Faraday’s law cannot be derived from the other fundamental principles we have studied Formal...
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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
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Transcript of Faraday’s law cannot be derived from the other fundamental principles we have studied Formal...
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 rule
Faraday’s Law
Differential form ofFaraday’s law:
∮𝐸 ∙𝑑 �⃗�=−𝑑𝑑𝑡 [∫ �⃗� ∙ �̂�𝑑 𝐴 ]
�⃗�× �⃗�=−𝜕 �⃗�𝜕𝑡
curl(
dt
demf mag
Two ways to produce curly electric field:1. Changing B2. Changing A
ABdt
d
dt
d mag
dt
dABA
dt
dB
Two Ways to Produce Changing
Constant voltage – constant I, nocurly electric field.
Increase voltage: dB/dt is notzero emf
d
NIB 0For long solenoid:
Change current at rate dI/dt:
20
1 Rd
NI
dt
d
dt
demf mag
dt
dIR
d
N 20 (one loop)
dt
dIR
d
Nemf 2
20
emfbat
Remfcoil
Inductance
emfbat
Remfcoil
dt
dIR
d
Nemf 2
20
EC
Increasing I increasing B
dt
demf mag
ENC
dt
dILemf ind
emfbat
R
emfind
L – inductance, or self-inductance
22
0 Rd
NL
Inductance
ENC
EC
emfbat
R
emfind
L
dt
dILemf ind
IremfV solindsol
22
0 Rd
NL
Unit of inductance L: Henry = Volt.second/Ampere
Inductance
Increasing the current causes ENC to oppose this increase
EC
dt
demf mag
ENC
emfbat
R
emfind
L
dt
dILemf ind
Conclusion: Inductance resists changes in current
Inductance: Decrease Current
Orientation of emfind depends on sign of dI/dt
202
1E
Volume
energy Electric
)()(dt
dILIemfIVIP
∫∫ f
i
I
I
IdILPdtEnergy
22
2
1
2
1LILIEnergy
f
i
I
I
22
0 Rd
NL
d
NIB 0
2
0
22
0
2
1
N
BdR
d
NEnergy
dR
BEnergy 2
0
2
2
1
VBEnergy 2
0
1
2
1
2
0
1
2
1B
Volume
energy Magnetic
Magnetic Field Energy Density?
L I2
202
1E
Volume
energy Electric
2
0
1
2
1B
Volume
energy Magnetic
2
0
20
1
2
1
2
1BE
Volume
Energy
Electric and magnetic field energy density:
Field Energy Density
0 inductorresistorbattery VVV
0dt
dILRIemfbattery
ctbeatI )(
0 ctctbattery LbceRbeRaemf
R
emfa battery LbcRb
L
Rc
tL
Rbattery beR
emftI
)(
If t is very long:R
emftI battery )(
Current in RL Circuit
tL
Rbattery beR
emftI
)(
If t is zero: 0)0( I
01)0( bR
emfI battery
R
emfb battery
tL
Rbattery eR
emftI 1)(
Current in RL circuit:
Current in RL Circuit
tL
Rbattery eR
emftI 1)(
Current in RL circuit:
Time constant: time in which exponential factor become 1/e
1tL
R
R
L
Time Constant of an RL Circuit
0 inductorcapacitor VV
0dt
dIL
C
Q
dt
dQI
02
2
dt
QdLCQ
ctbaQ cos
0coscos 2 ctbcLCctba
a=0LC
c1
LC
tbQ cos
LC
tQQ cos0
Current in an LC Circuit
LC
tQQ cos0
dt
dQI
LC
t
LC
QI sin0
Current in an LC circuit
Period: LCT 2
Frequency: f 1 / 2 LC
Current in an LC Circuit
0 Rinductorcapacitor VVV
0dt
dILRI
C
Q
Non-ideal LC Circuit
Initial energy stored in a capacitor:C
Q
2
2
At time t=0: Q=Q0C
QUcap 2
20
At time t= : Q=0LC2
2
2
1LIU 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
Q
2
20
At time t= :LC2
mageltotal UUU 2max2
1LI
C
QLI
22
1 202
max LC
QI 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
dt
demf
temfemf cos0
tR
emf
R
emfI cos
220
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
