lecture 18-1 ways to change magnetic flux changing the magnitude of the field within a conducting...
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Lecture 18-Lecture 18-11 Ways to Change Magnetic Flux
• Changing the magnitude of the field within a conducting loop (or coil).
• Changing the area of the loop (or coil) that lies within the magnetic field.
• Changing the relative orientation of the field and the loop.
motor generator
cosB BA
http://www.wvic.com/how-gen-works.htm
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Lecture 18-Lecture 18-22
N
1. define the direction of ; can be any of the two normal direction, e.g. point to right
2. determine the sign of Φ. Here Φ>0
3. determine the sign of ∆Φ. Here ∆Φ >0
4. determine the sign of using faraday’s law. Here <0
5. RHR determines the positive direction for EMF • If >0, current follow the direction of the curled
fingers. • If <0, current goes to the opposite direction of
the curled fingers.
n
n
How to use Faraday’s law to determine the induced current direction
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Lecture 18-Lecture 18-33Eddy Current
• The presence of eddy current in the objectresults in dissipation of electric energy that is derived from mechanical motion of the object.
• The dissipation of electric energy in turncauses the loss of mechanical energy ofthe object, i.e., the presence of the field damps motion of the object.
A current induced in a solid conducting object, due to motion of the object in an external magnetic field.
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Lecture 18-Lecture 18-44 Self-Inductance
• As current i through coil increases, magnetic flux through itself increases.
This in turn induces counter EMF in the coil itself
• When current i is decreasing, EMF is induced again in the coil itself in such a way as to slow the decrease.
Self-induction
BLi
BN
Li
(if flux linked)
Faraday’s Law:Bd dI
dtL
dt
2 / /L T m A A HWb (henry)
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Lecture 18-Lecture 18-55
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Lecture 18-Lecture 18-66Solenoid: Archetypical Inductor
0
Nr l B i
l
Current i flows through a long solenoid of radius r with N turns in length l
2 20B
NA r BA i r
l For each turn
222 2
0 0BN N N
L r l ri l l
For the solenoid
or2
0L n Al
Inductance, like capacitance, only depends on geometry 0 /H m
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Lecture 18-Lecture 18-77Potential Difference Across Inductor
00 dt
dILIR
V I r
V
+
-
I
internal resistance
• Analogous to a battery
• An ideal inductor has r=0
• All dissipative effects are to be included in the internal resistance (i.e., those of the iron core if any)
00 dt
dILIR
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Lecture 18-Lecture 18-88RL Circuits – Starting Current
0dI
IR Ldt
2. Loop Rule:
3. Solve this differential equation
/( / ) /( / )1 ,t L R t L RL
dII e V L e
R dt
τ=L/R is the inductive time constant
2 2/ /
//
T m A T m AL R s
V A
1. Switch to e at t=0
As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I.
+
-
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Lecture 18-Lecture 18-99
Starting Current through Inductor vs Charging Capacitor
/( / )t L RL
dIV L e
dt
/(1 )t RCq C e
/t RCI eR
/( / )1 t L RIR e
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Lecture 18-Lecture 18-1010
R1
R2 LV
Which of the following statement is correct after switch S is closed ?
1. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/R1
2. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V.
3. At t = 0, the potential drop across the inductor is 0; When t = ∞, the current through R1 is V/(R1+R2)
4. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/R2
Warm-up
S
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Lecture 18-Lecture 18-1111Remove Battery after Steady I already exists in RL Circuits
3. Loop Rule: 0dI
IR Ldt
4. Solve this differential equation
/( / )
/( / )
t L R
t L RL
I eR
dIV L e
dt
I cannot instantly become zero!
Self-induction
like discharging a capacitor
1. Initially steady current Io is flowing: 0 1I R R
-
+
2. Switch to f at t=0, causing back EMF to oppose the change.
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Lecture 18-Lecture 18-1212Behavior of Inductors
• Increasing Current
– Initially, the inductor behaves like a battery connected in reverse.
– After a long time, the inductor behaves like a conducting wire.
• Decreasing Current
– Initially, the inductor behaves like a reinforcement battery.
– After a long time, the inductor behaves like a conducting wire.
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Lecture 18-Lecture 18-1313Energy Stored By Inductor
1. Switch on at t=0
0dI
IR Ldt
2. Loop Rule:
3. Multiply through by I
As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. +
-
2 dII I R LI
dt
Rate at which battery is supplying energy
Rate at which energy is dissipated by the resistor
Rate at which energy is stored in inductor L
mdU dILI
dt dt 21
2mU LI
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Lecture 18-Lecture 18-1414Where is the Energy Stored?
• Energy must be stored in the magnetic field!
Energy stored by a capacitor is stored in its electric field
20L n Al0 ,B nI• Consider a long solenoid where
2
2 2 20
0
1 1 1
2 2 2m
BU LI n Al I Al
area A
length l2
0
1
2m
m
U Bu
Al
• So energy density of the magnetic field is
20
1
2Eu E (Energy density of the electric field)
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Lecture 18-Lecture 18-1515Physics 241 –Quiz A
The switch in this circuit is initially open for along time, and then closed at t = 0. What is themagnitude of the voltage across the inductorjust after the switch is closed?
a) zero
b) V
c) R / L
d) V / R
e) 2V
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Lecture 18-Lecture 18-1616Physics 241 –Quiz B
The switch in this circuit is closed at t = 0. What is the magnitude of the voltage across the resistor a long time after the switch is closed?
a) zero
b) V
c) R / L
d) V / R
e) 2V
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Lecture 18-Lecture 18-1717Physics 241 –Quiz C
The switch in this circuit has been open for a long time. Then the switch is closed at t = 0. What is the magnitude of the current through the resistor immediately after the switch is closed?
a) zero
b) V / L
c) R / L
d) V / R
e) 2V / R