Time varying fields & Maxwell’s equations

(TOPIC 5)beyond the steady state

A time varying magnetic field generates an electric field (“induction”), and a time varying

l t i fi ld t ti fi ld (“di l t t”)

(TOPIC 5)

electric field generates a magnetic field (“displacement current”)

The first concept was demonstrated by one of the greatest experimenters of all time, the

second one was first introduced by one of the greatest mathematical physicists of history

h d d d l d d f ldWhy do we study in detail time dependent fields?

1) Only with time varying fields it is possible to derive the formula of the magnetic field energy

2) Only with time varying fields is the connection between electricity and magnetism evident

3) El t i hi ( t t d t f ) k ith ti i fi ld3) Electric machines (generators, motors and transformers) work with time varying fields

4) Time varying charges and currents generate electromagnetic radiation:

light is a time varying electric & magnetic field (electromagnetic wave)

Contents of TOPIC 5:

Electromotive force with static & time varying magnetic fields: Lorentz force;

Faraday’s law in integral & differential form; relation Lorentz force & Faraday’s lawFaraday s law in integral & differential form; relation Lorentz force & Faraday s law

Magnetic energy and magnetic forces, applications: generator, motor, transformer

Time varying electric fields: Ampère Maxwell’s law and displacement currents

Maxwell’s equations, Poynting vector and Poynting’s theorem, irradiance

Electromagnetic waves, refractive index for perfect insulators, Snell’s law

Lorentz force law & Electromotive forcesLorentz force on moving charges BvqEqF

rrrrLorentz force on moving charges BvqEqFLorentz

Lorentz force “density” :

(force per unit volume) ),(),(),(),(

),(),(),(),(),(

rtBrtJrtErt

rtBrtvrtErtrt drift rrrrrrr

rrrrrrrrrf

Polar aurora:

Example: motion in (quasi) uniform B (and E) fields

),(),(),(),( rtBrtJrtErt

lr

d1DEF: electromotive force =

(electromotance, emf ,

= work per unit charge) force per unit charge

ld

qFr

lrr

lrr

dFq

df1

E

magelecLorentz dBvdEdq

Fdf EEE l

rrrlrr

lr

r

lrr

o ce pe u t c a ge

If the applied force

is the Lorentz force:

there are 2 ways to generate a current:

electric electromotance: batteries & fuel cells

magnetic electromotance generators

qis the Lorentz force:

V

magnetic electromotance: generators

IMPORTANT : the B field does no work on moving charges, but it does work on magnetic dipoles

(and thus on magnetic materials). Moreover, energy must be spent to create magnetic fields.

Magnetic emf in a moving loopBvvqBvqFrrrrrr

surface delimited by C(t + t)

BvvqBvqFmag wiredriftcharge

vdrift is the speed of the charge with respect

to the wire: it is parallel to j and thus to d

)t

tvloopmag

vdBvvdB

Bvvd

iid ift

wiredrift

rlrrrr

lrr

rrrlr

EE

B t i th t d di t d l t f ti f th ibb HdAd ˆr

lr

trdvd wire

rlrr

lr

)tlooploopvdBvvdB wirewiredrift ll

, where is the displacement of wire segment during trr

lr

d

But is the outward directed area element of a section of the ribbon. Hence:

E = ( )/ t , where ( ) is the flux of B through the lateral surface (the integration is only

performed on d since in the other direction the thickness is infinitesimal). Now, the flux of B

through a surface enclosing a volume must be zero; hence ( ) + (C(t + t)) + (C(t)) = 0

dAnrdl

through a surface enclosing a volume must be zero; hence ( ) + (C(t + t)) + (C(t)) = 0,

where the 2nd and 3rd terms are the flux through the upper and lower surfaces in the figure.

In the definition of the flux, the direction of is outwards. But here it is more convenient to

define it always in the same direction with respect to the loop Let’s say it is outwards (paralleln̂

define it always in the same direction with respect to the loop. Let s say it is outwards (parallel

to ) for the surface S(t + t); then it is antiparallel to for S(t); with this definition the sign

of (C(t)) must be changed, to give:dt

tCd

dt

tCdttCdtd

))(())(())(()(E

n̂ n̂

dtdt

This magnetic electromotive force (which does not stem from any battery/voltage difference)

is non conservative (the line integral of Fmag around the closed loop is different from zero).

Applications of Lorentz’s law and magnetic emfHow can one tell which way a magnetic emf goes? Look at the direction of the magnetic force,How can one tell which way a magnetic emf goes? Look at the direction of the magnetic force,

or remember Lenz’s law: the current generated by a magnetic emf always flows so as to

oppose the change in external flux.

