Transcript 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 2015
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 Law
  0 id s  r̂
dB 
4 r 2
B due to 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
Ampere’s Law
 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 2015
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 2015
Changing magnetic flux induces EMFs
•
EMF / current is induced in a loop if there is
relative motion between loop and magnet - the
magnetic flux inside the loop is changing
•
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)
•
Any relative motion that changes the flux works
EMF/current is induced in the loop whenever magnetic flux
through the loop is changing. (We mean flux rather than just field)
Induced current creates it’s own induced B field and flux,
opposing the changing flux B (Lenz’ Law)
Copyright R. Janow – Fall 2015
CHANGING magnetic flux induces EMFs
and currents in wires
Key
Concepts:
• Magnetic flux B


B  B  A

B
n̂
• Faraday’s Law of Induction
E
induced EMF
dB
in closed loop  
dt
• Lenz’s Law
An induced EMF and current creates
it' s own magnetic field, opposing
changes in the existing flux
Basis of generator principle:
• Loops rotating in B field generate EMF and current.
Copyright
R. Janow – Fall 2015
• Lenz’s Law  Applied torque is needed (energy
conservation)
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 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
•
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 without batteries.
• Induced current flows only if the circuit is
completed.
Fm
v
Fe
Eind
-
Copyright R. Janow – Fall 2015
Flux approach to Motional EMF in a moving wire
Eind   BLv
via Lorentz force
Connection with Flux:
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
Flux arguments apply generally, for example:
E is induced directly in 15 turn coil by changing
flux in solenoid
• B   0in inside solenoid
• B = 0 outside, at location of charges forming
the current in the coil.
• Lorentz force doesn’t explain induced current
• Changing magnetic flux through coil creates
electric field that drives induced current in
loop that creates induced B field
• Flux is proportional to solenoid’s
Copyright R. Janow – Fall 2015
cross-section area (not the coil’s)
A rectangular loop is moving in a uniform B field
b
+
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+
+
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+
+
DOES CURRENT FLOW?
c
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v
a
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-
d
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Segments a-b & d-c create equal
but opposed EMFs in circuit
No EMF from b-c & a-d
or
FLUX is constant
d m
No current flows
dt
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
b
+
0
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 2015
changing
flux
Direction of induced fields and currents

v
Replace magnet with circuit below
Ammeter
• No actual motion
• Current i1 creates field B1
• Flux change in loop 1 causes induction
• Changing current i1 --> changing flux in loop 2.
• Flux is constant if current is constant
• Current i2 flows only while i1 (flux 1) 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 2015
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 2015
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
• Slider acts like a battery
• 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
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 2015
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 2015
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 2015
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 tries to keep the original magnetic flux
through the loop from changing.
front of loop
front of loop
Copyright R. Janow – Fall 2015
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 2015
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 2015
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 2015
AC Generator
DC Generator
Back-torque =xB in rotating loop,  ~ N.A.iind
Copyright R. Janow – Fall 2015
AC Generator, continued
DC Generator, continued
No reversal of output E
Reversal of output E
Copyright R. Janow – Fall 2015
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)
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 external B field produces induced field Bind
• INDUCED field/flux produces its own EMF.
• The BACK EMF opposes current change – analogous to inertia
Copyright R. Janow – Fall 2015
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 2015
Changing magnetic flux directly induces electric fields
A thin solenoid, cross section A, n turns/unit length
• zero B field outside solenoid
B   0In
• inside solenoid:
Flux through a
conducting loop:
conducting loop, resistance R
  BA   0nIA
Current I varies with time  flux varies  EMF is
induced in wire loop:
d
dI
   0nA
dt
dt
E
I'  ind
induced in the loop is:
R
Eind  
Current
Bind
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?
•
•
•
•
•
•
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 conductor (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 around a closed path is not zero
 Eind 

dB
E

d
s


loop ind
dt
Generalized Faradays’ Law
Janow – Fall 2015
(hold loopCopyright
path R.
constant)
Example: Find the electric field induced by
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 within the circular
shaded area. E must be tangential: Gauss’ law says any
normal component of E would require charge enclosed.
|E| is constant on the circular integration path due to
symmetry.


 Eind   E  ds   Eds  E ds  E(2r )
For r > R:
B  BA  B(R 2 )
dB
E(2r )  dB / dt  R 2
dt
For r < R:
B  BA  B(r 2 )
R 2 dB
E
2r dt
r dB
E
2 dt
dB
dt
The magnitude of induced electric field grows linearly
with r, then falls off as 1/r for r>R
E(2r )  dB / dt  r 2
Copyright R. Janow – Fall 2015
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
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  
w, q
1
BwR 2
2
+
USING MOTIONAL EMF FORMULA:
Emf induced across
conductor length ds
 

dE  v  B  ds
Eind  
0
(Equation 29.7)
Conductor moving transversely sees vXB as electric field E’
For points on rotating disk: v = wr,
R
Radial conductor length

 
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 2015