General Case :05

Practice on Enclosed Currents Checkpoint 1a Checkpoint 1b Checkpoint 1c Ienclosed = 0 Ienclosed = I Ienclosed = I Ienclosed = 0 For which loop is B·dl the greatest? A. Case 1 B. Case 2 C. Same For which loop is B·dl the greatest? A. Case 1 B. Case 2 C. Same For which loop is B·dl the greatest? A. Case 1 B. Case 2 C. Same :08

Cylindrical Symmetry Checkpoint 2a Enclosed Current = 0 An infinitely long hollow conducting tube carries current I in the direction shown. Cylindrical Symmetry Checkpoint 2a X Enclosed Current = 0 X X X 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

Line Integrals I into screen :12

Ampere’s Law dl B dl B dl B :14

Ampere’s Law dl B dl B dl B :16

Ampere’s Law dl B dl B dl B :16

Which of the following current distributions would give rise to the B Which of the following current distributions would give rise to the B.dL distribution at the right? A C B :18

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Match the other two: B A :21

Checkpoint 2b :22

Simulation :23

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. n = # turns/length :28

Example Problem I Conceptual Analysis 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. I a x Conceptual Analysis 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 For circular path concentric w/ shell The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Calculate B for the three regions separately: 1) r < a 2) a < r < b 3) r > b :31

Example Problem I so (A) (B) (C) r y I r a b x What does |B| look like for r < a ? so (A) (B) (C) :33

Example Problem I I (A) (B) (C) r y I r What does |B| look like for r > b ? a b x I (A) (B) (C) :35

Example Problem dl I LHS: B RHS: (A) (B) (C) y What does |B| look like for r > b ? dl I r LHS: B a b x RHS: (A) (B) (C) :36

Example Problem j = I / area I (A) (B) (C) y I What is the current density j (Amp/m2) in the conductor? a b x (A) (B) (C) j = I / area :40

Example Problem I (A) (B) (C) r y I r What does |B| look like for a < r < b ? a b x (A) (B) (C) :43

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

Example Problem y 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. I a x b :48

Follow-Up B will be zero if total current enclosed = 0 I I0 X y 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 ? X I a I0 x A) |I0| > |I| AND I0 into screen b B) |I0| > |I| AND I0 out of screen C) |I0| < |I| AND I0 into screen D) |I0| < |I| AND I0 out of screen E) There is no current I0 that can produce B = 0 there B will be zero if total current enclosed = 0 :48