AC generator &AC generator &

electric motorMagnetic brake

(“Eddy currents”)

tBAdAB cos

tBA sinE

i l ti l l tsimple particle accelerator

(cyclotron)

Other applications of the Lorentz force:

DC electric motor mass spectrometer

Hall effect (metal/semiconductor) & polarization of dielectric moving in B field

pp

Faraday’s law: induction(1) (2)(2)

The t o sit ations (1) & (2) are the same onl described b obser ers mo ing ith relati e speedThe two situations (1) & (2) are the same, only described by observers moving with relative speed v :

In (1), the charges in the loop are moving in a static B field and are

subject to the Lorentz force, which acts as electromotive force (topic 4):dt

d BE

In (2) the same net electromotive force must be present the very

existence of the Lorentz force by itself implies a modification of the field

equations for E: the line integral of E cannot be zero in (2), it must be equal to: dt

ddE Blrr

q g ( ) q

In order for the equation of the e.m.f to holds for case (2) no matter what the shape of the loop, it must be:

Faraday’s law: a changing magnetic field

d l t i fi ld (& th i d d f)B

E

rrr

produces an electric field (& thus an induced e.m.f)t

E

In the most general case, the total e.m.f. in a moving loop which is not connected to a power

supply is the sum of two terms one due to the Lorentz force and the other due to induction:supply is the sum of two terms, one due to the Lorentz force, and the other due to induction:

loopS

dBvadt

Blrrrr

r

ES

adBdt

d

dt

d rrE. The total e.m.f. is always equal to:

Implications of Faraday’s lawIn the most general case, the total e.m.f. in a moving loop which is not connected to a power

supply is the sum of two terms, one due to the Lorentz force, and the other due to induction:

lS

dBvadt

Blrrrr

r

ES

adBdt

d

dt

d rrE. The total e.m.f. is always equal to:

electromagnetic induction = generation of an E field by a B field that varies with time

BE

rrr

loopSt S

Faraday’s law

Si l li ti Fi d th ti i B fi ld th t i d th l t i fi ld

tEFaraday s law

(actually discovered independently by Faraday and Henry in 1830 1831)

Simple application: Find the time varying B field that induces the electric field:

Answer:tyzxyaxE cos)ˆˆ( 432 tzyaxxzB

sin)ˆ3ˆ4( 223

r

Since , we can write Faraday’s law as: or:t

AE

rrrr

ABrrr

0t

AE

rrr

The last equality implies that the field in brackets is a conservative field, hence one can write:q y p ,

Vr

rr

t

AE

1) E is no longer conservative:A

E

rrr

V

This equation has two extremely important consequences:

t ) gt

E V

2) In the presence of time variations, the energy U = qV associated with electrical interactions

e.g. inside a battery, goes not only in the creation of electrical fields, but also of magnetic fields

Technological applications of Faraday’s lawBr

Applications of Faraday’s discoveries:

t

BErr

1) induced emf

2) Magnetic energy and self inductance, inductors

3) Foucault (or Eddy) currents: induction stove (heating due to Joule effect and

hysteresis loss of ferromagnetic iron pan which also amplifies the magnetic field

– you need a special pan for an induction kitchen!)

4) induction motor, transformer, …

induction motor transformerinduction cooker

AC input

current

Michael Faraday’s curriculum vitaeScientific terms and concepts introduced by M.F. :

Force field, force lines (electric and magnetic lines), ion (anion, cation), voltmeter,

electrode (anode/cathode), electrolyte, electrolysis, dielectric, dia & paramagnetic

Most important scientific discoveries

1) Faraday’s induction law

2) Linear dielectric and magnetic response:

diamagnetism

dielectric constant

3) Faraday’s laws of electrochemistry

4) Faraday rotation (magneto optic effect)

Inventions

Electric motor (1821)

Electric generator and precursor of dynamo (1831)Electric generator and precursor of dynamo (1831)

precursor of transformer (1831)

Units named after M.F.:Units named after M.F.:

farad (capacitance) ; faraday (electrochemistry: charge of NA electrons)

Energy balance in a circuitConsider a circuit in which the current density is increased from zero toConsider a circuit in which the current density is increased from zero to

some value Jf . The power source (e.g. battery) needs to supply energy to

accelerate the electrons; such energy is not only to overcome friction, but

also the effect of the opposing Faraday field generated by the increment

Take an element d of the circuit, of cross section da . Let’s calculate the work done by the

also the effect of the opposing Faraday field generated by the increment

of J, that is, the corresponding variation of .

