Faraday’s law Faraday: A transient current is induced in a circuit if A steady current flowing in...

Faraday’s lawFaraday: A transient current is induced in a circuit if

• A steady current flowing in an adjacent circuit is switched off• An adjacent circuit with a steady current is moved• A permanent magnet is moved into or out of the circuit

No current flows unless

• the current in the adjacent circuit changes or • there is relative motion of circuits

• Faraday related the transient current flow to a changing magnetic flux

1830) (circaLaw sFaraday' dt

d.dforce motive Electro

.dflux Magnetic

E

SB

S

Faraday’s law

Total or convective derivative:

0 B vvB

vBBvBvvBvB

BvBBB

AvAA

v

v

...

identity. ... x x

field inductionmagnetic x x t

dt

d

t

dt

d

f field scalar f t

f

dt

df

derivative total t

dt

d

.

.

. )

t

x

T(x,t)

∂T(x,t)/∂t

dx/dt . ∂T(x,t)/∂x

∂T(x,t)/∂t + dx/dt. ∂T(x,t)/∂x

Faraday’s lawConsider two situations:

(1) Source of B field contributing to f is moving

(2) Surface/enclosing contour on which f is measured is moving

Which situation applies depends on observer’s rest frame

Situation (1)

Rest frame of measured circuit(unprimed frame)

B is changing on S becausesource circuit is moving at v

SS

SS

SB

SBE

BB

.dt

.ddt

d.d

t

dt

d

Sv

Faraday’s law

Situation (2)

Rest frame of source circuit (primed frame)

B’ is changing because measured circuit is moving at v

Theorem Stokes'by '.d'x '.dx '

'.dx 'x ''.ddt

d

rest at is source since 0 t

' x 'x

dt

'd

''

''

''

CC

SS

SS

B vvB

SvBSB

BvB

B

S’v

Faraday’s law

Situation (2)Rest frame of source circuit (primed frame)B’ is changing because measured circuit is moving at v

'

''

''

''

C

CC

SS

SS

''.d 'q

' q charge on force 'x q '

Theorem Stokes'by '.d'x '.dx '

'.dx 'x ''.ddt

d

rest at is source since 0 t

' x 'x

dt

'd

EEF

B vF

B vvB

SvBSB

BvB

B

S’v

Lenz’s Law

dt

d.d

E

Minus sign in Faraday’s law is incorporation of Lenz’s Law which states

The direction of any magnetic induction effect is such as to oppose the cause of the effect

It ensures that there is no runaway induction (via positive feedback) or non-conservation of energy

Consider a magnetic North Pole moving towards/away from a conducting loop

SN v

dSB

Bind

SN v

dSB

Bind

B.dS < 0Flux magnitude increasesdf/dt < 0

B.dS < 0Flux magnitude decreasesdf/dt > 0

Motional EMF

Charges in conductor, moving at constant velocity vperpendicular to B field, experience Lorentz force, F = q v x B. Charges move until field established which balances F/q.No steady current established.

B

v

F = q(vxB)

-

+

Completing a circuit does not produce a steady current either

B

v

F = q(vxB)

-

+

-

+

Motional EMFemf in rod length L moving through B field, sliding on fixed U shaped wire

Charge continues to flow while rod continues to move

resistance R R

vBL Current

vBL.d.d q

1emf

B

A

B

A

I

EF

Law. sLenz' field.magnetic external the opposes

current inducedby produced field that implies sign

.dflux Magnetic

1830) (circaLaw sFaraday' dt

d.d

vLdt

d(area) since BvL

dt

d

.dflux Magnetic

S

S

SB

E

SB

I

F = q(vxB)

+ +

- -

B

v

emf induced in circuit equals minus rate of change of magnetic flux through circuit

Faraday’s Law in differential form

- potential scalarby generated is field of part This 0x NB

form aldifferenti inLaw sFaraday' t

-x

.dx t

.dx .ddt

d

Theorem Stokes' .dx .d

.ddt

d.d

notEE

BE

SEB

SESB

SEE

SBE

0

S

SS

SC

SC

. particular a is choice One fields. and

same the produce which, of choices ealternativ are There

potential vector of curl as field Representx

waythis field general Represent t

--

tx-

t

x-

t-x

- be cannot This zero!y Identicall x-

Law sFaraday' t

-x

potential of gradient as drepresente fieldtic Electrosta -

gaugeBE

A

AB A B

EA

E

A ABE

E

BE

E

0

Electric vector potential

Inductance

H or mH~typically More

0.5H turns 5000 radius, 5cm withcoil cm 50

) V/(As1 1H henry 1

inductance (self) LRN

dt

d

dt

dLRN

dt

demf

L length in loops current NL linking''flux .NL R .N

flux magnetic loops of er.area.numbB

1-

22o

22o

NL

2o

solenoidNL

L

L

L

L

II

I

Self-Inductance in solenoid

Faraday’s Law applied to solenoid with changing magnetic flux implies an emf

Area of cross section = pR2

N loops (turns) per unit length

B

IL

InductanceWork done by emf in LR series circuit

t

0

t

00

t

0

t

0

RL

dtR2

1

dtRddt' R dt'

ddt'V W

sdifference potential R Vdt

dV

'II

'II'I'I

II

II

22

2I

L

LL

LVo

L

R

First term is energy stored in inductor B fieldSecond term is heat dissipated by resistorsolenoid inductance L = moN2pr2L solenoid field B = moNIW = ½ LI2 = ½ moN2pr2L I2 = ½ (moN I)2 pr2L/mo = ½ B2 volume/mo

E

dv

dW

B

dv

dW

dvE W dvBW

2electric

2magnetic

2electric

2magnetic

22

22

1

o

o

o

o

VV

elastic exchange

of field energy

Inductance

LCR series circuit driven by sinusoidal emf

m

k

C

1 at Resonance

t)cos(m

Fxxx t)cos(

VQ

C

1Q

RQ

t)cos(Fkxxmxm t)cos(VC

QRQQ

Q RQ VC

Q VQV

R VC

Q V

dt

dV

oo

o2o

o

oo

RCL

RCL

L

LLL

L

L

L

I

II

Vo

L

C R

elastic exchange of kinetic and potential energy

Displacement currentAmpere’s Law

currentssteady -non for t

1

1

0.

0..

j

Bj

BjjB

o

oo

Problem!

Steady current implies constant charge density so Ampere’s law consistent with the continuity equation for steady currents (only).

Ampere’s law inconsistent with the continuity equation (conservation of charge) when charge density is time dependent.

Continuity equation

Displacement current

Add term to LHS such that taking Div makes LHS also identically equal to zero:

The extra term is in the bracket

extended Ampere’s Law (Maxwell 1862)

0..

0..

jj

?j

B?j

or

1

o

jE

E

EE

...

..

ttt oo

oo

t

t

ooo

oo

EjB

BE

j

1

Displacement current in vacuum (see later)

Displacement current

Relative magnitude of displacement and conduction currents

magnitude of order same currents if rangeray - xhard in isFrequency

s rad 10~10

8.854x10

Cu for m10~ty conductivielectric

t)(s t)cos(

1-198

12

1-1-8

j

E

Ej

in EE

EE

EjB

oo

t

t

t

o

oo

σ σ

Maxwell Equations in Vacuum

Maxwell equations in vacuum

Law s Ampere'Extended

Law sFaraday'

monopolesmagnetic of Absence .

Law Gauss' .o

t

t

ooo

EjB

BE

B

E

0