Download Lecture11: Faraday`s Law of Induction

Physics 121 - Magnetism
Lecture 11 - Faraday’s Law of Induction
Y&F Chapter 29, Sect. 1-5
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Magnetic Flux
Motional EMF: moving wire in a B field
Two Magnetic Induction Experiments
Faraday’s Law of Induction
Lenz’s Law
Rotating Loops – Generator Principle
Concentric Coils – Transformer Principle
Induction and Energy Transfers
Induced Electric Fields
Summary
Copyright R. Janow – Fall 2016
Previously:
Current-lengths in a magnetic field feel forces and torques

 
Fmag  qv  B

 
Fm  i L  B
Force on charge and wire carrying current

  N i A n̂
  
  B
 
Um     B
torque and potential energy of a dipole
 q
B
Current-lengths (changing electric fields) produce magnetic fields
Biot-Savart
  0 id s  r̂
dB 
4 r 2
Ampere
B outside a long straight wire carrying a current i:
B at center of circular loop carrying a current i :
Long Solenoid field:
B outside = 0
Binside  0in
B on the symmetry axis of a
current loop (far field):
 
B
  ds  0ienc
 0i
2 r
 i
B 0
2R
Current loops are
elementary dipoles
B
B inside a torus carrying
a current i :
 0iN
B


0 
B(z) 
2 z 3
2 r

  NiAˆ
Next: Changing magnetic flux induces EMFs
and currents in wires
Copyright R. Janow – Fall 2016
Magnetic Flux:
Electrostatic
Gauss Law
S

 q
E  dA  enc
0

 
dB  B  dA  B  n̂dA
over surface (open
 or
 closed)
B   B  dA
Magnetic
Gauss Law
S


B  dA  0
defined analogously to flux of electric field

B
n̂
Flux Unit: 1 Weber = 1 T.m 2
n̂
n̂
q
n̂
n̂
Copyright R. Janow – Fall 2016
Changing magnetic flux induces EMFs
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EMF / current is induced in a loop if there is
relative motion between loop and magnet - the
magnetic flux inside the loop is changing
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Induced current stops when relative motion
stops (case b).
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Faster motion produces a larger current.
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Induced current direction reverses when
magnet motion reverses direction (case c
versus case a)
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Any relative motion that changes the flux works
Changing magnetic flux through area of a boundary loop causes
an induced EMF (or current) in the loop. (We mean flux rather than just field)
An induced current creates it’s own induced B field and flux that
opposes the changing flux B (Lenz’ Law)
Copyright R. Janow – Fall 2016
CHANGING magnetic flux induces EMFs
and currents in wires
Key
Concepts:
• Magnetic flux B


B  B  A
Area bounded
By a path

B
n̂
• Faraday’s Law of Induction
dB
E  induced EMF in a loop  N
dt
• Lenz’s Law
An induced current creates it' s own induced
magnetic field, opposing changes in the existing flux
Basis of generator principle:
• Loops rotating in B field generate EMF and current.
• Lenz’s Law  Applied torque does work
Copyright R. Janow – Fall 2016
Motional EMF: Lorentz Force on moving charges in conductors
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
Uniform magnetic field points away from viewer.
FB
Wire of length L moves with constant velocity v
perpendicular to the field
Electrons feel a magnetic Lorentz force and migrate to
the lower end of the wire. Upper end becomes positive.
Charge separation  an induced electric field Eind inside
wire
Charges come to equilibrium when the forces on charges
balance:
qEind  qvB
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or
 

FE  q E
 q v B
+
L
v
Eind  vB
Electric field Eind in the wire corresponds to potential
difference Eind across the ends of wire:
-
Eind  EindL  BLv
•