source (battery) to increase the current across d from zero to a finite value If . The source

maintains a fixed voltage drop across any two points of the circuit. If at a given instant the

current through the circuit is I(t), the instantaneous power supplied by the source is equal to :

If the two points are an infinitesimal distance d away, the voltage drop between them is dV,

which is the variation of the scalar potential field across d : lrr

dd VV

ItIabbattery V)()VV(

rrHence, using , we get:

Now, we saw that Faraday’s law implies . We thus get:

dJIdIdd f V V)V(rr

lrr

Vr

rr

t

AE

dJId f

rlr

rt

(1) h i l

dJt

AdJEd ff

rr

rr

ff Jt

AdJEd

rr

rr

The first term is the mechanical power absorbed by the charges, which is finally dissipated as

Joule heat. The second one is instead stored as magnetic energy as we show in the next slide

(1) mechanical power (2) Variation of magnetic energycharges

Magnetic energyff J

t

AdJEd

rr

rr

(1) (2)charges

The first term is the mechanical power required to accelerate the charges and keep them

dtdUmag

charges driftdriftdriftfdriftff vFvFdvEdqvEdJEdrrrrrrrrrr

(1)

dd

moving against the viscous friction drag responsible for the resistance of the wire:

Notice that we can extend the integrals to all space, since and are zero outside the wire

The 2nd term is the power used up against the induced e m f created by the time variation of

ff dqd

driftvr

fJr

The 2nd term is the power used up against the induced e.m.f. created by the time variation of

the magnetic field. It is the time variation of a magnetic energy stored in the magnetic field:

BHd

AHd

AHdH

AdJ

Ad f

rr

rrr

rrrrr

rr

r

(2)ttttt

f

In linear media . Hence:

( )

HB r

rr

0at 0ad

t

AH

rr

r

dt

dUud

dt

dBd

dt

dB

dt

dd

t

BBd

mag

mag 2

2

)2(0r

2

0r

2

0r

rr

. Therefore:

dU B rr12

with and

Umag is the energy stored in the magnetic field (= used to build up the field against induction)

dt

dUsource

mag

charges magmag udU BHB

umag

rr

2

1

2 0r

Self inductance and magnetic energyThe magnetic energy in linear media can also be written as In fact it is:AJdU

rr1The magnetic energy in linear media can also be written as . In fact it is:AJdU fmag

2

1

fmag JAdHAdHAdAHdBHdUrrrrrrrrrrrrr

2

1

2

1

2

1

2

1

2

1

According to Biot Savart’s law, the B field

of a loop carrying a uniform current I is: 3

0

3

0loop

)()()( rrIdrrrJdB

rrlrrrrrr

02

1 atadHA

rrr

of a loop carrying a uniform current I is:

We see that the field is linearly proportional to I . In the same way, the flux of B through the

surface delimited by the loop is also proportional to the current:

33loop||4||4 rrrrrrrr

y p p p

Th i li ffi i L b h fl f B d I i h

ILadrr

rrdIadBB

||

)(

4 3

0 rrr

rrlr

rrILB

The proportionality coefficient L between the flux of B and I is the

(self )inductance of the closed loop. In terms of L, Faraday’s law becomes:dt

dIL

dt

d BE

(this is the formula you use in circuit theory)

Similarly, we find for the magnetic energy of a loop (using ):

Bfmag IadBIadAIAdIAJdU 2

1

2

1

2

1

2

1

2

1 rrrrrrlrrr

dJId f

rlr

Stokes’ theorem

Magnetic energy of a loop:L

LIIU BBmag

22

1

2

12

2

Magnetic vs electric energy formulasIt is interesting to compare the energy formulas we just found with those for electrostatics:It is interesting to compare the energy formulas we just found with those for electrostatics:

AJHBurrrr 11

VEDu11 rr

AJHBu fmag22

VEDu fel22

IL

V

QC

f2

2

1

2

1LIIUmag

2

2

1

2

1CVQVU el

fI V22g 22

extBmU extEpU

*Superconductors: London eq., Meissner effectSuperconducting currents are not really two dimensional: they are not strictly confined to the

surface of the medium but actually penetrate inside it a short distance, called London

penetration depth. In a normal metal, Ohm’s law holds. The fact that J (a constant

times vdrift) is proportional to E is a consequence of electron collisions. But in a superconductor

EgJ f

rr

Edsuch collisions do not occur; rather, electrons have an accelerated motion:

