faraday’s law cannot be derived from the other fundamental principles we have studied formal...
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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
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