Mutual Inductance - Physics & Astronomy

51
24 P21- Mutual Inductance 1 12 12 2 1 12 12 2 N M I N M I Φ Φ = 2 12 12 dI dt M ε 12 21 M M M = = A current I 2 in coil 2, induces some magnetic flux Φ 12 in coil 1. We define the flux in terms of a “mutual inductance” M 12 :

Transcript of Mutual Inductance - Physics & Astronomy

Page 1: Mutual Inductance - Physics & Astronomy

24P21-

Mutual Inductance

1 12 12 2

1 1212

2

N M INM

I

Φ ≡Φ

→ =

212 12

dIdt

Mε ≡ −

12 21M M M= =

A current I2 in coil 2, induces some magnetic flux Φ12 in coil 1. We define the flux in terms of a “mutual inductance” M12:

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7P24-

Self InductanceWhat if we forget about coil 2 and ask about putting current into coil 1?There is “self flux”:

1 11 11 1 N M I LINLI

Φ ≡ ≡Φ

→ =

dILdt

ε ≡ −

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Figure 30-3

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8P24-

Calculating Self Inductance

NLIΦ

= V s1 H = 1 A⋅

Unit: Henry

1. Assume a current I is flowing in your device2. Calculate the B field due to that I3. Calculate the flux due to that B field4. Calculate the self inductance (divide out I)

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Figure 30-4

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Group Problem: Solenoid

Calculate the self-inductance L of a solenoid (n turns per meter, length , radius R)

REMEMBER1. Assume a current I is flowing in your device2. Calculate the B field due to that I3. Calculate the flux due to that B field4. Calculate the self inductance (divide out I)

L N I= Φ

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Inductor Behavior

IdILdt

ε = −

L

Inductor with constant current does nothing

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dILdt

ε = −Back EMF

I

0, 0LdIdt

ε> <

I

0, 0LdIdt

ε< >

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Inductors in CircuitsInductor: Circuit element which exhibits self-inductance

Symbol:

When traveling in direction of current:

dILdt

ε = −Inductors hate change, like steady stateThey are the opposite of capacitors!

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14P24-

LR Circuit

0ii

dIV Ldt

IRε= =− −∑

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LR Circuit

0 dI L dIL Idt R dt R

IR εε ⎛ ⎞= ⇒ = − −⎜ ⎟⎝ ⎠

− −

Solution to this equation when switch is closed at t = 0:

( )/( ) 1 tI t eR

τε −= −

: LR time constantLR

τ =

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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.

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LR CircuitReadings on Voltmeter

Inductor (a to b)Resistor (c to a)

c

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.

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18P24-

General Comment: LR/RCAll Quantities Either:

( )/FinalValue( ) Value 1 tt e τ−= − /

0Value( ) Value tt e τ−=

τ can be obtained from differential equation (prefactor on d/dt) e.g. τ = L/R or τ = RC

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Group Problem: LR Circuit

1. What direction does the current flow just after turning off the battery (at t=0+)? At t=∞?

2. Write a differential equation for the circuit3. Solve and plot I vs. t and voltmeters vs. t

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21P24-

Non-Conservative Fields

R=100ΩR=10Ω

Bddd tΦ

⋅ = −∫ E s

I=1A

E is no longer a conservative field –Potential now meaningless

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Figure 30-6

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Figure 30-8

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Figure 30-9

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Figure 30-16

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Figure 30-19

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Figure 30-9

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4P25-

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.

( )/( ) 1 tI t eR

τε −= −

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5P25 ­

LR Circuit: AC Output Voltage

0.00 0.01 0.02 0.03 0.00

0.05

0.10

0.15

0

1

2

3

)

)

Current -3 -2 -1 0 1 2 3

Vol

l

I (A

Time (s

Indu

ctor

(V) tmeter across L

Out

put (

V) Output Vo tage

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10P25-

Mass on a Spring

2

2

2

2 0

d xF kx ma mdt

d xm kxdt

= − = =

+ =

0 0( ) cos( )x t x tω φ= +

(1) (2)

(3) (4)

What is Motion?

0 Angular frequencykm

ω = =

Simple Harmonic Motion

x0: Amplitude of Motionφ: Phase (time offset)

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Mass on a Spring: Energy

0 0( ) cos( )x t x tω φ= +

(1) Spring (2) Mass (3) Spring (4) Mass

Energy has 2 parts: (Mass) Kinetic and (Spring) Potential

22 2

0 0

2 2 20 0

1 1 sin ( )2 21 1 cos ( )2 2s

dxK m kx tdt

U kx kx t

ω φ

ω φ

⎛ ⎞= = +⎜ ⎟⎝ ⎠

= = +

0 0 0'( ) sin( )x t x tω ω φ= − +

Energy sloshes back

and forth

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Simple Harmonic Motion

Amplitude (x0)

0 0( ) cos( )x t x tω φ= −

1Period ( )frequency ( )

2angular frequency ( )

Tfπ

ω

=

=

Phase Shift ( )2πϕ =

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Analog: LC Circuit

Mass doesn’t like to accelerateKinetic energy associated with motion

Inductor doesn’t like to have current changeEnergy associated with current

22

2

1;2

dv d xF ma m m E mvdt dt

= = = =

22

2

1;2

dI d qL L E LIdt dt

ε = − = − =

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Analog: LC Circuit

Spring doesn’t like to be compressed/extendedPotential energy associated with compression