Since (where n = density of superconducting electrons) we get, instead of Ohm’sm

eE

dt

dva d

d

dt

dvne

dt

J d

law, the London equation: (with ). Taking the curl of both sides and

i F d ’ l bt i H t f t t th t

Em

Ene

t

J 2

BJ )(

dtdt

m

ne2

using Faraday’s law we obtain: . Hence apart from a constant that

we can discard, we have . Combining this with Ampère’s law for B,BJrrr

t

BE

t

J )(

JBrrr

0

(valid since a superconductor is nonmagnetic), we get:

, which yields, using :0B0B

rr

BrBBrrrr

0

with = skin depth

h l h ( )

20

2

L

BBBrr

2

0ne

mL xeBxB 0)(

xeBxB )(The solution to this equation is (in 1D) :

This shows that B decays inside a superconductor in a short

distance . Hence B = 0 inside : this is the Meissner effect !superconductor

L

air

eBxB 0)(

*Condensation energy of a superconductorWe saw in topic 4 that a magnetic field destroys the superconducting state. We can use this toWe saw in topic 4 that a magnetic field destroys the superconducting state. We can use this to

calculate the energy density of the superconducting state at a given temperature T . Take a long

cylinder made of a superconducting material, initially in the absence of a magnetic field. As we

turn on a weak magnetic field, it fills all space except for the region occupied by the cylinder,g , p p g p y y ,

since superconducting currents arise that screen the interior of the cylinder from the field. When

the applied field reaches the value of the critical field at the temperature T, the

superconducting state ceases to exist because it is energetically more favorable to have the

region it occupies filled by the magnetic field. Hence we can calculate the so called

“condensation energy” density of the superconductor as the equivalent magnetic energy

density. The energy density associated with a magnetic field (or, which is the same, with the2

the superconducting current density needed to expel it) is

0

2

22

1 BHBumag

rr

Here we considered that the normal and superconducting states are basically non magnetic, sop g y g ,

that (the normal metal is in fact a Pauli paramagnet, but also in this case )HBrr

01r

2 )(TB

The condensation energy density uSC of the superconductor corresponds to the value of the

ti d it h th B fi ld i l t it iti l l

h l f l f h d d d ll l l d f

0

2

2

)(TBuu c

BSC c

magnetic energy density when the B field is equal to its critical value:

The simple formula for the superconducting condensate energy density is actually only valid for

a cylinder. For other geometries the amount of energy required to expel the magnetic field turns

out to be equal to the magnetic field energy times a geometrical factor.

Mechanical force in an electric circuitConsider a closed circuit of resistance R and inductance L, connected to a source that provides a

constant current I (with I = const, the power transferred to the electrons is dissipated as heat:

). Suppose that we move or deform the circuit in such a way as to vary the

inductance L and thus the flux through the circuit. The voltage drop across the resistance R is:

IR Joulecharges

The e.m.f supplied by the source that maintains the current

constant must overcome the induced e.m.f. due

IRdt

dV B induced EEE

IRIdL

IRILd

IRd B

Eto he change of L, as well as the Joule losses Hence: IRIdt

IRdt

IRdt

E

On the other hand, by energy conservation, the energy provided by the source must be equal to

the change in energy stored in the magnetic field, plus the heat lost by Joule heating, plus the

dt

dW

dt

dUI mecmag

source Joule Emechanical work W to move the coil:

Assuming that the work is done by a force F across a distance dx, we find:g y ,

FdxQdLIdWQILdIdt JoulemecJoule

22

2

1

2

1 E

Plugging the electromotive force found above in the last equation we finally get:1

(note: QJoule is heat, not charge)

FdxQdLIdtIRIdLIdt Joule

222

2

1 E . Since Joule’s heat is , this gives:dtIRQJoule

2

dLIdLIdLIFdx 222

2

1

2

1. Hence:

d

dLIF 2

2

1

22 dx2

This last expression for the force can also be written, using , as:L

U Bmag

2

2

const

mag

B

dx

dUF

Inductance, energy & force in a magnetic circuitWe saw in topic one that in a magnetic circuit Hopkinson’s law applies: M = N I =We saw in topic one that in a magnetic circuit Hopkinson s law applies:

Here core is the flux through a cross section of the circuit; for a coil of N turns, the total flux

crossing it is . Neglecting the magnetic field outside the circuit, we have:

M N I core

coreB N

N 2

NNI B

core , or: . The self inductance of a

coil of N turns wrapped around a magnetic circuit of reluctance is thus:

ILIN

B 2N

L

The magnetic energy of a magnetic circuit with only one coil around it carrying a current I is :

corecoremag IN

LIU 2

1

2

1

2

1

2

1

2

1 22

22

2 MM

corecoremag22222

If there is a gap in the magnetic circuit, there is a force between the opposite poles of the

electromagnet across the gap, given by:

Alth h l l t d thi f i th i lid i t t t th f

dx

d

dx

dNI

dx

dLIF

2

2222

2

11

2

1

2

1 M , that is:

dx

dF core

2

2

1

Although we calculated this force in the previous slide assuming a constant current, the force

between the poles cannot depend on how the power supply works, but it is always the same for

a given pole density, that is, for a given magnetic flux through the core. This is true even if the

magnetomotive force is provided by a permanent magnet instead of a coil Notice that suchmagnetomotive force is provided by a permanent magnet instead of a coil. Notice that such

force can be written, using the expression for the energy given above, as:

const

mag

core

dx

dUF

Electric machines & the electric gridClassification of electric machines:

1) Machines that convert mechanical energy/work into electric energy: GENERATORS

2) Machines that convert electric energy into mechanical energy/work: MOTORS

3) Machines that inter convert electric energy: TRANSFORMERS

In cases 1) and 2), the machine contains at least a coil moving in an external magnetic field.

Whenever this happens, there is:

a) an induced e.m.f./back e.m.f. in the coil, as a result of the time varying flux;a) an induced e.m.f./back e.m.f. in the coil, as a result of the time varying flux;

b) a mechanical boost/friction force due to the Lorentz force

It is impossible to have one without the other; however:

generators are designed in such a way as (also) minimizing the back e.m.f. that would reduceg g y ( ) g

the output voltage;

motors are designed in such a way as (also) maximizing the current through the loop.

In both cases part of the task is achieved by ensuring that the magnetic field generated by the

current in the loop is negligible with respect to the external field

OVERVIEW of an

ELECTRIC GRID

(electric power(electric power

delivery system)

AC generators (power plants)

tBAdAB cos

tBAd

sinE tBAdt

sinE

Hydroelectric / Thermal / nuclearHydroelectric /

tidal power plant

/

power plant

Solar energySolar energy

concentration

windmills

(Ideal) transformer

The primary coil consists of

NP turns, the secondary one

of NS turns. Suppose an AC

lt i li d t thvoltage is applied to the

primary coil. By Faraday’s

law, the induced e.m.f. in the

secondary coil is:secondary coil is:

Since the same magnetic flux

goes through the primary

coil, one also must have:

Hence for an ideal transformer where a is the winding turns ratioHence for an ideal transformer , where a is the winding turns ratio

Motors(brushed or brushless)A motor has a moving part

( ) d f d ( ) DC motor: a DC current,

whose direction reverses

every half turn, flows

(rotor) and a fixed part (stator).

There are many types of motors:

A

through the rotorReluctance motor : AC

power (usually with

different phases) is supplied A

C Bto the windings (A, B, C) of

the stator. The total

reluctance of the magnetic

A

CBcircuit made of stator and

rotor depends on their

relative orientation

Induction (asynchronous)

motor and synchronous motor:

AC power is supplied to the

stator windings, producing a B

field which causes an induced

d i f icurrent and a magnetic force in

the rotor, which contains either

windings or a “squirrel cage”

Ampère Maxwell law & displacement currentAmpère’s law can’t hold in general. In fact :

fJHrrr

a) Taking the divergence of Ampère’s law we get: . This can’t be true

always, since it violates the charge conservation law:

p g

0)( HJ f

tJ

f

f

fJH

(we see that Ampère’s law only holds for magnetostatics )

b) Ampère’s law in integral form, applied to the

charging of a capacitor, gives conflicting results on

tf

different surfaces enclosed by the same loop:

!!! 0

a bS S

ff adJIadJdHrrrr

lrr

r

Maxwell “fixed” Ampère’s law by adding an extra term, obtaining:t

DJH f

rrrr

a)“Displacement

current density”

b) Charging of a

ffft

JDt

JH )()(

EDrr

capacitor at a

constant rate

(current): Qf(t) = If t

f

S

f

SS

f Iadt

EIad

t

DadJdH

111

10

11

rrrrlrr

0 1

t

Jg

r

H1 = H2f

SS

f ISDdt

dad

t

DadJdH 22

22

22

r

rrr

lrr

22

2

S

tI

S

QD

ff

f

Displacement current densitydisplacement You saw the 1st term in Fisica 2 it is calledPEDdisplacement

current density

You saw the 1st term in Fisica 2, it is called

“vacuum” displacement current density

We encountered the 2nd term previously, it is the bound charge current density Jb

t

P

t

E

t

D0

When can the displacement current be neglected?

