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The story so far…
dI
dB
r
Magnetic field
generated by current
element: Biot-Savart
I
closed
path
Ampere’s law
 B ds   I
o
o Ids  rˆ
dB 
4 r 2

surface
bounded by path
Exam 2 results
Phy208 Exam 2
35
Ave=69
30
Count
25
20
15
10
5
0
10
20
30
Mon. Mar. 31, 2008
40
50 60 70
SCORE
80
Grade cutoffs:
A: 86
AB: 79
B: 66
BC: 58
C: 37
D: 23
90 100
Physics 208, Lecture 18
2
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
Mon. Mar. 31, 2008
Physics 208, Lecture 18
3
“Ampere’s law” in electrostatics
 E  ds  ?
path
Work done by E-field = W AB 
B
F
Coulomb
A
So
 ds 
B
 qE  ds
A
 E  ds
path

is work per unit charge to
bring charge back to where it started.
This is zero.
Mon. Mar. 31, 2008
Physics 208, Lecture 18
4
Gauss’ law in electrostatics

Electric flux through surface
 charge enclosed
What about magnetic flux?
Mon. Mar. 31, 2008
Physics 208, Lecture 18
5
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 )
Mon. Mar. 31, 2008
Physics 208, Lecture 18
6
Magnetic flux
What is that magnetic flux through this
surface?
A. Positive
B. Negative
C. Zero
Mon. Mar. 31, 2008
Physics 208, Lecture 18
7
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.

Mon. Mar. 31, 2008
Basic magnetic element
is the dipole
Physics 208, Lecture 18
8
Comparison with electrostatics
Gauss’ law
Ampere’s law
Electrostatics
Magnetostatics
Mon. Mar. 31, 2008
Physics 208, Lecture 18
9
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
Mon. Mar. 31, 2008
Physics 208, Lecture 18
10
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

Mon. Mar. 31, 2008
Physics 208, Lecture 18
11
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
Mon. Mar. 31, 2008
Physics 208, Lecture 18
12
Faraday’s law

d
d
  E ds   B  
dt
dt
EMF around loop
 B  dA
Magnetic flux through
surface bounded by path

EMF no longer zero
around closed loop
Mon. Mar. 31, 2008
Physics 208, Lecture 18
13
Quick quiz
Which of these conducting loops will
have currents flowing in them?
I(t) increases
Constant I
Constant v
Constant v
Constant I
Mon. Mar. 31, 2008
Constant I
Physics 208, Lecture 18
14
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
Mon. Mar. 31, 2008
Physics 208, Lecture 18
15
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.
Mon. Mar. 31, 2008
Physics 208, Lecture 18
16
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
Mon. Mar. 31, 2008
Physics 208, Lecture 18
17
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
Mon. Mar. 31, 2008
XXXXXXXXXXXX
XXXXXXXXXXXX
X X X X X X X vX X X X X
XXXXXXXXXXXX
x
Physics 208, Lecture 18
18
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.
Mon. Mar. 31, 2008
Physics 208, Lecture 18
19
Motional EMF
Conductor moving in uniform magnetic field
+ / - charges in conductor are moving.
Magnetic field exerts force.



Charges pile up at ends
FB
L

-
Static equilibrium: E-field
generated canceling
magnetic force
qE  qvB
v
v
Solid
conductor


Mon. Mar. 31, 2008

EMF  vBL
Physics 208, Lecture 18
20