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Clicker Question: Non-uniform B field
A current loop is oriented so that its magnetic dipole is orientedalong direction “2”. In which direction is the force on the loop?
A) In direction 1B) In direction 2C) In direction 3D) In direction 4E) No force
0
ˆ
insideqdAnE
0 ldE
pathinsideIldB _0
Gauss’s law for electricity
Gauss’s law for magnetism
Incomplete version of Faraday’s law
Ampere’s law(Incomplete Ampere-Maxwell law)
First two: integrals over a surfaceSecond two: integrals along a path
Incomplete: no time dependence
Maxwell’s Equations (incomplete)
0ˆ AnB
Solenoid: insided
NIB 0
outside 0B
Constant current: there will be no forceson charges outside (B=0, E=0)
What if current is not constant in time? Let B increase in time
E~1/r E~dB/dt
Non-Coulomb ENC !
Changing Magnetic Field
1. Coulomb electric field: produced by charges rr
qE ˆ
4
121
01
2. Non-Coulomb electric field:using changing magnetic field
Same effect on charges: 1221 EqF
Two Ways to Produce Electric Field
Field outside of solenoid
𝐸1
𝑑𝐵1
𝑑𝑡1𝑟
Right hand rule:
Thumb in direction offingers: ENC
dt
Bd 1
Exercise:
Magnetic field points downfrom the ceiling and is increasing.
What is the direction of E?
Direction of the Curly Electric Field
The direction of the induced magnetic field due to a change in flux is such it attempts to keep the flux constant.
Lenz’s Rule
𝐵
𝐵𝑜𝑢𝑡𝑎𝑛𝑑𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
𝐸𝑁𝐶
𝐵
𝐵𝑜𝑢𝑡𝑎𝑛𝑑𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
𝐸𝑁𝐶
• The direction of the induced current (and is what is needed to produce the induced magnetic field.
𝐵𝑖𝑛𝑑
X 𝐵𝑖𝑛𝑑
ENC causes current torun along the ring
What is the surface chargedistribution?
What is emf and I?
ldEemf NC
22 rENC
R
rEI NC 22
Driving Current by Changing B
Ring has resistance, R
ldEemf NC
22 rENC
1. Change radius r2 by a factor of 2.
2/1~ rENC
22 rL 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
Effect of the Ring Geometry
ldEemf NC
A
D
NC
D
C
NC
C
B
NC
B
A
NC ldEldEldEldEemf
=0 =0
D
C
NC
B
A
NC ldEldEemf
0emf for any path notEnclosing solenoid!
Round-Trip Not Encircling the Solenoid
+ _
Can observe experimentally:
I=emf/R ENC~emf
1. ENC~dB/dt2. ENC~ cross-section of a solenoid
emf d
dtB1r1
2
Quantitative Relationship Between B and EMF
emf d
dtB1r1
2
211 rB - magnetic flux mag on the area encircled by the circuit
Magnetic flux on a small area A:
ABAnB ˆ
Definition of magnetic flux:
dABdAnBmag ˆ
Magnetic Flux
This area does not enclose a volume!
dt
demf mag
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 ruleMichael Faraday
(1791 - 1867)
Faraday’s Law
∮𝐸𝑁𝐶 ∙𝑑 𝑙=−𝑑𝑑𝑡 [𝐵 ∙ �̂�𝑑 𝐴 ]
Can we use total E in Faraday’s law?
ldEEldE CNCtotal
ldEldE CNC
=0
Including Coulomb Electric Field
∮𝐸𝑁𝐶 ∙𝑑 𝑙=−𝑑𝑑𝑡 [𝐵 ∙ �̂�𝑑 𝐴 ]
∮𝐸 ∙𝑑 𝑙=−𝑑𝑑𝑡 [𝐵 ∙ �̂�𝑑 𝐴 ]
dt
demf mag
mag B1 r12
Example: B1 changes from 0.1 to 0.7 T in 0.2 seconds; area=3 cm2.
What is the ammeter reading? (resistance of ammeter+wire is 0.5)
A Circuit Surrounding a Solenoid
V
V downwards
𝐸𝑁𝐶
If we increase current through solenoid what will be ammeter reading?
A Circuit Not Surrounding a Solenoid
Each loop is subject to similarmagnetic field emf of loops in series:
dt
dNemf mag
The EMF for a Coil With Multiple Loops
Time varying B can beproduced by moving coil:
by moving magnet:
by rotating magnet:(or coil)
Moving Coils or Magnets
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?
3. The bar magnet is rotated. What will ammeter read?
dt
demf mag
Exercise
+
_
_
𝐵𝐵
?
‘Magnetic force’ approach:
BvqEqFtot
vBE LvBemf I
Use Faraday law:
dt
demf mag
ABmag tLvB
LvBt
emf mag
t
0lim I
Faraday’s Law and Motional EMF
𝑣 ∆ 𝑡
𝐿
𝐿
B1 B2
vL
R
dt
demf mag
mag B2A B1A B2 B1 A
Lvt tLvBBmag 12
120
lim BBvLt
emf mag
t
R
BBvL
R
emfI 12
I
Example
dt
demf mag
AnB ˆ
cosBwh
t Bwh cos t
tdt
dBwh
dt
demf mag cos
tBwhemf sin
I
Faraday’s Law and Generator