1) In magnetostatics, always (no time variation of the fields)

2) In general for slow variations (quasi static approximation):

p y, g y b

2) In general for slow variations (quasi static approximation):

3) Especially inside metals at all frequency attainable in electronics (below infrared/visible):

)()( tJtH f

rrr“slow varying field”

approximationt

DJH f

rrrr

3) Especially inside metals, at all frequency attainable in electronics (below infrared/visible):

Example: time varying current in a metal. If , then:EgJ f

rrtEE sin0

g

E

gE

D

|J|

tEt

E

t

P

t

E

t

D

tgEJf

f 18

000

0max

0000

0 10

cos

sin

r

r

ttttt

max=0 in a metal

I ntdisplaceme 180 10 I t l ll IIgI free

p 180 10 In a metal, normally: freentdisplaceme II

the slow varying approximation holds in metals at electronic frequencies

Sources of B and HThe Ampère Maxwell’s equation besides implying the local form of the charge

DJHThe Ampère Maxwell s equation , besides implying the local form of the charge

conservation law, together with Gauss’s law for H (or B) it entails that there are four types of

sources of H and B:

tJH f

rrr

vacuum displacement

0

mag

ff

MH

t

P

t

EJ

t

DJH

rrrr

rrr

rrrr

0

0

0

B

t

E

t

PMJ

Bf

rr

rrrrr

rr

Jr

We see that the sources of B are indeed 4 :

Free currents, (1)

vacuum displacement

current density

mag 0B

fJr

eJbJr

the curl of M, which we know from Ampère’s equivalence theorem corresponds to an

equivalent current density (2)

the displacement current density , which is the sum of :tDr

MJ e

rrr

f

p y ,

the “proper” (vacuum) displacement current density, (3)

the bound charge current density, (4)t

Pt

Er

0

The last term is not surprising since we already know that is the current of bound

charges, which is a current of real charges and as such a source of B. Some books define a

t

bJt

P rr

“total” current density as , so that one can write an equation similar to the

Ampère Maxwell equation in vacuum you saw in Física 2:

bfftot JJJJrrrr

t

EJB tot

rrrr

000

Summary: axioms of electromagnetismin the absence of material media, we

E ED,D totftotf

rrrrrr I) 0

in the absence of material media, we

get back the equations of Física 2:

BE

BE

BBr

rrr

rr

rrrr

III)

0 0 II)

0

t

EJB

HB

ED

t

DJH

tE

tE

ff

rrrr

rr

rrrrrr

IV)

III)

000

BJEfJBJJEf

tHBt

ffLorentzbbbfbfLorentz

rrrrrrrrrr ,0 V)

0

EMAG =

Maxwell’s equationsS

+

Lorentz force

+

Some consequences:

B and E fields are interrelated !!!

Poynting’s theorem (e.m. energy theorem)

Existence of e m waves !!!+

Constitutive relations

Existence of e.m. waves !!!

Uniqueness theorem for EMAG

MAXWELL’s EQUATIONS

George Street,

Edinburgh

e.m. energy (Poynting’s) theorem in vacuum

If k h d d f E i h A è M ll’ iE

B

rrr1

If we take the dot product of E with Ampère Maxwell’s equation we get:t

B 0

0

t

EEBE

rrrrr

0

0

1. Using the general vector identity , it is :BEEBBE

rrrrrrrrr

(we also used )t

BE

0000

0

111 BE

t

BBBEEB

t

EE

BErr

rWith the definition of Poynting’s vector , the last equation can be written as:

0

BESr

SBEB

BE

E 22

00

111. Since:

1

densityenergy electric vacuum2

1 2

0E

tttt 0

0

0

022

densityenergy magnetic vacuum2

1 2

0

B

0SuPoynting’s theorem (in vacuum)

ti f l t ti0Sut

em

Sem SdVUt

In integral form (integrating over the volume) :

conservation of electromagnetic energy

the variation of electromagnetic energy (in vacuum) is equal to the flux of Poynting’s vector;

Poynting’s vector takes e.m. energy across the boundary of a volume

t

Def: time average of the modulus of Poynting vector = irradiance ( ) =

= (electromagnetic) energy flux per unit area and unit time

Poynting’s theorem in linear mediaWe define the Poynting’s vector as . Taking the divergence of the Poynting vector,HES

we find (using the product rule for derivatives):

BE

HEEHHESrrrrrrrrrrr

Using Faraday’s law and Ampère Maxwell’s law this entails:t

DJH f

rrrr

t t

t

DE

t

BHJE

t

DEJE

t

BHS ff

rr

rrrr

rrrr

rrrr

where:Since for a linear mediumt

uu

t

BH

t

DE

t

BH

t

DE

magel

rrrrrr

rr

2

1

2

1

densityenergymagnetic1

densityenergy electric 2

1

BHu

DEuelmagelf uu

tJESrrrr

We then get : , or:

densityenergy magnetic 2

BHumag

U

SJEt

uf

merrrr

..

