Faraday’s Law
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Faraday’s law cannot be derived from the other fundamental principles we have studied Formal version of Faraday’s law: n: given by right hand rule Faraday’s Law Differential form of Faraday’s law: ∮ ∙ =− [∫ ∙ ^ ] × = − curl( dt d emf mag
description
Faraday’s Law. 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. Differential form of Faraday’s law:. c url(. Two Ways to Produce Changing . Two ways to produce curly electric field: - PowerPoint PPT Presentation
Transcript of Faraday’s Law
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 rule
Faraday’s Law
Differential form ofFaraday’s law:
∮𝐸 ∙𝑑 �⃗�=− 𝑑𝑑𝑡 [∫ �⃗� ∙ �̂�𝑑 𝐴 ]
�⃗�× �⃗�=− 𝜕 �⃗�𝜕𝑡
curl(
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
Constant voltage – constant I, nocurly electric field.
Increase voltage: dB/dt is notzero emf
dNIB 0For long solenoid:
Change current at rate dI/dt:
20
1 RdNI
dtd
dtd
emf mag dtdIR
dN 20
(one loop)
dtdIR
dNemf 2
20
emfbat
Remfcoil
Inductance
emfbat
Remfcoil
dtdIR
dNemf 2
20
EC
Increasing I increasing B
dtd
emf mag
ENC
dtdILemf ind
emfbat
R
emfind
L – inductance, or self-inductance2
20 RdNL
Inductance
ENC
EC
emfbat
R
emfind
L
dtdILemf ind
IremfV solindsol
22
0 RdNL
Unit of inductance L: Henry = Volt.second/Ampere
Inductance
Increasing the current causes ENC to oppose this increase
EC
dtd
emf mag
ENC
emfbat
R
emfind
L
dtdILemf ind
Conclusion: Inductance resists changes in current
Inductance: Decrease Current
Orientation of emfind depends on sign of dI/dt
202
1 EVolume
energy Electric
)()(dtdILIemfIVIP
∫∫ f
i
I
I
IdILPdtEnergy
22
21
21 LILIEnergy
f
i
I
I
22
0 RdNL
dNIB 0
2
0
22
0
21
NBdR
dNEnergy
dRBEnergy 2
0
2
21
VBEnergy 2
0
121
2
0
121 B
Volume
energy Magnetic
Magnetic Field Energy Density?
L I2
202
1 EVolume
energy Electric
2
0
121 B
Volume
energy Magnetic
2
0
20
121
21 BE
VolumeEnergy
Electric and magnetic field energy density:
Field Energy Density
0 inductorresistorbattery VVV
0dtdILRIemfbattery
ctbeatI )(
0 ctctbattery LbceRbeRaemf
Remf
a battery LbcRb LRc
tLR
battery beR
emftI
)(
If t is very long:R
emftI battery )(
Current in RL Circuit
tLR
battery beR
emftI
)(
If t is zero: 0)0( I
01)0( bR
emfI battery
Remf
b battery
tLR
battery eR
emftI 1)(
Current in RL circuit:
Current in RL Circuit
tLR
battery eR
emftI 1)(
Current in RL circuit:
Time constant: time in which exponential factor become 1/e
1tLR
RL
Time Constant of an RL Circuit
0 inductorcapacitor VV
0dtdIL
CQ
dtdQI
02
2
dt
QdLCQ
ctbaQ cos
0coscos 2 ctbcLCctba
a=0LC
c 1
LCtbQ cos
LCtQQ cos0
Current in an LC Circuit
LCtQQ cos0
dtdQI
LCt
LCQI sin0
Current in an LC circuit
Period: LCT 2
Frequency: f 1 / 2 LC
Current in an LC Circuit
0 Rinductorcapacitor VVV
0dtdILRI
CQ
Non-ideal LC Circuit
Initial energy stored in a capacitor:C
Q2
2
At time t=0: Q=Q0 CQUcap 2
20
At time t= : Q=0LC2 2
21 LIU sol
System oscillates: energy is passed back and forth between electric and magnetic fields.
Energy in an LC Circuit
1/4 of a period
What is maximum current?
At time t=0:
mageltotal UUU C
Q2
20
At time t= :LC2
mageltotal UUU 2max2
1 LI
CQLI22
1 202
max LCQI 0
max
Energy in an LC Circuit
Frequency: f 1 / 2 LC
Radioreceiver:
LC Circuit and Resonance
Varying B is created by AC current in a solenoid
What is the current in this circuit?
tmag sin0
dtdemf
temfemf cos0
tR
emfR
emfI cos22
0
Advantage of using AC: Currents and emf ‘s behave as sine and cosine waves.
Two Bulbs Near a Solenoid
Add a thick wire:
Loop 1
Loop 2
I1
I2
I3
Loop 1: 02211 IRIRemf
Loop 2: 022 IR 02 I
Node: 321 III 31 II
11 R
emfI
Two Bulbs Near a Solenoid
Add a thick wire:
Loop 1
Loop 2
I1
I2
I3
Loop 1: 02211 IRIRemf
Loop 2: 022 IR 02 I
Node: 321 III 31 II
11 R
emfI
Two Bulbs Near a Solenoid
Exercise
Exercise
