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Chapter 23
Faraday’s Law
Predicting Possible Field Configuration
Is the following “curly” pattern of electric field possible?
f
DV = - ò E · dl
dl
i
E is always parallel to dl
dl
dl
DVAA = - ò E · dl = -El = -E2p R ¹ 0
This “curly” pattern of electric field is impossible to produce by
arranging any number of stationary point charges!
Faraday’s Experimentation
Set wire (connected to ammeter) around a solenoid.
Can a current be observed in this wire?
Changing Magnetic Field
Solenoid: inside
outside
B
0 NI
B0
d
Constant current: there will be no forces
on charges outside (B=0, E=0)
What if current is not constant in time? Let B increase in time
E~dB/dt
Non-Coulomb ENC !
Two Ways to Produce Electric Field

1. Coulomb electric field: produced by charges E1 
2. Non-Coulomb electric field:
using changing magnetic field
Field outside of solenoid


Same effect on charges: F21  q2 E1
1
q1
rˆ
2
40 r

 dB1 1
E1 ~
dt r
Direction of the Curly Electric Field
Right hand rule:

dB
Thumb in direction of  1
dt
fingers: E
NC
Exercise:
Magnetic field points down
from the ceiling and is increasing.
What is the direction of E?
“Current opposes change in B”
Driving Current by Changing B
ENC causes current to
run along the ring
What is emf and I?


emf   ENC  dl  ENC 2r2
I
E NC 2r2
R
Ring has resistance, R
Effect of the Ring Geometry


emf   ENC  dl  ENC 2r2
1. Change radius r2 by a factor of 2.
ENC ~ 1 / r2
L  2r2
emf does not depend on radius of the ring!
2. One can easily show that emf will be the same for any circuit
surrounding the solenoid
Round-Trip Not Encircling the Solenoid


emf   ENC  dl
=0
=0
B
 C 
 D
 A


emf   ENC  dl   ENC  dl   ENC  dl   ENC  dl
A
B
C
 D


emf   ENC  dl   ENC  dl
B
A
C
emf  0 for any path which does not
enclose the solenoid!
D
Exercise
Is there current in these circuits?
Quantitative Relationship Between B and EMF
Can observe experimentally:
I=emf/R
1.
2.
ENC~emf
ENC~dB/dt
ENC~ cross-section of a solenoid
d
emf 
B1 r12
dt


Magnetic Flux
d
emf 
B1 r12
dt

B r 
1
2
1

-magnetic flux mag on the area encircled by the circuit
(wire loop)
Magnetic flux on a small area A:

B  nˆA  B A
Definition of magnetic flux:

 mag   B  nˆdA   BdA
This area does not enclose a volume!
Faraday’s Law
emf  
d mag
dt
Faraday’s law cannot be derived from the
other fundamental principles we have studied
Formal version of Faraday’s law:
Sign: given by right hand rule
“Current opposes change in B”
Michael Faraday
(1791 - 1867)
Various ways of making changing
magnetic field
Change current through the solenoid
Time varying B can be
produced by moving coil:
by moving magnet:
by rotating magnet:
(or coil)
Clicker
ammeter
+
-
Solenoid has circumference 10 cm. The
current I1 is increasing causing
increasing B1 as in the figure.
Solenoid is surrounded by a wire with
finite resistance.
When length of the wire is changed
from 30 to 20 cm what will happen to
detected current?
A. It will decrease
B. Increase
C. Does not change
A Circuit Surrounding a Solenoid
emf  
d mag
dt
 
mag  B1  r12
Example:
B1 changes from 0.1 to 0.7 T
in 0.2 seconds; area=3 cm2.


4
2
 0.6 T 3  10 m
emf 

 9  10 4 V
t
0.2 s
What is the ammeter reading? (resistance of ammeter+wire is 0.5)
I  emf / R  1.8  103 A
Clicker: A Circuit Not Surrounding a
Solenoid
If we increase current
through solenoid what will
be ammeter reading?
A. positive current
B. negative current
C. zero
The EMF for a Coil With Multiple Loops
Each loop is subject to similar
magnetic field  emf of loops
in series:
emf   N
d mag
dt
Exercise
1. A bar magnet is moved toward a coil.
What is the ammeter reading (+/-)?
2. The bar magnet is moved away from the coil.
What will ammeter read?
emf  
d mag
dt