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Gauss' Law: Examples
(Ampere's Law)
1
Ampere’s Law in Magnetostatics
Biot-Savart’s Law can be used to derive another relation: Ampere’s Law
The path integral of the dot product of magnetic field and unit
vector along a closed loop, Amperian loop, is proportional to
the net current encircled by the loop,
 B dl   B  dl   I
c
t
0 (i1  i2 )
c
0 c
• Choosing a direction of
integration.
• A current is positive if it
flows along the RHR normal
direction of the Amperian
loop, as defined by the
direction of integration.
2
iClicker question (last class)
Use Ampere’s law to calculate the
magnetic field inside a solenoid. (n
is number of wraps per unite
length).
A.
B.
• Ampere’s Law:
C.
 B  dl  Bh
I C  nhI
n windings
per unit length
B  μ0nI
D.
B  2 μ0 nI
B  μ0nI
B  3 μ0 nI
B  4 μ0 nI
3
Today
Ampere's Law
Faraday’s law
4
Example: A Non-Uniform Current Distribution
Long, hollow cylindrical current of current density:
J  cr 2  r 2
Insider the cylinder, the total current encircled by the Amperian loop is
r
r
a
a
I   JdA   cr 2 (2 rdr )  2 c  r 3dr 
 c( r 4  a 4 )
2
B(2r ) 
 B
0c
2
r 4  a 4 

r 4  a4 
4r
0c
5
Example: Magnetic field of a long wire
outside the wire

C
B d l 2 rB  0 I
I
B 0
2 r
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iClicker Question
outside the wire

C
B d l 2 rB  0 I
I
B 0
2 r
Assume uniform current density, what’s the
magnetic field vs. r inside the long wire.
A).
 0 I 
B
2 

R


 0 I 
B). B  
r
2 
 2 R 
C).
0 I
B
2 r
D).
0 I
B
r
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iClicker Question
outside the wire

C
B d l 2 rB  0 I
I
B 0
2 r
inside the wire

C
B d l  2 rB
 r2
 0 I
 R2
 0 I 
B
r
2 
 2 R 
8
Moving Bar and Energy Conservation
P=IV=I(emf)
Are we getting something for nothing?
Bar – current I:
FI
F
Fm
FI = IDl ´ B = -F
FI = ILB
Work:
emf = vBL
x
W = FDx = ILBDx
W
Dx
= ILB
Power: P =
Dt
Dt
Main principle of electric generators:
Mechanical power is converted to electric power
P = ILBv
P = I (emf )
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Right now, can you answer the following questions?
The magnetic field is decreasing, what’s the direction of the
induced currents in the closed rectangular loop?
A. Clockwise
B. Counterclockwise
C. No induced currents.
Faraday’s Law: Electromagnetic
Induction
• We have seen that an electric current produces a magnetic
field.
 Can magnetic fields produce electric currents?
•
An electric field is produced when there is
a changing magnetic field.
•
In a closed electric circuit, that means
current is generated due to the changing
magnetic field.
11
approaching
6D-04 Earth Magnetic Field
Inductor
moving away
12
Magnetic Flux
B
 B  BA2 cos  B nA
1 Wb = 1 T m2
B 
B
ndA
S
Bi
Gauss’s Law
for Magnetism

S
B ndA  0
over closed surface
 B  NBA cos
(N turns)
Faraday’s Law
emf  
d mag
dt
Faraday’s law cannot be derived from the
other fundamental principles we have studied
Formal version of Faraday’s law:
𝑑
𝐸 ∙ 𝑑𝑙 = − [ 𝐵 ∙ 𝑛𝑑𝐴]
𝑑𝑡
Sign: given by right hand rule
Michael Faraday
(1791 - 1867)
Differential form of
Faraday’s law:
𝜕𝐵
𝛻×𝐸 =−
𝜕𝑡
Two Ways to Produce Changing 
emf  
d mag
dt
Two ways to produce curly electric field:
1. Changing B
2. Changing A
d mag
dt
d
dB
dA
 B A 
A  B
dt
dt
dt
Inductor Radio (6D-15)
16
UHF Transmitter and Dipole
Receiver (6D-17)
17
Faraday’s Law and Motional EMF
‘Magnetic force’ approach:


 
Ftot  qE  qv  B
E  vB
emf  vB L
I
Use Faraday law:
emf  
d mag
dt
 mag  B A  B Lvt
emf  lim
t 0
 mag
t
 vB L
I
Faraday’s Law and Generator
emf  
d mag
dt

  B  nˆA
  Bwh cos
 t   Bwh cos  t 
emf  
d mag
dt
d
  Bwh cost 
dt
emf  Bwh sin t 
I
Exercise
A uniform time-independent
magnetic field B=3 T points 30o to
the normal of the rectangular loop.
The loop moves at constant speed v
1. What is the emf?
2. In 0.1 s the loop is stretched to be
0.12 m by 0.22 m. What is average
emf during this time?

d mag
emf  
  B  nˆA  BAcos 30
dt

emf 

t
3 T0.0264 m 2  0.02 m 2 0.87
0.1 s
 0.166 V
Example
R
L
emf  
d mag
I
B1
dt
v
B2
mag  B2 A  B1A  B2  B1 A
 mag  B2  B1 Lvt
emf  lim
t 0
 mag
t
Lvt
 vLB2  B1 
I
emf vLB2  B1 

R
R
Faraday’s Law of Induction (More Quantitative)
The magnitude of the induced EMF in conducting loop is
equal to the rate at which the magnetic flux through the
surface spanned by the loop changes with time.
dΦB
ε
dt
where  B 

S
B ndA
N
Minus sign indicates the sense of EMF: Lenz’s Law
• Decide on which way n goes
Fixes sign of B
• RHR determines the
positive direction for EMF 
N
How to use Faraday’s law to determine the
induced current direction

1.
define the direction of n ; can be any of the two
normal direction, e.g. n point to right
2.
determine the sign of Φ. Here Φ>0
N
3.
determine the sign of ∆Φ. Here ∆Φ >0
4.
determine the sign of  using faraday’s law. Here  <0
5.
RHR determines the positive direction for EMF 
• If >0, current follow the direction of the curled
fingers.
• If <0, current goes to the opposite direction of
the curled fingers.
Conducting Loop in a Changing Magnetic Field
Induced EMF has a direction such that it opposes
the change in magnetic flux that produced it.
approaching
 Magnetic moment 
created by induced currrent
I repels the bar magnet.
Force on ring is repulsive.
moving away
 Magnetic moment 
created by induced currrent
I attracts the bar magnet.
Force on ring is attractive.
Induced Electric Field from Faraday’s Law
• EMF is work done per unit charge:
ε W /q
• If work is done on charge q, electric field E must be present:
ε
E
nc
W   q Enc ds
ds
Rewrite Faraday’s Law in terms
of induced electric field:
E
nc
dΦB
ds  
dt
This form relates E and B!
B
• Note that  E  ds  0for E fields generated by charges at rest
(electrostatics) since this would correspond to the potential difference
between a point and itself. => Static E is conservative.
• The induced E by magnetic flux changes is non-conservative.
iClicker Question
The magnetic field is decreasing, what’s the direction of the
induced currents in the closed rectangular loop?
A. Clockwise
B. Counterclockwise
C. No induced currents.
6D-11 Jumping Ring
Is there any
differences in the
two rings ?
Why one can
jump up, the
other can’t ?
http://www.youtube.com/watch?v=
ZL4kbBIf39s
27
iClicker Question
The magnetic field is fixed, what’s the direction of the induced
currents in the closed rectangular loop?
A. Clockwise
B. Counterclockwise
C. No induced currents.
Example
 At 1, 3, and 5, B is not changing.
So there is no induced emf.
 At 4, B in decreasing
into page. So current is
 At 2, B is increasing into page. So clockwise.
emf is induced to produce a
counterclockwise current.
iClicker Question
A current directed toward the top of the page and a rectangular
loop of wire lie in the plane of the page. Both are held in place
by an external force. If the current I is decreasing, what is the
direction of the magnetic force on the left edge of the loop?
a. Toward the right
b. Toward the left
c. Toward top of page
d. Toward bottom of page
e. No force acts on it.
I
iClicker Question
A current directed toward the top of the page and a circular
loop of wire lie in the plane of the page. If a clockwise
current is induced in the loop by the current I, what can you
conclude about it?
I
a. I is increasing
b. I is decreasing
c. I remains constant
d. I is discontinuous
e. Nothing can be said.