Capacitor doesn’t like to be charged (+ or -)Energy associated with stored charge

21;2

F kx E kx= − =

21 1 1;2

q E qC C

ε = =

1; ; ; ;F x q v I m L k Cε −→ → → → →

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16P25 ­

LC Circuit

resistor, and battery. 1. Set up the circuit above with capacitor, inductor,

2. Let the capacitor become fully charged. 3. Throw the switch from a to b 4. What happens?

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LC Circuit It undergoes simple harmonic motion, just like a mass on a spring, with trade-off between charge on capacitor (Spring) and current in inductor (Mass)

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LC Circuit

0 ; Q dI dQL IC dt dt

− = = −

2

2

1 0 d Q Qdt LC

+ =

0 0( ) cos( )Q t Q tω φ= + 01LC

ω =

Q0: Amplitude of Charge Oscillationφ: Phase (time offset)

Simple Harmonic Motion

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LC Oscillations: Energy

222 01

2 2 2E BQQU U U LI

C C= + = + =

2220

0cos2 2E

QQU tC C

ω⎛ ⎞

= = ⎜ ⎟⎝ ⎠

22 2 2 20

0 0 01 1 sin sin2 2 2B

QU LI LI t tC

ω ω⎛ ⎞

= = = ⎜ ⎟⎝ ⎠

Total energy is conserved !!

Notice relative phases

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Damped LC Oscillations

Resistor dissipates energy and system rings down over time

Also, frequency decreases: 2

2 0 '

2 R L

ω ω ⎛ ⎞ = − ⎜ ⎟⎝ ⎠

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22P24-

This concept (& next 3 slides) are complicated.

Bare with me and try not to get confused

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Kirchhoff’s Modified 2nd RuleB

ii

dV d Nd tΦ

∆ = − ⋅ = +∑ ∫ E s

0Bi

i

dV Nd tΦ

⇒ ∆ − =∑If all inductance is ‘localized’ in inductors then our problems go away – we just have:

0ii

d IV Ld t

∆ − =∑

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Ideal Inductor• BUT, EMF generated

in an inductor is not a voltage drop across the inductor!

d ILd t

ε = −

i n d u c t o r 0V d∆ ≡ − ⋅ =∫E sBecause resistance is 0, E must be 0!

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Energy Stored in InductordIIR Ldt

ε = + +

2 dII I R L Idt

ε = +

( )2 212

dI I R L Idt

ε = +Battery

SuppliesResistor

DissipatesInductorStores

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Energy Stored in Inductor

212LU L I=

But where is energy stored?

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Example: Solenoid Ideal solenoid, length l, radius R, n turns/length, current I:

0B nIµ= 2 2oL n R lµ π=

( )2 2 2 21 12 2B oU LI n R l Iµ π= =

22

2Bo

BU R lπµ

⎛ ⎞= ⎜ ⎟⎝ ⎠

EnergyDensity

Volume

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Energy Density

Energy is stored in the magnetic field!2

2Bo

Buµ

= : Magnetic Energy Density

2

2o

EEu ε

= : Electric Energy Density

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35P24-

Group Problem: Coaxial Cable

XI I Inner wire: r=aOuter wire: r=b

1. How much energy is stored per unit length? 2. What is inductance per unit length?

HINTS: This does require an integralThe EASIEST way to do (2) is to use (1)

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26P21-

TransformerStep-up transformer

Bs s

dNdt

ε Φ=

Bp p

dNdt

ε Φ=

s s

p p

NN

εε =

Ns > Np: step-up transformerNs < Np: step-down transformer

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Transmission of Electric Power

Power loss can be greatly reduced if transmitted at high voltage

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Example: Transmission linesAn average of 120 kW of electric power is sent from a power plant. The transmission lines have a total resistance of 0.40 Ω. Calculate the power loss if the power is sent at (a) 240 V, and (b) 24,000 V.

5

2

1.2 10 5002.4 10

P WI AV V

×= = =

×(a)

83% loss!!2 2(500 ) (0.40 ) 100LP I R A kW= = Ω =

5

4

1.2 10 5.02.4 10

P WI AV V

×= = =

×(b)

0.0083% loss2 2(5.0 ) (0.40 ) 10LP I R A W= = Ω =

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31P21-

Recall Polar Dielectrics

Dielectric polarization decreases Electric Field!

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32P21-

Para/Ferromagnetism

Applied external field B0 tends to align the atomic magnetic moments

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Para/Ferromagnetism

The aligned moments tend to increase the B field

0mκ=B BEκ

= 0EECompare to:

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Para/Ferromagnetism

Paramagnet: Turn off B0, everything disordersFerromagnet: Turn off B0, remains (partially) ordered

This is why some items you can pick up with a magnet even though they don’t pick up other items

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35P21-

Magnetization Vector

M=0 M>0Useful to define “Magnetization” of material:

1

1 N

iiV V=

= =∑ µM µ 0 0µ= +B B M

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36P21-

Hysteresis in FerromagnetsThe magnetization M of a ferromagnetic material depends on the history of the substance

Magnetization remains even with B0 off !!!