Clicker Question: Non-uniform B field
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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
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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
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Chapter 23
Faraday’s Law
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Solenoid: insidedNIB 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
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1. Coulomb electric field: produced by charges rrqE ˆ
41
21
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
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Right hand rule:
Thumb in direction offingers: ENC
dtBd 1
Exercise:
Magnetic field points downfrom the ceiling and is increasing.
What is the direction of E?
Direction of the Curly Electric Field
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ENC causes current torun along the ring
What is the surface chargedistribution?
What is emf and I?
ldEemf NC
22 rENC
RrEI NC 22
Driving Current by Changing B
Ring has resistance, R
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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
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ldEemf NC
A
DNC
D
CNC
C
BNC
B
ANC ldEldEldEldEemf
=0 =0
D
CNC
B
ANC ldEldEemf
0emf for any path notEnclosing solenoid!
Round-Trip Not Encircling the Solenoid
+ _
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Is there current in these circuits?
Exercise
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Can observe experimentally:
I=emf/R ENC~emf
1. ENC~dB/dt2. ENC~ cross-section of a solenoid
emf ddt
B1r12
Quantitative Relationship Between B and EMF
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emf ddt
B1r12
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!
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dtd
emf 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 ]
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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 ]
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dtd
emf 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
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If we increase current through solenoid what will be ammeter reading?
A Circuit Not Surrounding a Solenoid
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Each loop is subject to similarmagnetic field emf of loops in series:
dtd
Nemf mag
The EMF for a Coil With Multiple Loops
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Time varying B can beproduced by moving coil:
by moving magnet:
by rotating magnet:(or coil)
Moving Coils or Magnets
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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?
dtd
emf mag
Exercise
+
_
_
𝐵𝐵
?
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‘Magnetic force’ approach:BvqEqFtot
vBE LvBemf I
Use Faraday law:
dtd
emf mag
ABmag tLvB
LvBt
emf mag
t
0
lim I
Faraday’s Law and Motional EMF
𝑣 ∆ 𝑡
𝐿
𝐿
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B1 B2
vLR
dtd
emf mag
mag B2A B1A B2 B1 A
Lvt tLvBBmag 12
120lim BBvL
temf mag
t
R
BBvLR
emfI 12
I
Example
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dtd
emf mag
AnB ˆ
cosBwh
t Bwh cos t
tdtdBwh
dtd
emf mag cos
tBwhemf sin
I
Faraday’s Law and Generator
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dtd
emf mag
Two ways to produce curly electric field:1. Changing B2. Changing A
ABdtd
dtd mag
dtdABA
dtdB
Two Ways to Produce Changing