FIELD
BOUNDARY
Potential difference Eind is maintained between the ends
of the wire as long as the wire continues to move
through the magnetic field.
+
FBD of free
charge q in a wire
• Induced EMF is created by the motion.
• Induced current flows only if the circuit is
completed.
Fm
v
Fe
Eind
-
Copyright R. Janow – Fall 2016
Complementary Flux view for
Motional EMF in a moving wire
The rate of flux change = [field B] x [rate of sweeping out area (B uniform)]
Rate of sweeping out area  dA/dt  Ldx / dt  Lv
d  d (BA)  BdA  BLv
so
dt dt
dt
Eind  
d B
dt
changing flux
Copyright R. Janow – Fall 2016
Flux change arguments work even without motion:
Simple Transformer: changing current
In solenoid causes current in coil
Inside solenoid:
B  0in
dB / dt  0n di / dt
Outside solenoid: B = 0 at location
of charges in the 15 turn coil.
So Lorentz force can’t cause current there
Magnetic flux through coil is proportional to solenoid’s crosssection area (not the coil’s).
Changing magnetic flux through coil must create electric field that
drives induced current in coil, that also creates its own B field
Copyright R. Janow – Fall 2016
A rectangular loop is moving in a uniform B field
DOES CURRENT FLOW?
b
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c
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v
a
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d
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DOES CURRENT FLOW NOW?
• B-field ends or is not uniform
• Segment c-d now creates NO EMF
• Segment a-b creates EMF as above
• Un-balanced EMF drives current
just like a battery
d m
• FLUX is DECREASING
0
dt
Segments a-b & d-c create equal but
opposed EMFs in circuit
No EMF from b-c & a-d
or
FLUX in b-c and a-b
grow at same rate
Net flux is constant
No current flows
b
+
d m
0
dt
c
v
a
-
Which way does current flow?
What is different when loop is entering field?
d
E
dB
dt
Copyright R. Janow
– Fall 2016 flux
changing
Directions of induced fields and currents

v
Replace magnet with circuit
Ammeter
• No actual motion
• Current i1 creates field B1 (RH rule)
• Flux is constant if current i1 is constant
• Changing current i1 --> changing flux in loop 2.
• Current i2 flows only while i1 is
changing (after switch S closes or opens)

B1
Induced current i2 creates it’s own induced field B2
whose flux 2 opposes the change in 1 (Lenz’ Law)
1 = B1A
Close S
dB1
 0 (B1 growing)
dt
i2

B2
2 = B2A
Open S later
dB1
 0 (B1 decreasing)
dt
i2
Induced Dipoles

B2
Copyright R. Janow – Fall 2016
Faraday’s Law: Changing Flux
“-” sign for Lenz’s Law
uniform B
Induced EMF
(Volts)
n̂

m  B  n̂A
dB
Eind  
dt
insert N for multiple turns in loop, loop of any shape
EMF=0
Several ways to change flux:
B(t)
• change |B| through a coil
• change area of a coil or loop
• change angle between B and coil
e.g., rotating coils  generator effect
Example:
uniform
dB/dt
Rate of flux change
(Webers/sec)
slope  EMF
B through a loop increases by 0.1 Tesla in 1 second: Loop
area A = 10-3 m2 is constant. Find the induced EMF
B
B
0.1 x 10-3
Eind  
 A

t
t
1
front

Bind
Eind  10-4 volts
iind  Eind / R
Current iind creates field Bind that opposes increase in B
Copyright R. Janow – Fall 2016
Slidewire Generator
Slider moves, increasing loop area (flux)
Induced EMF:
• Slider: constant speed v to right
• Uniform external field into the page
• Loop area & FLUX are increasing
Eind  
+
dm
dA
dx
 B
 BL
 BLv
dt
dt
dt
• Lenz’s Law says induced field Bind
is out of page.
• RH rule says current is CCW
iind
iind  Eind / R R  total resistance in circuit
• Slider acts like a battery (Lorentz)
FM DRAG FORCE on slider

 
Fm  iind L  B  | Fm |  iindLB
is due to the induced current:
• Opposes the motion of the wire
FEXT (an external force) is needed
to keep wire from slowing down
L
-
x
FEXT needed is opposed to Fm
Eind
B 2L2 v
Fext  Fm  iind LB 
LB 
R
R
Mechanical power supplied & dissipated:
B2L2v2
Eind2
Power  Fextv 

