lecture physical optics
TRANSCRIPT
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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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3.1.4 Ampres Circular Law
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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3.1.4 Ampres Circular Law
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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Field Due to a Long Straight Wire
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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Field Due to a Long Straight Wire
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: