Dr. Ray Kwok sjsu · Heinrich Friedrich Emil Lenz (1804-1865) Italy Lenz’s Law (1834) “Back...

Electrodynamics

Dr. Ray Kwok

sjsu

Electrodynamics – Dr. Ray Kwok

Static Fields (stationary charges, steady current)

=×∇

=⋅∇

=×∇

ρ=⋅∇

f

f

JH

0B

0E

D

rr

r

r

r

µ=⋅

=⋅

=⋅

ε=⋅

∫∫∫

∫

to

o

t

IdB

0adB

0dE

QadE

lrr

rr

lrr

rr

)MH(HHB

PEEED

oro

ororrrrr

rrrrr

+µ=µµ=µ=

+ε=εε=ε=

Gauss’s Law

Conservative

No magnetic charge

Ampere’s Law

f

r

fbft

b

o

t

P

E

ρ≤ερ

=ρ+ρ=ρ

ρ−=⋅∇

ερ

=⋅∇

r

rlinear, homogeneous, isotropic


Orsted’s Discovery

toIdB µ=⋅∫ lrr

V

S

R

compass

I

André-Marie Ampère(1775–1836) France

SI unit for current

Hans Christian Ørsted(1777–1851) Danmark

cgs unit for B

4/21/1820

Danmark

Ampere’s Law, 9/18/1820,

after he learned about Orsted’s

discovery on 9/11/1820 !!


Faraday’s Experiment

Michael Faraday(1791–1867) England

SI unit for capacitance

Joseph Henry(1797–1878) USA

SI unit for inductance

With stationary magnet,

no current induced (1831)


Magnetic induction


Faraday’s Law

∫

∫⋅=Φ

Φ−=⋅≡

adB

dt

ddEVemf

rr

lrr

R

uB

R

VI

VuBdt

dxB

dt

d

xBdxB

emf

emf

x

0

l

ll

ll

==

−===Φ

==Φ ∫

magnetic flux

oppose the “change” of

direction given by Lenz’s Law

u

B

IR

x

l


u

B

IR

x

Heinrich Friedrich Emil Lenz

(1804-1865) Italy

Lenz’s Law (1834)“Back emf” to oppose the “change” of magnetic flux


Example – moving magnet


Example - jumping ring

V

S

R

Al ring

I

FB B

I

FB


e.g. magnet in a copper tube

N

S

Bind


Eddy current - disc brake

BLIFBrrr

×=

Lenz’s Law

metal plate


To reduce eddy current

transformer


Example – metal detector


Example – guitar pickup


Question

Right, 0, Left, Left, Left

V

S r

R

Find the direction of the current in the resistor R shown in Figure at

each the following steps: (a) at the instant the switch is closed, (b)

after the switch has been closed for several minutes, (c) when the

variable resistance r increases, (d) when the circuit containing R

moving to the right, away from the other circuit, and (e) at the instant

the switch is opened.


Moving Conductor

u

B

+FB

−−−−FB

l

l

rrr

rrr

rr

uBV

uBV

BuE

0BuqEq

0FF BE

−=

−=

×−=

=×+

=+

saturate when

emf

u

B+

-

IR

induced E

similar to Hall Effect

induced

separate the charges

direction agrees w/ Lenz’s Law

emf is “+” u x B

induced current

I = |V/R| = B u/Rl


Generalized Faraday’s Law

∫∫

∫∫∫∫

∫∫

∫∫

⋅∂∂

−⋅×=Φ

−=

⋅×−⋅∂∂

=⋅

××∇−

∂∂

=⋅

⋅×=⋅≡

⋅∂∂

−=Φ

−=⋅≡

adt

BdBu

dt

dV

dBuadt

Bad)Bu(

t

BadB

dt

d

dBudq

FV

adt

B

dt

ddEV

emf

Bemf

emf

rr

lrrr

lrrrr

rrrr

rrr

lrrr

lr

r

rr

lrr

changing B(t), stationary loop

fixed B, moving loop

induced emf in a moving loop w.r.t. “stationary” B(t)

u

loop

magnet

Einstein’s Relativity:

move the loop, Lorentz force (magnetic) [motional emf]

move the magnet, induced emf – electric [transformer emf]

hw


Example – rotating loop

tsinBAtsinBh2

w2

h)tsinuB(2dBuVemf

ωω=ω

ω=

ω=⋅×= ∫ lrrr

tsinBA)tcosBA(dt

d

adBdt

dVemf

ωω=ω−=

⋅−= ∫rr

θ = ωt uniform B

OR

Direction of current (at this instance)??


Group Exercise

Find the induced emf in a

rectangular loop rotating at

an angular velocity ω in a

magnetic field Bosinωt.

& the direction of induced

current?

(Can you do this problem in

2 ways?)

−ωBoAcos2ωt


tsinBAN

dt

dNVemf

ωω=

Φ−=

AC generator


DC generator


Electromotor

To turn faster, should we

1. use thicker wire?

2. use more turns?

3. make bigger loop?

4. use stronger magnet


Answer… (not counting friction)

e.g. flash lights

l

l

rr

r

rrrrr

rlrr

'AVm

'AR

I

nNIAm

BmFr

BIF

R

BAN|I|

tsinBANdt

dNV

B

B

emf

ρ=ρ=

ρ=

α=τ

=

×=×=τ

×=

ω=

ωω=Φ

−= 1. use thicker wire?

2. use more turns?

same.

e.g. half R, double mass, same α

same.

double N, double R, same I,

double mm, but double inertia, same α.

