MIT电磁学幻灯片23

1 0A0 1.0
0 5A0 0.5
A = A0 e
0τ 1τ 2τ 3τ 4τ
−t τ
A
A0/e = 0 368 A0 0.368 0.0A0
5τ
6τ
14
Time t
Exponential Behavior dA 1 Slightly modify diff. eq.: = − A − Af dt τ A “decays” to A :
1. 2 2. 3. 4 4. 5. 6.
ε /R ε /2R - ε /R - ε /2R Zero I don’t know
27
MIT OpenCourseWare
8.02SC Physics II: Electricity and Magnetism
Fall 2010
I 1A I=1A
G G dΦB E ⋅ d s = − ∫ dt
E is no longer a conservative field –
Potential now meaningless
6
Kirchhoff’s Modified 2nd Rule
G G dΦB Δ Vi = − v E ⋅ d s =+ ∑ ∫ dt i
Δ Vi n d u c t o r
r r ≡ −∫ E ⋅ d s = 0
dI ε = −L dt
Because resistance is 0, E must be 0!
9
Circuits: Applying Modified Kirchhoff’s (Really Just Faraday’s Law)
For the above circuit sketch the currents through the two bottom branches as a function of time (switch closes at t = 0, 0 opens at t = T). State values at t = 0+, T-, T+
26
Concept Q.: RL Circuit
In the circuit at right the switch S has been closed a very long time. At t = 0, , the switch is opened. p Taking downward current as positive, immediately after the switch is opened the current in the inductor is equal to
1. 1 2. 3. 4. 5. 6. 7. 8. 9. +10 V -10 V +100 V -100 V +110 V -110 V +90 V -90 V None of the above
A R=10Ω R=100Ω B
5
Non-Conservative Fields
R=10 R 10Ω R=100Ω
22
Non-Ideal Inductors
Non-Ideal Non -Ideal (Real) Inductor: Not only L but also some R
=
In direc− IR dt
23
Experiment 8: Part 1 Inductance & LR Circuits
4
Concept Question: Faraday Circuit
A magnetic field B penetrates this circuit outwards, and is increasing at a rate such that a current of 1 A is induced in the circuit (which direction?) direction?). The potential difference VA-VB is:
Module M d l 23: 23 LR Circuit
1
Module 23: Outline
LR Circuits Expt. p 8: Part 1: LR Circuits
2
Think Harder about Faraday
3
Concept Question Question: Faraday y in Circuit
For information about citing these materials or our Terms of Use, visit: /terms.
dΦB ⇒ ∑ Δ Vi − =0 dt i
If all inductance is ‘localized’ in inductors then our problems go away – we just have:
dI ∑ Δ Vi − L d t = 0 i
7
Inductors in Circuits
16
LR Circuit
⎞ dI 1 ⎛ ε =− I − ⎜ ⎟ dt L R⎝ R⎠
Solution to this equation when switch is closed at t = 0:
I (t ) =
1− e ) ( R
−t /τ
ε
L τ = : time constant R
(units: seconds) ( )
(
)
1 0Af 1.0
f
A
0.5Af
A = Af 1 − e
0τ 1τ 2τ 3τ 4τ
(
−t τ
)
6τ
15
0.0Af
5τ
Time t
This is one of two differential equations we expect you to know how to solve (know the answer to). t ) The other is simple p harmonic motion (more on that next module)
19
Concept Question Question: Voltage g Across Inductor
20
Concept Question: Voltage A Across I Inductor d t
In the circuit at right the switch is closed at t = 0. A voltmeter lt t hooked h k d across the inductor will read:
1. 1 2. 3. 4.
VL = ε e −t /τ VL = ε (1 − e ) VL = 0
I don’t know
21
−t /τ
LR Circuit
c
Readings on Voltmeter Inductor (a to b) Resistor (c to a)
t=0+: Current is trying to change. Inductor works as hard as it needs to to stop it t=∞: Current is steady. Inductor does nothing.
17
LR Circuit
t=0+: Current is trying to change. Inductor works as hard as it needs to to stop it t=∞: Current is steady. Inductor does nothing.
18
Problem: Circuits
Inductor: Circuit element with self-inductance Id ll it h Ideally has zero resistance i t
Symbol:
8
Ideal Inductor
• BUT, EMF generated by an inductor is not a voltage drop across the inductor!
24
Concept Question Questions: LR Ci Circuits it
25
Concept p Question: Inserting g a Core
When you insert the iron core what happens?
1. B Increases so L does too 2. B Decreases so L does too 3 B Increases so L Decreases 3. 4. B Decreases so L Increases 5. I don’t know
10
LR Circuit
dI ∑Vi = ε − IR − L dt = 0 i
11
LR Circuit
⎞ dI dI 1 ⎛ ε ε − IR − L = 0 ⇒ =− I − ⎜ ⎟ dt dt L R⎝ R⎠
12
Review Some Math: Exponential Decay
13
Exponential Decay y dA 1 Consider function A where: =− A dt τ A decays exponentially: