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The story so far…
dB
dI
Magnetic field
generated by current
element: Biot-Savart
r
o Ids  rˆ
dB 
4 r 2
I
Ampere’s law
 B ds   I
closed
path
o
surface
bounded by path
Exam 2 results
Phy208 Exam 2
35
Ave=70
30
Count
25
20
15
10
5
0
20
40
60
80
100
Grade
boundaries:
A:83
AB: 76
B: 67
BC: 57
C: 43
D: 23
Score
Thur. Oct. 30, 2008
Physics 208, Lecture 18
2
Magnetic field from a current loop

One loop:
field still loops around the wire.

Thur. Oct. 30, 2008
Many loops: same effect
Physics 208, Lecture 18
3
Building a solenoid
Thur. Oct. 30, 2008
Physics 208, Lecture 18
4
Ampere’s law for the solenoid
 B  ds   I
o through
 oNI
 B  ds  BL
Bsolenoid 
oNI
L
 onI

Thur. Oct. 30, 2008
Physics 208, Lecture 18
5
Ampere’s law
Sum up component of B around path
Equals current through surface.
Component of B
along path
 B ds   I
o

Ampere’s law
B
I
closed
path

surface
bounded by path
Thur. Oct. 30, 2008
Physics 208, Lecture 18
6
Gauss’ law in electrostatics

Electric flux through surface
 charge enclosed
What about magnetic flux?
Thur. Oct. 30, 2008
Physics 208, Lecture 18
7
Magnetic flux



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 )
Thur. Oct. 30, 2008
Physics 208, Lecture 18
8
Magnetic flux
What is that magnetic flux through this
surface?
A. Positive
B. Negative
C. Zero
Thur. Oct. 30, 2008
Physics 208, Lecture 18
9
Gauss’ law in magnetostatics

Net magnetic flux through any closed
surface is always zero: magnetic  0

Compare to Gauss’ law
for electric field
electric 
Qenclosed
o
No magnetic ‘charge’,
so right-hand side=0 for mag.
 Basic magnetic element
is the dipole
Thur. Oct. 30, 2008
Physics 208, Lecture 18
10
Time-dependent fields
Up to this point, have discussed only magnetic
and electric fields constant in time.



E-fields arise from charges
B-fields arise from moving charges (currents)
Faraday’s discovery

Another source of electric field

Time-varying magnetic field creates electric field
Thur. Oct. 30, 2008
Physics 208, Lecture 18
11
Measuring the induced field



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

Thur. Oct. 30, 2008
Physics 208, Lecture 18
12
Current but no battery?


Electric currents require a battery (EMF)
Faraday:
Time-varying magnetic field creates EMF
Faraday’s law:
EMF around loop = - rate of change of mag. flux
Thur. Oct. 30, 2008
Physics 208, Lecture 18
13
Faraday’s law

d
d
  E ds   B    B  dA
dt
dt
EMF around loop
Magnetic flux through
surface bounded by path

EMF no longer zero
around closed loop
Thur. Oct. 30, 2008
Physics 208, Lecture 18
14
Quick quiz
Which of these conducting loops will
have currents flowing in them?
B.
A.
I(t) increases
Constant I
C.
Constant I
Thur. Oct. 30, 2008
Constant v
D.
Constant v
Constant I
Physics 208, Lecture 18
15
Faraday’s law

Faraday’s law



Biot-Savart law


Time-varying B-field creates E-field
Conductor: E-field creates electric current
Electric current creates magnetic field
Result

Another magnetic field created
Thur. Oct. 30, 2008
Physics 208, Lecture 18
16
Thur. Oct. 30, 2008
Physics 208, Lecture 18
17
Lenz’s law

Induced current produces a magnetic field.


Interacts with bar magnet just as another bar magnet
Lenz’s law

Induced current generates a magnetic field
that tries to cancel the change in the flux.

Here flux through loop due to bar magnet is increasing.
Induced current produces flux to left.

Force on bar magnet is to left.
Thur. Oct. 30, 2008
Physics 208, Lecture 18
18
Demonstration: Faraday & Lenz
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Quick quiz
What direction force do I feel due to Lenz’ law when I
push the magnet down?
A. Up
B. Down
Strong magnet
C. Left
D. Right
Copper
Thur. Oct. 30, 2008
Physics 208, Lecture 18
20
Quick Quiz


A conducting rectangular loop moves with constant
velocity v in the +x direction through a region of
constant magnetic field B in the -z direction as shown.
What is the direction of the induced loop current?
A. CCW
y
B. CW
C. No induced current
Thur. Oct. 30, 2008
XXXXXXXXXXXX
XXXXXXXXXXXX
X X X X X X X vX X X X X
XXXXXXXXXXXX
x
Physics 208, Lecture 18
21
Quick Quiz
•Conducting rectangular loop moves with constant
velocity v in the -y direction away from a wire with a
constant current I as shown.
What is the direction of the induced loop current?
I
A. CCW
B. CW
C. No induced current
v
B-field from wire into page at loop
Loop moves to region of smaller B, so flux decreases
Induced loop current opposes this change, so must create a field in same
direction as field from wire -> CW current.
Thur. Oct. 30, 2008
Physics 208, Lecture 18
22
AC Generators
The AC generator consists of
a loop of wire rotated by some
external means in a magnetic field



N turns of same area rotating in a
uniform B
B = BA cos q = BA cos wt
emf
max=NABw
Thur. Oct. 30, 2008
In USA & Canada:
f=w/(2)=60 Hz
In Europe f=50 Hz
Physics 208, Lecture 18
23

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