3. use bigger loop? (same N)

same.

bigger A, more Φsomewhat higher R, but still more I ~A/R~ r

mm~A2/R~r3, inertia~r3, same α

torque

magnetic moment

I = moment of inertia

4. use stronger magnet?

YES

more I, m, τ, α

resistivity

mass density


Another important application – telephony / loudspeaker

(?? before him)

Alexander Graham Bell(1847 –1922) UK

(practical improvement)

Thomas Alva Edison

(1847 –1931) USA


Ideal Transformer (AC)

Φ is confined in the core (µ = ∞)

L

2

2

1

2

2

1

2

2

1

1in

2

1

1

2

2

1

2211

21

2

1

2

1

22

11

RN

N

N

N

I

V

I

VR

N

N

I

I

V

V

IVIV

PP

N

N

V

V

dt

dNV

dt

dNV

=

==

==

=

=

=

Φ−=

Φ−=

RL


Self-Inductance

dt

dIL

dt

dNV

I

NL

−=Φ

−=

Φ≡

back emf causes I lags V

V = IZ = I(R + jωL) = I |Z|ejθ

RL

V


Ideal Solenoid

ideal:

• large N tightly wound

• no end effect

• uniform internal B

• zero external B in the vicinity

AnI

nIAN

I

NL

nIAadB

nINI

B

2

oo

o

oo

l

rrl

µ=µ

=Φ

=

µ=⋅=Φ

µ=µ

=

∫

Ampere’s Law:


π

µ=

Φ=

π

µ=φ⋅φ

πµ

=⋅=Φ

φπ

µ=

µ=⋅

∫∫

∫

a

bln

2

h

IL

a

bln

2

Ihhdrˆˆ

r2

IadB

ˆr2

IB

IdB

o

o

b

a

o

o

o

rr

r

lrr

Coaxial Cable


LC Resonator (Lenz’s Law, later resonator…)


Mutual-Inductance

1

1212

IM

Φ≡

Φ12 is the flux through loop 2 due to the

B generated by loop 1

21

12

21o

1

1212

21

12

1o12

oo

21212112

Mr

dd

4IM

ddr

I

4

'd|'rr|

)'r(I

4'dV

|'rr|

)'r(J

4)r(A

dAadAadB

∫∫

∫∫

∫∫

∫∫∫

=⋅

πµ

=Φ

=

⋅π

µ=Φ

−πµ

=−π

µ=

⋅=⋅×∇=⋅=Φ

lr

lr

lr

lr

lr

rrrr

rrrr

lrrrrrr

more than 1 turn?

1

12212

I

NM

Φ=


Transformer - Primary / Secondary coils

same cross-section?

12

12

2

1

1

2

2

2

1

1

M

I

N

I

N

tN

V

N

V

Φ==

∂Φ∂

−==

same equations different cross-section? h.w.


Magnetic Energy

21

2

22

2

11m

222121

111212

2222

1111

122222211111m

12222

21111

2211m

IMIIL2

1IL

2

1W

MIIM

MIIM

IL

IL

I2

1I

2

1I

2

1I

2

1W

I2

1I

2

1W

++=

==Φ

==Φ

=Φ

=Φ

Φ+Φ+Φ+Φ=

Φ+Φ=Φ

Φ+Φ=Φ

Φ+Φ=

For coupling circuits 1 & 2…For an inductor …

Φ=

=

µ= ∫

I2

1W

LI2

1W

dVB

2

1W

m

2

m

o

2

m


Faraday’s Law - differential

Stokes’ theorem

E is induced in moving medium

B is measured in stationary frame

curl operates in moving frame

t

)r(B)r(E

))r(Bu('t

)r(B)'r(E

)Bu(t

BE

ad)Bu(t

BadE

adBdt

ddEVemf

∂∂

−=×∇

××∇+∂

∂−=×∇

××∇+∂∂

−=×∇

⋅

××∇−

∂∂

−=⋅×∇

⋅−=⋅≡

∫∫

∫∫

rrrr

rrrrr

rr

rrr

r

rrrr

rr

rrlrr

for stationary loop (rest frame)

or localized, RF EM wave (microscopic)

relativity, low f


Electrodynamics

=×∇

=⋅∇∂∂

−=×∇

ρ=⋅∇

f

f

JH

0Bt

BE

D

rr

r

rr

r

µ=⋅

=⋅

Φ−=⋅

ε=⋅

∫∫

∫

∫

to

o

t

IdB

0adB

dt

ddE

QadE

lrr

rr

lrr

rrGauss’s Law

Faraday’s Law

No magnetic charge

Ampere’s Law


Electric Potential

0V

VE

0E

=∇×∇

−∇≡

=×∇r

rstatic

define

such that

t

AVE

Vt

AE

0t

AE

Att

BE

∂∂

−−∇=

−∇≡∂∂

+

=

∂∂

+×∇

×∇∂∂

−=∂∂

−=×∇

rr

rr

rr

rr

rdynamic

Need both V and A to find E !!!!!

0A

AB

0B

=×∇⋅∇

×∇≡

=⋅∇

r

rr

r

define

such that


Group Exercise

I1(t)

I2

1 m

3 m

5 m

N = 20

Rectangular loop of 20 turns is placed at

1 m away from a long current-carrying

wire as shown.

I1(t) = 2 cos(60t) (A)

Resistivity of the wire in the loop is 4 Ω/m.

Find I2.

+2.6 sin(60t) µA

independent of N

Dr. Ray Kwok sjsu · Heinrich Friedrich Emil Lenz (1804-1865) Italy Lenz’s Law (1834) “Back...

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Transcript of Dr. Ray Kwok sjsu · Heinrich Friedrich Emil Lenz (1804-1865) Italy Lenz’s Law (1834) “Back...