Lecture physical optics

7/29/2019 Lecture physical optics

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EM I&II

The energy transported by a large number ofphotonsis, on the average,equivalent to the energy transferred by a classicalelectromagnetic wave.

The dual nature of light is evidenced by the fact that itpropagates throughspace in awave-like fashion and yet displaysparticlelike behavior duringthe processes ofemission and absorption.


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3.1 Basic Laws of Electromagnetic Theory

Charge & Fields

Electric Field E

A point charge qexperiences a force , theEfield at the position of the charge is:

EF

EqFE

Magnetic Field B

A movingcharge experience force , whichis depended on its velocity and magnetic

field as:

MF

BVqFM

q

V

B

Moving charge in both Eand Bfields experiences:

BVqEqF

Electric

current

Time-varying

EfieldElectric

charge

Time-varying

Bfield

Efield Bfield

E E


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3.1.1 Faradays Induction Law

Convert magnetism into electricity

---- Michael Faraday 1822

Swing of galvanometer

when switch close/open

A changing magnetic

field generates a current

(induced electromotiveforceemf).

When constant,

When constant,A

tBAemf /

B

tABemf /


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3.1.1 Faradays Induction Law

Flux of the magnetic fieldthrough the wire loop

cosBABAABM

When constant, When constant,A

tBAemf /B

tABemf /

Thefluxof the field: the product of field andarea where the penetration is perpendicular.

More generally, ifBvaries in space

AM SdB

The induced emf developed around the loop

dt

demf M

Very generally, an emfis a potential difference per unit charge,which corresponds to work done per unit charge, which is force per

unit charge times distance, which is electric field times distance

c

ldEemf

Thus,

c A

SdBdt

dldE

c A

Sd

t

BldE

or


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3.1.2 Gauss's Law - Electric

The relationship between the flux of theelectric fieldand the sourcesof that flux, charge.

The flux ofEfield through an imaginary

closed area A is:

AE SdE

If no sources or sinks of the Efield withinthe region encompassed by A, the net flux

through the surface equals zero.

qin vacuum centered inside a sphericalsurface of radius r.

A AE EdSdSEEis constant overA AE rEdSE

24

By Coulombs Law:2

04

1

r

qE

so

0q

E Electric flux of single point-charge

Multiple charges: qE0

1

and A qSdE

0

1

Gausss Law

Continuously distributed charges: A V dVSdE 01

V: volumeenclosed by A

E


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Electric Permittivity

2212

0 /108542.8 mNC

Electric permittivity offreespace(vacuum)

Conceptually, the permittivity embodies the electrical behavior of the

medium: in a sense, it is a measure of the degree to which the material

is permeated by the electric field in which it is immersed.

Indeed, permittivity is often measured by a procedure in which thematerial under study is placed within a capacitor.

Relative permittivity ordielectric constant:

0 EK KE is defined as /0 and it is unitless.


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3.1.3 Gauss's Law - Magnetic

NO isolated magnetic poles

Any closed surface in a region ofmagnetic field would accordingly have

an equal number of lines of entering

and emerging from it because there is

no monopoles.

B

The flux of magnetic field throughsuch a surface is zero.

A

M SdB 0


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3.1.4 Ampres Circular Law

Magnetic field of a straight

wire carrying a current iis:

riB 2/0

Suppose magnetic charge qm,analogy to electric charge

Magnetic force = qmB in the direction ofB.

The work done by carrying the monopole alongl: lBqW m

The total work done is: lBqm

In this case, riBB 2/0 so rBqlBqlBq mmm 2 Substitute riB 2/0 work done becomes iqm 0

Hence, ilB 0 To be summed over any closed path surrounding i.

Hecht

???


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ilB 0 0l C ildB 0

Ampres Law

It relates line integral of tangent to a closed curve C, with

the total current ipassing within the confines ofC.

B

For the current with a nonuniform cross section:

C A SdJldB

0 here open surface A is bounded by C.

The quantity 0 is called the permeability of free space

and it is defined as 227 /104 CsN

The permeability of a medium where the current

imbedded in:

0 MK KM: the dimensionless relative permeability.

Hecht


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C A SdJldB

0A1: 0JA2: no net current0J

???

Moving charges are not the only source of a magnetic field.

