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Faraday:
d
B
dt
Changing flux generates EMF
Motional EMF
Moving conductor generates
electric potential difference to
cancel motional EMF
Thur. Nov. 5, 2009
Physics 208, Lecture 19
1
Motional EMF
Charges in metal
feel magnetic
r r
force qv B
Charges move,
build up at ends
of metal
r
r
FE FB qE qV /l qvB
Tue. Nov. 2, 2009
Equilibrium:
electric force
cancels
magnetic force
so V vl B
Physics 208, Lecture 18
2
Question
Two identical bars are moving through a vertical magnetic field.
Bar (a) is moving vertically and bar (b) is moving
horizontally.
Which of following statements is true?
A. motional emf exists for (a), but not (b)
B. motional emf exists for (b), but not (a)
C. motional emf exists for both (a) and (b)
D. motional emf exists for neither (a) nor (b)
Tue. Nov. 2, 2009
Physics 208, Lecture 18
3
Coil in magnetic field
Uniform B-field
increasing in time
Flux in z-direction
increasing in time
B(t)
Induced
current
Lenz’ law:
Induced EMF would produce current to oppose change in flux
Thur. Nov. 5, 2009
Physics 208, Lecture 19
4
But no equilibrium current
Charges cannot flow out end
Build up at ends,
makes Coulomb
electric field
Cancels Faraday
electric field
- -B(t)
+++
Increasing
with time
End result
No current flowing
Electric potential difference
from one end to other
Opposes Faraday EMF
ΔV = - EMF
Thur. Nov. 5, 2009
Compare
motional EMF
Physics 208, Lecture 19
5
Coil can generate it’s own flux
Uniform field inside solenoid Binside
Change current -> change flux
o N
I
N turns
Wire turns
Thur. Nov. 5, 2009
Surface for flux
Physics 208, Lecture 19
6
‘Self’-flux in a solenoid
Bsolenoid
oN
I
N=# of turns,
=length of solenoid
o NA
I
Flux through one turn BsolenoidA
o N 2 A
I
=Flux through entire solenoid NBsolenoid A
inductance
LI
Lsolenoid o N 2 A /
Thur. Nov. 5, 2009
Physics 208, Lecture 19
7
Inductance: a general result
Flux = (Inductance) X (Current)
LI
Change in Flux
= (Inductance) X (Change in Current)
LI
Thur. Nov. 5, 2009
Physics 208, Lecture 19
8
Question
The current through a solenoid is doubled.
The inductance of the solenoid
A. Doubles
B. Halves
C. Stays the same
Inductance is a geometrical
property, like capacitance
Thur. Nov. 5, 2009
Physics 208, Lecture 19
9
Question
A solenoid is stretched to twice its length
while keeping the same current and same
cross-sectional area. The inductance
Bsolenoid
A. Increases
B. Decreases
C. Stays the same
oNI
Field, hence flux,
have decreased for
same current
LI
Lsolenoid o N A /
2
Thur. Nov. 5, 2009
Physics 208, Lecture 19
10
Fixed current
through ideal inductor
L
For fixed resistor value
Current through inductor I = Vbatt/R
Flux through inductor = LI
Constant current -> Flux through
inductor doesn’t change
No induced EMF
Voltage across inductor = 0
I
Vbatt
R
Ideal inductor:
coil has zero resistance
Thur. Nov. 5, 2009
Physics 208, Lecture 19
11
L
Try to change current
You increase R in time:
Current through inductor
starts to decrease
Vb
Flux LI through inductor
starts to decrease
Faraday electric fields
in inductor wires
R
Va
I
Vbatt
Induces current to oppose flux decrease
Drive charges to ends of inductor
R(t)
Charges produce
Coulomb electric field
dI
Electric potential diff VL Vb Va L
Thur. Nov. 5, 2009
time
dt
Physics 208, Lecture 19
12
Question
The potential at a is higher than at b. Which
of the following could be true?
A) I is from a to b, steady
B) I is from a to b, increasing
C) I is from a to b, decreasing
D) I is from b to a, increasing
E) I is from b to a, decreasing
Thur. Nov. 5, 2009
e,.g. current from a to b: current
increases, flux to right increases.
sign of induced emf such that it
would induce current to produce
flux to left to oppose change in flux.
Electric potential difference
opposite to induced EMF, so Va>VB
Physics 208, Lecture 19
13
Energy stored in ideal inductor
Constant current (uniform charge motion)
No work required
to move charge through inductor
Increasing current:
Work VLq VL Idt
required
to move charge across induced EMF
dI
dW V Idt L
Idt LIdI
L
dt
Energy is stored in inductor:
dUL dW
Total stored energy
UL
I
0
Thur. Nov. 5, 2009
1 2 Energy stored
LIdI LI
in inductor
2
Physics 208, Lecture 19
14
Magnetic energy density
1 2
Energy stored in inductor U L LI
2
N 2A
Solenoid inductance Lsolenoid o
o NI
1
A
Energy stored in solenoid Usolenoid
2o
Bsolenoid
2
Energy density
U
B2
solenoid
A
Thur. Nov. 5, 2009
Physics 208, Lecture 19
solenoid
2 o
15
Question
A solenoid is stretched to twice its length
while keeping the same current and same
cross-sectional area. The stored energy
Bsolenoid
A. Increases
oNI
B decreases by 2
B. Decreases
C. Stays the same
2
U solenoid Bsolenoid
A
2 o
Energy density decr by 4
Thur. Nov. 5, 2009
Volume increases by 2
Physics 208, Lecture 19
16
Inductor circuit
Induced EMF extremely high
Breaks down air gap at switch
Air gap acts as resistor
Thur. Nov. 5, 2009
Physics 208, Lecture 19
17
Question
Here is a snapshot of an inductor circuit at
a particular time. What is the behavior of
the current?
A. Increasing
B. Decreasing
Va
I
C. Nonzero Constant
Vb
VL Vb Va L
D. Must be zero
dI
dt
R
VL IR 0 dI /dt I
L
Thur. Nov. 5, 2009
E. Need more info
Physics 208, Lecture 19
18
Perfect inductors in circuits
I
Constant current flowing
I?
-
Voltage needed to drive
current thru resistor
+
Thur. Nov. 5, 2009
All Voltage drops = 0
-IR + VL = 0
IR LdI /dt 0
dI /dt I R /L
Physics 208, Lecture 19
19
RL circuits
-
I?
dI
R
I
dt
L
R
dI I dt
L
+
Current decreases in time
Slow for large inductance
Slow for small resistance
(inductor fights hard, tries to keep constant current)
(little inductor voltage needed to drive current)
Time constant L /R
Thur. Nov. 5, 2009
Physics 208, Lecture 19
20
RL circuits
-
I(t)
I Ioet /(L / R ) Ioet /
+
Time constant
L /R
Thur. Nov. 5, 2009
Physics 208, Lecture 19
21