electromagnetic inductionhep0.okstate.edu/khanov/phys1214/ch21.pdf · electromagnetic induction 7...

Electromagnetic Induction

Induced EMF We already know that moving charge (=current)

causes magnetic field

It also works the other way around: changing magnetic field (e.g. moving permanent magnet) causes current

Electromagnetic Induction 2

•it’s called induced current •since there is a current, there is electromotive force called induced emf

Induction Experiments Stationary permanent magnet does not

induce current, the moving one does

Stationary current in the first coil does not induce the current in the second coil, moving the coil in and out or switching the current on and off does


What matters is change of magnetic field

Magnetic Flux


B

AAB

if B is not perpendicular to surface then

AB

AB

cos

cos

if Bcosϕ is the same everywhere then

cosBA

A is like a vector: you need to pick one of the two possible directions

Faraday’s Law


t

E

induced emf

rate of flux change

for a coil of N turns,

tN

E

[Φ] = weber, 1 Wb = 1 T∙m2=1 V∙s

notice the “−” sign in the formula

A Slide-Wire Generator


B

v

tv

BLv

t

AB

tLvA

constB

E

L

Lenz’s Law The direction of induced current is such as to oppose

the direction of the phenomenon causing it


Lenz’s law is a particular case of Le Chatelier's principle: in a stable equilibrium, any deviation would cause a force which drives the system back to the equilibrium (that’s why it’s stable!)

Lenz’s Law and a Slide-Wire rod


B

v

I induced current

F

direction of induced current is such that force F acting on it due to B tries to slow down the moving rod

induced field is directed opposite to B

otherwise we would get infinite acceleration without external force

inducedB

Motional Electromotive Force


B

v

I

separate from wires (no current)

qvBFB

qEFE

charges keep accumulating until electric force compensates the magnetic force

vBLV

vBE

ab

a

b

Generator =device which converts mechanical energy to

electricity


tABt

tAB

t

sin

cos

E

need calculus to derive it

Alternating and Same Sign emf


brush

slip ring

brush

commutator

Mutual Inductance =coupling between two coils


•when current through the main coil changes, the secondary coil gets induced emf •the question is, how large is it?

tN

2

22E

Φ2 is proportional to the current in the main coil:

||1

121

2

2 iMN

for convenience

t

iM

1

212E

Mutual Inductance


t

iM

1

212Et

iM

2

121E2112 MMM

it turns out that

even for different coils

[M]= henry, 1 H = 1 Wb/A=1 V∙s/A=1 Ω∙s

mutual inductance M depends on coils’ geometry and magnetic properties of the material

Transformers =two coils that share magnetic flux


tN

11E

tN

22E

1

2

1

2

N

N

E

E

Self-Inductance =mutual inductance applied to the main coil itself


t

iL

E

L

Capacitors vs inductors:

t

iLv

t

vCi

DC: i=0 (R=∞) DC: v=0 (R=0) Inductor:

Magnetic Field Energy


t

iLv

Power supplied during time Δt:

iLitPU

t

iLiviP

i

i

2

2

1LIU

Capacitors vs Inductors


2

2CVU

2

2LIU

2

2

0

Eu

0

2

2

Bu

Energy

Energy density

Mechanical analog

spring (potential energy)

mass (kinetic energy)

Resistance-Inductance Circuits


S

E

R L

a b cI

0

0

bcab VV

I



S

E

R L

a b cI

t

iLiR

E Switch closed )0( t

Lt

i

V

i

initial

ab

E

0

0



S

E

R L

a b cI

t

iLiR

E Switch closed )0( t

iL

R

Lt

i

V

i

ab

E

0

0

current increases current change rate decreases



S

E

R L

a b cI

t

iLiR

E Switch closed )( t

Ri

t

i

final

final

E

0

Current vs Time

Current, Resistance, and Direct-Current Circuits 22

– need calculus to derive it )1( / LtReR

i E

i 71828.2e

t

)/11( ei final

R

L

finali

L/R = time constant

Current Decay in LR Circuit


SE

R L

a b cI

0i

R

L

i

t

LtReii /

0

The L-C Circuit


E

L

a b cI

C

22

22 LICVEtotal LC

1

electromagnetic inductionhep0.okstate.edu/khanov/phys1214/ch21.pdf · electromagnetic induction 7...

Documents