R
R
Copyright R. Janow – Fall 2016
Slidewire Generator: Numerical Example
+
B = 0.35 T
L = 25 cm
v = 55 cm/s
a) Find the EMF generated:
Eind  
iind
L
v
-
dm
 BLv  (0.35)(. 25)(. 55)  48 mV
dt
iind
DIRECTION: Bind is into slide, iind is clockwise
b) Find the induced current if R for the whole loop = 18 W:
E
iind  ind  48 mV / 18 W  2.67 mA
clockwise
R
c) Find the thermal power dissipated:
2
Eind
(48 x 103 ) 2
P

 1.28 x 10- 4 Watts
R
18
2
P  iindR  (2.67 x 10-3 )2 x 18  1.28 x 10-4 Watts
d) Find the power needed to move slider at constant speed
Pmech  Fv  iLBv  2.67 x 10-3 x 0.25 x 0.35 x 0.55  1.28 x 10-4 Watts
Copyright
!!! Power dissipated via R = Mechanical power
!!!R. Janow – Fall 2016
Induced Current and Emf
11 – 1: A circular loop of wire is in a uniform magnetic field covering
the area shown. The plane of the loop is perpendicular to the field
lines.
Which of the following will not cause a current to be induced in the
loop?
A. Sliding the loop into the field region from the far left
B. Rotating the loop about an axis perpendicular to the field lines.
C. Keeping the orientation of the loop fixed and moving it along
the field lines.
D. Crushing the loop.
E. Sliding the loop out of the field region from left to right
B
Eind  
dB
d
  {BA cos(q)}
dt
dt
Copyright R. Janow – Fall 2016
Lenz’s Law Example:
A loop crossing a region of uniform magnetic field
The induced current and EMF create induced magnetic flux that
opposes the change in magnetic flux that created them
B = 0
B = uniform
v
d
0
dt
iind  0
v
d
is
dt
into slide
B = 0
v
d
0
dt
iind  0
v
v
d
is
dt
out of slide
Bind is out
Bind is down
iind is CCW
iind is CW
d
0
dt
iind  0
Copyright R. Janow – Fall 2016
Lenz’s
Law
The induced current and EMF create induced
magnetic flux that opposes the change in
magnetic flux that created them
front of
loop
induced dipole in loop
opposes flux growth
induced dipole in loop
opposes flux decrease
The induced current and field try to keep the original
magnetic flux through the loop from changing.
front of loop
front of loop
Copyright R. Janow – Fall 2016
Direction of induced current
11-2: A circular loop of wire is falling toward a straight
wire carrying a steady current to the left as shown
B
What is the direction of the induced current
in the loop of wire?
 0i
2 r
A. Clockwise
B. Counterclockwise
C. Zero
D. Impossible to determine
E. I will agree with whatever the majority chooses
v
11-3: The loop continues falling until it is
below the straight wire.
Now what is the direction of the induced
current in the loop of wire?
A. Clockwise
B. Counterclockwise
C. Zero
D. Impossible to determine
E. I will oppose whatever the majority chooses
I
Copyright R. Janow – Fall 2016
Generator Effect (Sinusoidal AC)
Changing flux through a rotating current loop
angular velocity w  2f in B field:
n̂
q

B
Rotation axis is
out of slide

q  wt i.e., n̂ is along B when t  0

B  B  n̂A  BA cos(q)  BA cos(wt)  max cos(wt)
peak flux magnitude when wt = 0, , etc.
EMF induced is the time derivative of the flux
Eind
dB

 BAw sin(wt )  E0 sin(wt )
dt
E0  BAw is the peak value of the induced EMF
Eind has sinusoidal behavior - alternating polarity over a cycle
maxima when wt = +/- /2
Copyright R. Janow – Fall 2016
AC Generator
DC Generator
No mechanical reversal of output E
Mechanical Reversal of output E
Copyright R. Janow – Fall 2016
AC Generator
DC Generator
Back-torque =xB in rotating loop,  ~ N.A.iind
Copyright R. Janow – Fall 2016
Numerical
Example
Flat coil with N turns of wire