Capacitor:

A

QE

as the charge varies, the electric field changes

Taking the derivative of both sides

A

i

t

E

Effective current density

Maxwell hypothesized the existence ofdisplacementcurrent density.

t

EJD

Rewrite Ampres Law as

C A

Sdt

EJldB

)(

Note: a time-varying field will be accompanied by a

field even whenE

B

0J


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3.1.5 Maxwells Equations

C A Sdt

EJldB

)(

A SdB 0

A V dVSdE 01

c A

Sdt

BldE

Free space

0

0

0

0

J

C A Sdt

E

ldB

00

A SdB 0

A SdE 0

c A

Sdt

BldE

Eaffects Bwhile Bin turn affects E

Symmetries of Math and Physics!

Youre beautiful!


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3.1.5 Maxwells Equations

Differential form see Appendix 1 Hecht

Forfree space, in Cartesian coordinates

C A

Sdt

EldB

00

A SdB 0

A

SdE 0

c ASd

t

BldE

Faraday

Ampere

Gauss

Magnetic field of a straight

wire carrying a current iis:

riB 2/0 ???


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3.2 Electromagnetic Waves: application of Maxwell equations

Time-varying E-field generatesperpendicularB-field

Time-varying B-field generatesperpendicularE-field

Consider an accelerating charge

At the instant the charge begins to move, the Efield is altered and the time-

varying Efield induces a Bfield.

The charge is accelerating. Hence the induced Bfield is time-dependent.

The time-varying Bfield generates an Efield.

The process continues with Eand Bcoupled in the form of pulse.

c A

Sdt

BldE

C A

Sdt

EldB

00


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3.2 Electromagnetic Waves: application of Maxwell equations

Second differential equations in Cartesian coordinates

Note: each and every component of the

electromagnetic field (Ex, Ey, Ez, Bx, By, Bz)obeys the scalar differential wave equation

2

2

22

2

2

2

2

21

tvzyx

With00

/1 v

22182732212

00/1012.11)/104)(/1085.8( msCkgmkgmCs

Thus, in free spacethe speed of all electromganetic waves would be

smv /1031 8

00

Light of speed in vacuum smc /1099792458.28

Latin: celer- fast

3 2 1 T W


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3.2.1 Transverse Waves

2

2

002

tEE

0

z

E

y

E

x

E zyx

Explain transverse wave property of light by EM theory

For example, plane wave propagating in the positive x-direction in vacuum

The electric field intensity should be a solution of

and is constant over each of an infinite set of planes perpendicular

to the x-axis. Hence, it is a function only ofxand t.E

),( txEE

Back to Maxwell Eqs

0

x

Ex),( txEE

0xE

xE constant

No such traveling wave advancing

in the + x-direction.

Thus, the EM wave has no Efield component in the direction ofpropagation. The E-field associated with the plane wave is transverse.

Transverse E-field Many directions (polarizations)

For plane or linearly polarized waves, E-vector is fixed. For example, y-axis.

),(

txEjE y

A SdE 0

3 2 1 T W


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3.2.1 Transverse Waves

),( txEjE y

t

B

x

Ezy

The time-dependent B-fieldonly have a component in

the z-direction.

In free space, the plane electromagnetic

wave is transverse.

Harmonic wave: ])/(cos[),( 0 cxtEtxE yy with a propagation speed ofc.

The associated magnetic flux density is:

dtx

EB

y

z

So, ])/(cos[1

),(])/sin[ 00

cxtEctxBordtcxt

c

EB yz

y

z

And zy cBE

Note:

Eyand Bzdiffer only by a scalar.

They have same time-dependence. They are in-phase at all points in space.

They are mutually perpendicular.

Their cross-product Ex Bpoints in the propagation direction.


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3.3 Energy and Momentum

EM transports energy and momentum. i.e. light from star

3.3.1 The Poynting Vector

Radian energy per unit volume orenergy density, u.

Energy density of the E-field (considering the plates of a charged capacitor)

20

2

EuE

Energy density of the B-field (considering a current-carrying toroid)

2

02

1BuB

For plane EM wave:00

/1

candcBE

So, we have BE uu The energy streaming through space in the form ofan electromagnetic wave is shared equally between

the constituent electric and magnetic fields.

Further,

2

0

2

0

1,, BuEuuuu

BE


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3.3.1 The Poynting Vector

To represent the flow of electromagnetic energy associated with a traveling

wave, let Ssymbolize the transport of energy per unit time (the power) acrossa unit area. Shas a unit of W/m2 SI system.

