Download Ch23.1-23.3, 23.9

Clicker Question: Non-uniform B field
A current loop is oriented so that its magnetic dipole is oriented
along direction “2”. In which direction is the force on the loop?
A)
B)
C)
D)
E)
In direction 1
In direction 2
In direction 3
In direction 4
No force
Maxwell’s Equations (incomplete)

 E  nˆdA 
q
inside
0
Gauss’s law for electricity

Gauss’s law for magnetism
 B  nˆA  0
 
Incomplete version of Faraday’s law
 E  dl  0
 
Ampere’s law
B

d
l


I

0
inside_ path

(Incomplete Ampere-Maxwell law)
First two: integrals over a surface
Second two: integrals along a path
Incomplete: no time dependence
Chapter 23
Faraday’s Law
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
E~1/r
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?
Lenz’s Rule
The direction of the induced magnetic field due to a
change in flux is such it attempts to keep the flux constant.
• The direction of the induced current (and 𝐸𝑁𝐶 ) is what
is needed to produce the induced magnetic field.
𝐸𝑁𝐶
𝐵X
𝐵𝑖𝑛𝑑
𝐵𝑜𝑢𝑡 𝑎𝑛𝑑 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
𝐸𝑁𝐶
𝐵
𝐵𝑖𝑛𝑑
𝐵𝑜𝑢𝑡 𝑎𝑛𝑑 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
Driving Current by Changing B
ENC causes current to
run along the ring
What is the surface charge
distribution?
What is emf and I?


emf   ENC  dl  ENC 2r2
E NC 2r2
I
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 not
Enclosing solenoid!
D
Exercise
Is there current in these circuits?
Quantitative Relationship Between B and EMF
Can observe experimentally:
I=emf/R
ENC~emf
1. ENC~dB/dt
2. 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
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:
𝑑
𝐸𝑁𝐶 ∙ 𝑑𝑙 = −
𝑑𝑡
𝐵 ∙ 𝑛𝑑𝐴
Sign: given by right hand rule
Michael Faraday
(1791 - 1867)
Including Coulomb Electric Field
𝑑
𝐸𝑁𝐶 ∙ 𝑑𝑙 = −
𝑑𝑡
𝐵 ∙ 𝑛𝑑𝐴
Can we use total E in Faraday’s law?





 Etotal  dl   ENC  EC  dl




  ENC  dl   EC  dl


=0
𝑑
𝐸 ∙ 𝑑𝑙 = −
𝑑𝑡
𝐵 ∙ 𝑛𝑑𝐴
A Circuit Surrounding a Solenoid
emf  
d mag
dt
 
mag  B1  r12
𝐸𝑁𝐶
Example:
B1 changes from 0.1 to 0.7 T
in 0.2 seconds; area=3 cm2.
𝑒𝑚𝑓 = −
∆Φ𝑚𝑎𝑔
Δ𝑡
=−
0.6T 3×10−4 m2
0.2s
= −9 × 10−4 V
What is the ammeter reading? (resistance of ammeter+wire is 0.5)
𝐼 = 𝑒𝑚𝑓/𝑅 = −1.8 × 10−3 V
𝑑𝐵
−
𝑑𝑡
downwards
A Circuit Not Surrounding a Solenoid
If we increase current
through solenoid what will
be ammeter reading?
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
Moving Coils or Magnets
Time varying B can be
produced by moving coil:
by moving magnet:
by rotating magnet:
(or coil)
Exercise
1. A bar magnet is moved toward a coil.
What is the ammeter reading (+/-)?
+
𝐵
𝐵
2. The bar magnet is moved
away from the coil.
What will ammeter read?
_
3. The bar magnet is rotated.
What will ammeter read?
_
emf  
d mag
𝑑𝐵
−
𝑑𝑡
dt
?
Faraday’s Law and Motional EMF
‘Magnetic force’ approach:


 
Ftot  qE  qv  B
E  vB
emf  vB L
𝐿
I
Use Faraday law:
emf  
d mag
𝑣∆𝑡
dt
 mag  B A  B Lvt
emf  lim
t 0
 mag
t
 vB L
I
𝐿
Example
R
L
emf  
d mag
I
B1
dt
v
B2
mag  B2 A  B1A  B2  B1 A
 mag  B2  B1 Lvt
emf  lim
t 0
 mag
t
Lvt
 vLB2  B1 
I
emf vLB2  B1 

R
R
Faraday’s Law and Generator
emf  
d mag
dt

  B  nˆA
  Bwh cos
 t   Bwh cos  t 
emf  
d mag
dt
d
  Bwh cost 
dt
emf  Bwh sin t 
I
Two Ways to Produce Changing 
emf  
d mag
dt
Two ways to produce curly electric field:
1. Changing B
2. Changing A
d mag
dt
d
dB
dA
 B A 
A  B
dt
dt
dt