Transcript Lec17

Lecture 17-1
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,

C
Bt dl 
0 (i1  i2 )

C
B d l  0 I 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.
Lecture 17-2
Magnetization and “Bound” Current in Matter
• Strong externally applied field Bapp aligns the
magnetic moments in matter.  Magnetization
M 

V

d 
M 

dV


Magnetic susceptibility
M  B app
 Bm  m Bapp
m
 M
B app
0
B  Bapp  0 M  (1  m ) Bapp
Relative permeability Km
B  K m B app
 B app 
  0 K m  


 0 
permeability
Lecture 17-3
Hysteresis for a Ferromagnet
Lack of retraceability shown is called hysteresis.
 Memory in magnetic disk and tape
 Alignment of magnetic domains
retained in rock (cf. lodestones)
Area enclosed in hysteresis loop

Energy loss per unit volume
 hard magnet: broad hysteresis loop
(hard to demagnetize, large energy loss,
highe memory)
 soft magnet: narrow hysteresis loop
(easy to demagnetize,…)
Lecture 17-4
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)
Lecture 17-5
Faraday’s Law of Induction
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
Lecture 17-6How
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.
Lecture 17-7
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.
Lecture 17-8
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.
Lecture 17-9
Warm-up quiz
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.
Lecture 17-10
Faraday’s and Lenz’s Laws
 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.
Lecture 17-11
Motional EMF of Sliding Conductor
Induced EMF:
 Lenz’s Law gives direction
counter-clockwise
Faraday’s Law
 FM decelerates the bar
dv
B 2l 2 v
m 
dt
R
dv
B 2l 2

dt
v
mR
v(t )  v  0  e
 B 2l 2 

t
mR


 
dB
dx
  Bl
  Blv
dt
dt
 This EMF induces current I

Blv
I  
R
R
 Magnetic force FM acts on this I
B 2l 2 v
FM  I lB 
R
Lecture 17-12
Ways to Change Magnetic Flux
 B  BA cos 
• Changing the magnitude of the field within a conducting loop (or coil).
• Changing the area of the loop (or coil) that lies within the magnetic field.
• Changing the relative orientation of the field and the loop.
motor
generator
Lecture 17-13
Other Examples of Induction
+
-
Switch has been
open for some time:
Switch is just closed:
Nothing happening
EMF induced in Coil 2
+
-
Switch is just opened:
EMF is induced again
Switch is just closed:
EMF is induced in coil
-
+
Back emf
(counter emf)
Lecture 17-14
PHYS241 - Quiz A
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
Lecture 17-15
PHYS241 - Quiz B
A current directed toward the top of the page and a
rectangular loop of wire lie in the plane of the page. If the
current I is increasing, what happens to the loop?
a. The loop is pulled toward the
top of the page
b. The loop is pulled toward the
current
c. A clockwise current is induced
in the loop.
d. A counterclockwise current is
induced in the loop.
e. Nothing happens to the loop
I
Lecture 17-16
PHYS241 - Quiz C
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.