An EM wave traveling with a speed cthroughan area A.

uctA

tAucS

Energy in cylindrical

volume

or EBS0

1

Poynting vector

BEcSorBES

0

2

0

1

Direction: the energy flows in the direction of the propagation of the wave.

Magnitude: the power per unit area crossing a surface whose normal is parallel to S

Example: Harmonic, linearly polarized plane wave traveling through free space inthe direction of k

)cos()cos(00 trkBBandtrkEE

)(cos20002 trkBEcS

Instantaneous flow of energy per unit areaper unit time

00/1 candcBE


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Averaging Harmonic Functions

More practically, due to the extremely rapid changing ofS, we need to take anaveraging procedure to measure S.

Time-averaged value of some function f(t) over an interval T:

2/

2/)(

1)(

Tt

TtTdttf

Ttf

For harmonic function:

)(1

))(1

11

2/2/

)2/(2/(

2/

2/

2/

2/

TiTiti

T

ti

TtiTti

T

ti

Tt

Tt

Tt

Tt

titi

T

ti

eeeTie

eeTi

e

eTi

dteT

e

ti

T

ti

eT

T

e

)2/

2/sin

(

sinc u

u = , 2, 3..


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3.3.2 Irradiance

The amount of light illuminating a surface is called irradiance, Ithe averageenergy per unit area per unit time.

How to measure I?The time-averaged value (T>>) of the magnitude of the Poynting vector,

T is a measure ofI.

Harmonic Wave:

)(cos20002 trkBEcS T Tfor

T

2/1cos2

000

2

2BE

cS T

or 20

0

2E

cSI

T

The irradiance is proportional to the square

of the amplitude of the electric field.

Alternatively,TT

EcIandBc

I 2020

Radian flux (optical power), W: the time rate of flow of radiant energy

Radiant flux density W/m2: radian flux incident on or exiting from a unit area surface

EBS0

1

00/1 candcBE

A di


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Magnetic Field due to Conduction Current

24

Io

dd

r

s r

B

Biot-Savart Law

Refers to magnetic field due tothecurrent-carrying conductor

o

= 4 x 10-7 T.m / A

permeability of free space

Unit vector

Appendix:

A di


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Magnetic Field due to Conduction Current

To find the total field, sum up thecontributions from all the currentelements Ids

24

Io d r

s rB

Integration over the entire

current distribution

Vector sum

Useful to high-symmetry currents

2

4

Io

dd

r

s rB

Appendix:

A di


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Example 1: B for a Circular Loop of Wire

Magnetic field at the centerof a current loop

r

I

r

IB

r

r

r

rds

2

2

4

2

00

22

rI

24

Io

d

r

s rB

Appendix:

A di


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Example 2: B for a Long, Straight Conductor

The thin, straight wire

is carrying a constant

current

Integrating over all the

current elements gives

2

1

1 2

4

4

Isin

Icos cos

o

o

B da

a

sin

d dx s r k

Appendix:

A di


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B for a Long, Straight Conductor, Direction

aI

aIB

22

400

2

1

1 2

4

4

Isin

Icos cos

o

o

B d

a

a

From previous results

Now let 1=0, 2=, we have

back

Appendix:

A di


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Energy density of the E-field (considering the plates of a charged capacitor) 20

2EuE

The work to move a charge element dq from the negative plate to the

positive plate is equal to Vdq, where Vis the voltage on the capacitor.

C=Q/V

(1)

(2)

(3)

(4)

(5)

Appendix:

Appendix:


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The instantaneous power which must be suppliedto initiate the current in the inductor is

The energy density (energy/volume)

Energy density of the B-field (considering a current-carrying toroid) 2

02

1BuB

(1)

(2)

(3)

back

Appendix:

Appendi


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Field Due to a Long Straight Wire

Want to calculate themagnetic field at adistance rfrom the center

of a wire carrying asteady current I

The current is uniformlydistributed through the

cross section of the wire

Appendix:

Appendix


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Outside of the wire, r> R

Inside the wire, we need I, the current insidethe amperian circle

2

2

( ) I

I

o

o

d B r

B

r

B s

2

2

2

2

2

( ) I ' I ' I

I

o

o

rd B r R

B r

R

B s

Appendix:

Appendix:


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The field is proportional

to rinside the wire

The field varies as 1/r

outside the wire Both equations are

equal at r= R

Appendix:

Lecture physical optics

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