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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,
• 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 B app aligns the
magnetic moments in matter. (M) Magnetization
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,
high memory)
 soft magnet: narrow hysteresis loop
(easy to demagnetize,…)
Lecture 17-4
BRIDGE OF NAILS
Lecture 17-5
INDUCTION
• Bar magnet approaches coil
S
N
v
Current induced in coil
• Reverse poles of magnet
N
S
Current in opposite
direction
N
• Bar magnet stationary
v
S
No induced current
v
• Coil moving around bar magnet
Same currents
induced in coil
S
N
What’s in common?: Change of Magnetic flux = EMF!
Lecture 17-6
Magnetic Flux
B
Bi
(N turns)
Lecture 17-7
MAGNETIC BRAKING 6D08
Lecture 17-8
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.
N
Minus sign indicates the sense of EMF: Lenz’s Law
• Decide on which way n goes
Fixes sign of ϕB
• RHR determines the
positive direction for EMF
N
Lecture 17-9
Induced Electric Field from Faraday’s Law
Rewrite Faraday’s Law in
terms of induced electric field:
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 17-10
JUMPING JACK 6D11
Lecture 17-11
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 Δϕ ind using faraday’s law. Here
Δϕ ind <0
5.
RHR determines the positive direction for EMF
• If Δϕ >0, current follow the direction of the
curled fingers.
• If Δϕ <0, current goes to the opposite direction of
the curled fingers.
Lecture 17-12
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 17-13
Faraday’s and Lenz’s Laws
 At 1, 3, and 5, ΦB is not changing.
So there is no induced emf.
 At 4, ΦB in decreasing
into page. So current is
 At 2, ΦB is increasing into page. So clockwise.
emf is induced to produce a
counterclockwise current.
Lecture 17-14
Motional EMF of Sliding Conductor
Induced EMF:
 Lenz’s Law gives direction
Faraday’s Law
 FM decelerates the bar
 This EMF induces current I
 Magnetic force FM acts on this I
Lecture 17-15
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 17-16
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)