B
N turns
 n̂
Bind

B
E
dB
Eind  N
dt
Each turn increases the flux
and induced EMF
• N = 1000 turns
• B through coil decreases from +1.0 T to -1.0 T in 1/120 s.
• Coil area A is 3 cm2 (one turn)
• Is B external or due to changing current in coil?
Find the EMF induced in the coil by it’s own changing flux
d tot
dt

B
 2.0
2
4 2
2
 Area  Number of turns 
 3 cm  10 m /cm  1000
t
1 / 120
d tot
dt
 72 Volts
Eind  72 Volts
Flux change due to B field produces induced field Bind
• INDUCED field/flux is due to induced EMF.
• The BACK EMF opposes current change – analogous to inertia
Copyright R. Janow – Fall 2016
Transformer Principle
Primary current is changing after switch closes
 Changing flux in primary coil
....which links to....
changing flux through secondary coil
 changing secondary current & EMF
PRIMARY SECONDARY
Iron ring strengthens
flux linkage
Copyright R. Janow – Fall 2016
Changing magnetic flux directly induces electric fields
conducting loop, resistance R
A thin solenoid, n turns/unit length
• Solenoid cross section A
• Zero B field outside solenoid
• Field inside solenoid:
B  
Flux through the wire loop:
In
  BA  0nIA
0
Changing current I  changing flux 
EMF is induced in wire loop:
d
dI
   0nA
dt
dt
Eind
I' 
induced in the loop is:
R
Eind  
Bind
Current
If dI/dt is positive, B is growing, then Bind opposes change and I’ is counter-clockwise
What makes the induced current I’ flow, outside solenoid?
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B = 0 outside solenoid, so it’s not the Lorentz force
An induced electric field Eind along the loop causes current to flow
It is caused directly by d/dt within the loop path
Eind is there even without the wire loop (no current flowing)
Electric field lines here are loops that don’t terminate on charge.
E-field is a non-conservative (non-electrostatic) field as the line
integral (potential) around a closed path is not zero
 Eind 

dB
E

d
s


loop ind
dt
Generalized Faradays’ Law
Janow – Fall 2016
(hold loopCopyright
path R.
constant)
Example: Electric field induced by a region of
changing magnetic flux
Eind 
 
 B  ds  0ienc

dB
Eind  ds  
loop
dt

Find the magnitude E of the induced electric field
at points within and outside the magnetic field.
Assume:
• dB/dt = constant across the circular shaded area.
• E must be tangential: Gauss’ law says any normal
component of E would require enclosed charge.
• |E| is constant on any circular integration path due to
symmetry.
For r > R:  
 Eind   E  ds   Eds  E ds  E(2r )
B  BA  B(R 2 )
R 2 dB
dB
E
E(2r )  dB / dt  R 2
2r dt
dt
For r < R:
B  BA  B(r 2 )
E(2r )  dB / dt  r 2
dB
dt
E
r dB
2 dt
The magnitude of induced electric field grows linearly
with r, then falls off as 1/r for r>R
Copyright R. Janow – Fall 2016
Example: EMF generated by Faraday Disk Dynamo
Conducting disk, radius R, rotates at rate w in uniform constant field B,
FLUX ARGUMENT:
Eind  
dB
dA
 B
dt
dt
evaluate areal velocity
dA = area swept out by radius vector in dq
= fraction of full circle in dq x area of disk
dq
R2
2
dA R 2 dq R 2
dA 
R 
dq



w
2
2
dt
2 dt
2
Eind  
1
BwR 2
2
w, q
+
MOTIONAL EMF FORMULA: (work/unit charge)
  
dE  v  B  ds
Radial conductor length
Emf induced across
(Equation 29.7)
length ds of conductor
Lorentz force/unit charge; vXB looks like electric field to moving charge
For points on rotating disk: v = wr,
Eind  
R
0

 
v  B  ds 

R
0
vXB = E’ is radially outward, so is ds = dr
Bwrdr 
1
BwR 2
2
current flows
radially out
Copyright R. Janow – Fall 2016