1. Physics 212
Lecture 15
Ampere’s Law

2. A classic for a classic day
Music
Who is the Artist?
Oscar Peterson
Kenny Barron
Dave Brubeck
Thelonius Monk
Marcus Roberts
A classic for a classic day
Physics 212 Lecture 17

3. Hour Exam II – THURSDAY Mar. 29
Your Comments
“Pretty neat stuff. I like how similar this is to Gauss’s law.”
“Need more explanation on the magnetic field directions. ”
Easier method to calculate magnetic fields
“You should probably look at getting some of the images up and running properly, its hard to deduce anything unless I try to imagine what youre asking about, which in any case i would be right all the time.”
“I find this kind of confusing. Also most of the checkpoint pictures didn’t work for me, so I can’t go back and study from them. It makes it more confusing when I can’t see the pictures to answer questions..”
Sorry!
We’ll go through the checkpoints
(with pictures)
Spend some time building the integrals
“Integrals are my greatest enemy ”
“Daaag man, my boy Gauss be gettin’ his style cramped by dat Ampere clown.”
Hour Exam II – THURSDAY Mar. 29 05 3

4. Infinite current-carrying wire
LHS:
RHS:
General Case
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5. 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
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6. 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
“If you point your thumb in the direction of I, your fingers curl CW.”
“Force is tangent to the cylinder so according to the RHR the magnetic field must be radailly towards the middle ”
“The enclosed current is zero, and if you take a non-zero closed path, then we zeo that the field MUST be zero in order for amperes law to hold true”
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7. Ampere’s Law (+ integrals + magnetic field directions)
I into screen
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8. Ampere’s Law dl B dl B dl B
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9. Ampere’s Law dl B dl B dl B
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10. Ampere’s Law dl B dl B dl B
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11. 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
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12. :19

13. :19

14. :19

15. Match the other two: B A
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16. A current carrying wire is wrapped around a cardboard tube as shown below.
Checkpoint 2b
In which direction does the magnetic field point inside the tube?
A. Left B. Right C. Up D. Down E. Out of screen
Use the right hand rule and curl your fingers along the direction of the current.
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17. Simulation
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18. Solenoid
Several loops packed tightly together form a uniform magnetic field inside, and nearly zero magnetic field outside. 1 2 4 3
From this simulation, we can assume a constant field inside the solenoid and zero field outside the solenoid, and apply Ampere’s law to find the magnitude of the constant field inside the solenoid !!
n = # turns/length
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19. Similar to the Current Sheet
Total integral around the loop

20. 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.
Strategic Analysis
Calculate B for the three regions separately:
1) r < a
2) a < r < b
3) r > b
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21. Example Problem I so (A) (B) (C) ry I r a b x
What does |B| look like for r < a ? so
(A) (B) (C)
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22. Example Problem I I (A) (B) (C) ry I r
What does |B| look like for r > b ? a b x I
(A) (B) (C)
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23. 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)
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24. 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
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25. Example Problem I (A) (B) (C) ry I r
What does |B| look like for a < r < b ? a b x
(A) (B) (C)
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26. 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)
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27. 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
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28. Example Problem How big is B at r = b? I Suppose I = 10 A, b = 1 mmy
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
How big is B at r = b? b
Suppose I = 10 A, b = 1 mm
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29. Follow-Up B will be zero if total current enclosed = 0 I I0 Xy
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
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30. Ampere’s Law dl B dl B dl B
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31. B Fields and Forces Check
What are the directions of the field and the force due to I3 at I1? F B F B B F F B F B A. B. C. D. E.
What are the magnitudes of the field and the force due to I3 at I1? A. B. C.
B1 due to I3; F on I1