Th l f l i i i l f i l h fl f

Sdt

Umechanical

em In integral form (integrating over a finite volume) :

Poynting’s theorem (e.m. energy theorem):

The loss of electromagnetic energy in a given volume of space is equal to the flux of

Poynting’s vector (which carries energy out of the volume) plus the power used to

accelerate charges. (the latter is equal to Joule’s heat for steady currents inside a conductor )

Maxwell’s eq

Electromagnetic waves (light) in vacuum

Using the vector identity: we get the:

Maxwell s eq.

(in vacuum)

Using the vector identity: we get the:

Speed of

propagation:smc 8

00

1031

e.m. wave

i

01

2

2

2

2

t

E

cE

rr

propagation:27132212 1041085.81 mkgCkgmCsc

00

)i ()( kA1 22 dd h i

equations0

12

2

2

2

t

B

cB

rr

)sin(),( 0kxtAtxy0

12

2

22

2

dt

yd

cdx

yd harmonic

solution:In 1D:

k d )i ()( kAYrrrrr

con /k = f = c

In 3D, k and y are vectors:

B

Err

rr

0

0

)sin(),( 0rktAtrYrr

EBBkEkrrrrrr

Bv

t

BE

Br

rr

0 EBBkEk , ,

kr

e.m. waves are transverse waves

in vacuum, Faraday’s law implies B = E/c

Er k

, y p /

the direction of E is called polarization of the e.m. wave (light)

E.m. waves in nonmagnetic linear dielectricswith no free charges nor currents (ideal dielectrics)

Maxwell equations for a nonmagnetic, nonconducting medium ( ):

D

with no free charges nor currents (ideal dielectrics)1r

f

BE

t

DJH

2

2

00 , D

ED

B

rrrr

rrr

D

B

tE

0(with Jf = 0)

200 ,tt

Using the linear constitutive relation for D, we get:

fD

n

ccv

rr

f

11

00

This is the wave equation of waves propagating with speed

h f i i d d h l ic

where = refraction index and = phase velocityrnn

cv f

From Faraday’s law we get00 BEk

t

BE

rrrr

rr En

EE

kB

00

00

rrrr

is the dielectric constant of electrostatics. If did not depend on frequency, n would be

constant. This is only true in the low frequency limit, while in general and n depend on .rr

t cv f

Light as electromagnetic energy: irradianceEnergy density ( u ) associated with E and B fields :

22

02

1 ;

2

1BuEu mre (with )1gy y ( )

Since , it is

0

022 r

mre

2

0..

2

02

2

0

22

0

)( 2

1

22

1tEuuuuE

c

EnBu rmemeerm

c

E

v

EB r

f

(with )1r

2

00

22

00

22

00

2

0

1)(sin)(sin)( EtEutEtEu rrmerrme < > = time average

Visible light: , 1015 Hz , the time variation of E can’t be measured, nor that of u !!! We can

only measure time averages such as that of the energy density < u >. For a harmonic wave:

Energy flux divided by area = energy transported by the wave per unit time and area.

BE

0000..000..2

)()()( rrmerrme < > time average

For a plane wave in vacuum:

1A

tvl

BE,

uemftAemtA

f

uvVoluU

tAvAVol

1,11,1

l

k

The flux is in the direction of k. In vector form : kHES u

rrrrtv fl

S = Poynting’s vector

I f i ll l drrr 22

r

The time averaged energy crossing a surface per unit time and unit surface is called irradiance ,

and it equals the time average value of the modulus of . For a monochromatic e.m. wave:S

In fact, is parallel to , andkHErr 2

00

2

0 tEvvtEEBEHtS rff

q g

2

0

0

..2

1 E

vuvS

f

mef

S

NOTE: light (e.m. wave) carries not only energy,

but also linear and angular momentum

Types of light: spectral ranges

harmonic waves travelling at speed c (or c/n): f = c (or c/n)

f is a constant (inside dielectric) = (vacuum)/n = 0/nf is a constant (inside dielectric) (vacuum)/n 0/n

