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Exam 2 covers Ch. 27-33,
Lecture, Discussion, HW, Lab
Exam 2 is Tue. Oct. 27, 5:30-7 pm, 145 Birge
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Chapter 27: The Electric Field
Chapter 29: Electric potential & work
Chapter 30: Electric potential & field
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Chapter 31: Current & Resistance
Chapter 32: Fundamentals of Circuits
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(exclude 30.7)
(exclude 32.8)
Chapter 33: The Magnetic Field
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(exclude 33.5-33.6, 33.9-10, & Hall effect)
1
Law of Biot-Savart
B out of page
r

ds

Each short length of current
produces contribution to
magnetic field.
I in
plane of page
dB
r
ds
r
r o Ids  rˆ
dB 
4 r 2
Field from very short
r
section of current ds
r = distance from current element
o  4 107 N / A 2 = permeability of free space
Tue. Oct. 27, 2009

Physics 208 Lecture 16
2
r r
CD
Vector cross product
r
r o Ids  rˆ
dB 
4 r 2
r
D


r
dB
r oI r
dB 
ds  rˆ
2
4 r
Dist. to
point at
which field
is evaluated
Short length
of current
r
r
Unit vector
toward point
at which field
is evaluated

r
C

r
ds

Tue. Oct. 27, 2009

Physics 208 Lecture 16
3
Field from a circular loop
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Each current element produce dB
All contributions add as vectors
Along axis, all
components cancel
except for x-comp
Tue. Oct. 27, 2009
Physics 208 Lecture 16
4
Magnetic field from loop of current
Looks like magnetic dipole
Tue. Oct. 27, 2009
Physics 208 Lecture 16
5
Building a solenoid
Tue. Oct. 27, 2009
Physics 208 Lecture 16
6
Solenoid: many current loops
Bsolenoid 
Tue. Oct. 27, 2009
oNI
L
 onI
Physics 208 Lecture 16
7
Magnetic Force on a Current
I
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
Force on each charge qv  B
Force on length ds of wire
Ids  B

 section of
Force on straight
wire, 
length L
 r
r r
F  I  BL
N
Current
S
Magnetic force
Magnetic
field
Tue. Oct. 27, 2009
Physics 208 Lecture 16
8
Quick Quiz
A current I flows in a square loop of wire y
with side length L.
A constant B field points in the x-direction,
perpendicular to the plane of the loop.
 What is the net force on the wire loop?
L
I
I
I
A. 4LB
I
B. 2LB
x
C. LB
D. 0
Tue. Oct. 27, 2009
Physics 208 Lecture 16
9
No force, but torque
Lever arm
Force
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
r r
Torque is r  F
Net torque can be nonzero even when
net force is zero.

Tue. Oct. 27, 2009
Physics 208 Lecture 16
10
Question on torque
Which of these loop orientations has the largest magnitude
torque? Loops are identical apart from orientation.
(A) a (B) b (C) c
a
b
c
12/09/2002
U. Wisconsin, Physics 208, Fall 2006
11
Quick Quiz
Which of these different sized current loops has
the greatest torque from a uniform magnetic
field to the right? All have same current.
A.
L
B.
2L
W
W/2
C.
L/2
D. All same
r
B
2W
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Physics 208 Lecture 16
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Torque on current loop
  r  F
F


  2 F sin  
2

I
F  IB    AIBsin 
A
2
B
r

=loop area
I
Torque proportional
to
• Loop area
• Current
F
B
• sinθ
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Physics 208 Lecture 16
13
Current loops & magnetic dipoles
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Current loop produces magnetic dipole field.
Magnetic dipole moment: 

  IA
current
Area of
loop


direction
magnitude
In a uniform magnetic field
Magnetic field exerts torque    B,    B sin 
Torque rotates loop to align  with B
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Physics 208 Lecture 16
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Works for any shape planar loop
r
  IA
r
 perpendicular to loop
Torque in uniform magnetic field
I
   B,    B sin 
r r
Potential energy of rotation: U   B  Bcos

Lowest energy aligned w/ magnetic field

Highest energy perpendicular to magnetic field

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Physics 208 Lecture 16
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Magnetic flux
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
Magnetic flux is defined
B  B dA
exactly as electric flux
(Component of B  surface) x (Area element)

zero flux
Maximum flux
SI unit of magnetic flux is the Weber ( = 1 T-m2 )
Tue. Oct. 27, 2009
Physics 208 Lecture 16
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Magnetic Flux
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Magnetic flux  through a surface:
(component of B-field  surface) X (surface area)
Proportional to
# B- field lines
penetrating surface
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Physics 208 Lecture 16
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Why perpendicular component?

Suppose surface make angle  surface normal
r
B  B|| sˆ  B nˆ
A  Anˆ
nˆ
Component || surface
Component  surface
Only  component
 ‘goes through’ surface


sˆ

B = BA cos 
B =0 if B parallel A
 B = BA (max) if B  A


Flux SI units are T·m2=Weber
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Physics 208 Lecture 16

r r
M  B  A
18
Total flux

E not constant

add up small areas
where it is constant
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Surface not flat

add up small areas
where it is ~ flat
r
r
   BiA i cos   Bi  Ai
i
B
r
r
Add them all up: B   B  dA
surface
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Physics 208 Lecture 16
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Magnetic flux
What is that magnetic flux through this
surface?
A. Positive
B. Negative
C. Zero
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Physics 208 Lecture 16
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Properties of flux lines
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Net magnetic flux through any closed
surface is always zero:  magnetic  0

For electric charges, and
electric flux
electric 
Qenclosed
o
No magnetic ‘charge’,
so right-hand side=0 for mag.

Tue. Oct. 27, 2009
Basic magnetic element
is the dipole
Physics 208 Lecture 16
21
Time-dependent fields
Up to this point, have discussed only magnetic
and electric fields constant in time.
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E-fields arise from charges
B-fields arise from moving charges (currents)
Faraday’s discovery
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Another source of electric field
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Time-varying magnetic field creates electric field
Tue. Oct. 27, 2009
Physics 208 Lecture 16
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Measuring the induced field
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A changing magnetic flux produces an EMF
around the closed path.
How to measure this?
Use a real loop of wire for the closed path.
The EMF corresponds to a current flow:
  IR

Tue. Oct. 27, 2009
Physics 208 Lecture 16
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