Unformatted text preview: Lecture 25
Chapter 23. Faraday’s Law Inductance
Constant voltage – constant I, no curly electric field. Increase voltage: dB/dt is not zero emf For long solenoid: B = µ0
NI d Change current at rate dI/dt: emfbat R emfcoil Inductance
ENC emfbat
emf = Inductance
ENC emfbat R L emfind
emf ind = L dI dt R emfcoil EC EC µ0 N 2 2 dI πR d dt
dI dt Increasing I increasing B
emf ind = L emf = − dΦ mag dt emfbat R emfind L – inductance, or selfinductance Unit of inductance L: Henry = Volt.second/Ampere Increasing the current causes ENC to oppose this increase 1 Inductance: Decrease Current
ENC EC emfbat R L emfind
emf ind = L dI dt RL Circuits
• At t=0, the switch is closed and the current I starts to flow. a I
b R I # L emf = − dΦ mag dt Conclusion: Inductance resists changes in current Orientation of emfind depends on sign of dI/dt
4/19/11 Initially, an inductor acts to oppose changes in current through it. A long time later, it acts like an ordinary connecting wire. 6 RL Circuits
• Find the current as a function of time.
I= a I
b R I RL Circuits ( on)
ε 1 − e− Rt / L R Current /1 R L/R 2L/R ε ε 1 − e− Rt / L = 1 − e− t /τ RL R R ( ) ( ) # L I= ( ) Q f( x ) 0.5 I 0 0 • What about potential differences? 0 1 2 x t/RC t 3 4 Voltage on L
VL = L dI = ε e− Rt / L dt 1 1 # f( x V ) 0.5 L 0.0183156
4/19/11 7 4/19/11 0 0 0 1 2 x t 3 4 4 8 2 RL Circuits
• Why does RL increase for larger L? a I
b R I RL Circuits
After the switch has been in position for a long time, redefined to be t=0, it is moved to position b. a I
b R I # L # L • Why does RL decrease for larger R? 4/19/11 9 4/19/11 10 RL Circuits ( off)
ε − Rt / L e R (on)
/R 11 L/R 2L/R /1 R L/R 2L/R
1 (off)
/R 1 L/R 2L/R Current I= f( x ) 0.5 I Q f( x ) 0.5 I I= ε 1 − e− Rt / L R ( )
4 f( x ) 0.5 I I= ε − Rt / L e R 0.0183156 0 0 1 2 x t 3 4 4
1 0 0 0 1 2 x t/RC t 0.0183156 3 0 0 1 2 x t 3 4 4 Voltage on L
VL = L dI = −ε e− Rt / L dt 1 00 #1 1 00 Q f( x ) V 0.5 L f( xV ) 0.5 L VL = L dI = ε e− Rt / L dt Q f( x ) 0.5 VL VL = L dI = −ε e− Rt / L dt 0 # 4/19/11 0.0183156 0 1 2 x t/RC t 3 4 0 11 0 1 2 x t 3 4 4 0 # 4/19/11 0 0 1 2 x t/RC t 3 4 12 3 Inductor in Series i L1 i L2 Inductor in Parallel
L1 i i1 i2 L2 4/19/11 13 4/19/11 14 LC Circuits
• Consider the LC and RC series circuits shown: • Suppose that at t=0 the capacitor is charged to a value of Q.
++++  LC Oscillations
Kirchoff’s loop rule
CR
++++  I Q
++  C L dI Q VL + VC = L + = 0 dt C C L Is there is a qualitative difference in the time development of the currents produced in these two cases. Why?? 15 4/19/11 4/19/11 16 4 x 0, r1 n .. r1 1.01 1 Q
f( x ) 0 Q = Q0 cos(ω0 t ) x 0, r1 n .. r1 LC Oscillations LC Oscillations: Energy Check
ω0 =
1 LC 0 1.01 VC 1 f( x ) 0 1.01 0 I 1 1.01 1 0 0 2 4 6
1.01 1 6.28 I = −ωx Q0 sin(ω 0 t ) 0 f( x ) 0 0 0 1.01 1 0 2 x 4 6 6.28 VL 1.01 1 0 1.01 1 0 f( x ) 0
2 x 4 6 6.28 0 • The other unknowns ( Q0, ) are found from the initial conditions. eg in our original example we took as given, initial values for the charge (Qi) and current (0). For these values: Q0 = Qi, = 0. • Question: Does this solution conserve energy? dI dt 1.01 1 f( x ) 0 0 0 0 2 tx 4 6 6.28 1.01 1 0 0 2 x 4 4/19/11 6.28 t6 17 4/19/11 18 Energy Check
Energy in Capacitor
12 U E (t ) = Q0 cos 2 (ω 0t + φ ) 2C x 0, r1 n .. r1
1 UB versus UE UE f( x ) 0.5 Energy in Inductor 1 2 0 U B (t ) = Lω 0 Q02 sin 2 (ω 0tx+ φ0), r1 .. 0 0 r1 2 n 1
ω0 =
LC
1 2 tx 4 6 U B (t ) =
Therefore, 12 Q0 sin 2 (ω 0t + φ ) 2C UB
f( x ) 0.5 U E (t ) + U B (t ) = Q02 2C 19 0 0 4/19/11 0 2 x t 4 6 20 4/19/11 5 LC Oscillations with Finite R
• If L has finite R, then energy will be dissipated in R and
x 0, r1 n Driven Oscillations
• An LC circuit is a natural oscillator.
n 100 the oscillations will become damped. 10 r1 r1 .. r1
x 0, n .. r1 ω resonance = 1 in absence of resistive loss LC +  +  R C
r1 10 n 100 L Q 1 Q
f( x ) 0 0 1 f( x ) 0 0 • In a real LC circuit, we must account for thex 0, r1 .. r1 resistance of the inductor. This resistance will n Q damp out the oscillations.
1 f( x ) 0 1 1 0 t5 x R=0 10 1 0 5 x 10 0 R=0 t5
x 10 t 21 4/19/11 4/19/11 22 6 ...
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 Spring '07
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 Current, Inductance, Inductor, RL circuit, LC circuit

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