Wave frequency (and vacuum wavelength) is associated with COLOR

Boundary conditions in opticsDr

rrr1n̂

f

t

BE

t

DJH

rrr

0Brr

21 BB

1

fD

B

t

rr

rr0

21 nn BB

2n̂

Br

t

BErr

21 tt EE

(the flux of B through the area da goes to zero in( g g

the limit that the segments d 1 and d 2 tend to

merge – they are taken just across the boundary)

Normally there is neither free surface charge nor free surface current on the interface. In suchy g

case the four boundary conditions that must be satisfied (for linear media) are:

Usually in optics one deals with nonmagnetic media. Even in the case of soft ferromagnets, the

magnetization cannot follow applied fields at frequencies higher than few MHz, so that the

t ib ti f M t B t IR ti l f i i li ibl H th b d diticontribution of M to B at IR or optical frequencies is negligible. Hence the boundary conditions

for optical fields become simply:B = const , E// = const ,

Boundary between media: light is partially reflected (R) and

Reflection & refraction at a planar interfacereflected n

I nTpartially transmitted (T) (that is, refracted). The E (or B) field

on the left hand side of the boundary is the sum of EI and ER ,

on the other side it is equal to ET R

reflected

transmitted

nI nT

There exists a fixed relation between the fields at all points

of the boundary. For a monochromatic incident wave, this

Boundary conditions in optics: B = const, E// = const

rI

RT

y ,

implies that the three waves (incident, reflected & transmitted)

must have equal total phase :

Setting the origin at a point on the boundary, at the origin , hence

trktrktrk TTRRII

rrrrrr

0r

r

ttt TRI

incident

( consistent with the definition of photon and with energy conservation)

For t = 0 : are coplanarTRI

TRI kkkrrr

and , rkrkrk TRI

rrrrrr

Taking to be coplanar with the wave vectors we get:

This implies:TTRRII kkk sinsinsin

rRRIIRI rkrkrkrk sinsin

rrrr

law of specular (incident and reflected waves propagate in the same medium

of refractive index they have the same value of k )IRInreflection

Snell’s law (the transmitted wave propagates in medium with

with same frequency as the incident wave changes)IITT nn sinsin

IT nn

Reflectance for normal incidenceBoundary conditions in optics: B = const, E// = const . E

ry p , //

For light impinging at normal incidence on a planar

boundary this implies:TRI

EEE 000

0

Br

IE

Ikr

Tkr

From Faraday’s law:TRI

BBB 000

En

k

EBBkE

t

BE 00

000

rrr

IB

nI

Rkr

nT

ckt

TTRIII

TRI

EnEnEn

EEE

000

000 The minus sign takes into account the change in relative

orientation of E & B in the reflected wave (as )IR kkrr

TTRIII 000

I En

E 00

2

Solving for the transmitted and reflected field amplitudes:

IR kk

We thus get:

I

TI

R

I

TI

T

Enn

nnE

Enn

E

00

00

REFLECTANCE

2

2

0

2

0

TI

TI

I

R

I

R

nn

nn

E

ER

0

0

TI

TI

I

R

nn

nn

E

E

g

TI nn 0 TIII

The transmittance is defined as . For energy conservation, it must be: 1TRTI

TT

NOTE: reflectance and transmittance are the same if light impinges from one side or the other.

For the air glass interface (e.g. at a clean window), R is about 4% at normal incidence, so a

window reflects at least 8% of the incoming light (R increases with increasing incidence angle)

Specular reflection vs Diffuse reflection

Total internal reflection: optical fibers, pentaprism

http://www.youtube.com/watch?v=hBQ8fh_Fp04&feature=related

y

Refraction in non homogeneous media: miragesy

Consider an inhomogeneous medium whose refractive index varies with the vertical coordinate

y . A light ray undergoes refraction wherever there is a change of refractive index. Due to Snell’s

x

y . A light ray undergoes refraction wherever there is a change of refractive index. Due to Snell s

law, since n is varying continuously with y, a light wave travelling through this medium

undergoes a continuous refraction mirages:

http://www youtube com/watch?v=d8zUHBbflmMhttp://www.youtube.com/watch?v=d8zUHBbflmM

GRIN waveguideThis effect leads to the phenomenon of

mirages and is also exploited commercially tomirages, and is also exploited commercially to

achieve so called graded index materials

(GRIN), used for lenses, waveguides and fibers.