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M A N M O H A N D A S H ,
P H Y S I C I S T , T E A C H E R
Physics for ‘Engineers and Physicists’
Lecture - 3
Electromagnetic Waves
I remember its close to 2 decades ago, when I saw the Maxwell's Equations and see those charge and currents and think to myself; if these things vanish why still we have the fields, I mean we read in the text books the sources of charges is what produces the electric fields and the currents are what produces magnetic fields, why the fields survive when the charges vanish. I was still thinking like the +2 (PU/American Highschool Senior) student that I was, because that's when I studied the fields being produced by the charges and currents. Somewhere down the line, when I became well versed with what these equations are actually doing, I understood; the fields are produced by charges and currents, that are not necessarily explicitly seen in these
equations, but the charges and currents that interact with these fields are explicitly placed in the equations. Its like we are produced by our parents who are not visible much in our lives although they impact us, but its our girlfriends and boyfriends who impact us explicitly, trying to pull the strings here and there ! It was just a passe analogy. But Physics is often fun if you imagine how they correspond to our day to day moorings.
Lets discover what electromagnetic phenomena are entailed by the Maxwell’s
equations.
“A concise course of important results”
Lectures delivered around 12.Nov.2009 + further content developments this week; 28 Aug - 02 Sept 2015 !
Electromagnetic Waves
Electromagnetic Waves are 3 dimensional propagation of vibrations of electric and magnetic fields. In the last two lectures we discussed what are vector fields. Electric fields and magnetic fields are vector fields.
>> Lecture-1 << and >> Lecture-2 <<
We discussed the analytical properties of vector and scalar fields in detail in those lectures. Have a look, they are linked below.
Transverse waves
Electromagnetic Waves are transverse in nature. Waves are basically two types transverse and longitudinal.
Note; vibrations are also called disturbances, undulation, oscillation, ripples and wiggles in various usage of waves. Electromagnetic waves are transverse undulations of electric and magnetic fields.
Transverse Waves are so called because in this type the direction of propagation of wave is perpendicular or transverse to the direction of vibration.
The example of longitudinal waves are acoustic waves (sound). They are longitudinal because the direction of propagation of wave is along the direction of vibration.
Transverse nature of electromagnetic waves. The vector field B vibrates along y-direction whereas the vector field E vibrates along x-direction. The wave cruises along the z-direction. Thus the E-Mwave is vector in nature in addition to being transverse. This results in photons being vector particle or spin-1 in nature, an advance concept we won’t discuss here.
Various types of em-waves
Electromagnetic Waves are a set of phenomena broadly categorized as “Gamma rays, X-rays, Ultraviolet Rays, Visible light, Infra-red Rays, Microwaves and Radio waves.
Spectrum of Electromagnetic Waves
Here is the quantitative spectrum of the electromagnetic waves.
E-M Waves from Maxwell’s Eqn
Electromagnetic Waves are comprehensively described by Maxwell’s Equations, these equations we discussed previously in Lecture-2 !
Lets rewrite the Maxwell’s Equations in a medium in presence of charges and currents. We will use the same to derive whats called wave equation (for EM-phenomena).
We saw in Lecture-2 ! D and H are defined with mu, eps.
medium
BH
E-M Wave equation in free space from Maxwell’s Eqn
In Lecture-2 we saw that for free space or vacuum Assuming no charges or currents, i.e. lets write the Maxwell’s Equations again, this time in freespace, free from sources of charges and currents.
0 ,0 j
0 )4(
0 )3(
0,0 )2 ,1(
00
t
EB
t
BE
BE
Lets take curl of the 3rd equation of this set and replace the curl of magnetic field B from equation 4, into the curl of 3rd.
Maxwell’s Equations, freespace
E-M Wave equation in free space, in Electric Field E.
In Lecture-1 we saw that !
0 )4(
0 )3(
,0,0 )2 ,1(
00
t
EB
t
BE
BE
In taking curl of the 3rd equation we get
AAA
2)(
0)(
)()( 2
t
BEE
t
BE
Maxwell’s Equations, freespace
Equation 4th and 1st ; t
EB
00
This leads to
0 E
02
2
00
2
t
EE
Freespace Wave-Equation, in electric field E
E-M Wave equation in free space, Magnetic Field B.
0 )4(
0 )3(
,0,0 )2 ,1(
00
t
EB
t
BE
BE
In taking curl of the 4th equation we get
0)(
)()(
00
2
00
t
EBB
t
EB
Maxwell’s Equations, freespace
Equation 3rd and 2nd; t
BE
This leads to
0 B
02
2
00
2
t
BB
Freespace Wave-Equation, in magnetic field B
Lets apply the exact same treatment to get the wave equation in terms of magnetic field B, apply curl on 4th equation !
E-M Wave equation in free space, general form.
In general any wave equation looks like (1); This gives speed of the E-M waves, as given above >>
00
1
emv0
1
x )1(
2
2
22
2
tv
The wave equations we obtained are vector equations which we can cast into individual scalar equations for 3 scalar components of the fields; x, y, z, for both fields; E and B.
0
0
0
2
z
2
00z
2
2
y
2
00y
2
2
x
2
00x
2
t
EE
t
EE
t
EE
0 ,0 ,02
z
2
00z
2
2
y
2
00y
2
2
x
2
00x
2
t
BB
t
BB
t
BB
SISI 104 , 1085.8 7
0
12
0
Wave equation in charge-free, non-conducting media
Its easy to see the forms in which they pertain to charge-free but non-conducting media, eg in (1), (2) ! Eps, mu; electro-magnetic properties of the media.
In last slide we saw the wave equations for the freespace, by freespace we meant charge-free and current-free space which are essential conditions of vacuum.
0 )2(
0 )1(
2
22
2
22
t
BB
t
EE
1
)1
( ,
00
00
v
c
vcvThere is a drop in speed of light when it enters any media from vacuum, speed of light in vacuua is maximum and drops by a factor eta known as refractive index of the media and the refractive index is related to eps and mu in freespace and in media.
Wave eqn, charge-free, conducting media; Telegraph Eqn
Sigma is called the conductivity; of the given medium.
‘Charge-free and Conducting’ media is a special case of the Maxwell’s Equations where we represent this condition by the equations on right !
Ej
,0
Lets take curl of the 3rd equation and interchange curl and time differentiation !
From the steady-state and time-varying Maxwell’s Equations given in slide 9, for any medium given by eps, mu and the conditions above, we have
Et
EB
t
BEBE
)4( ,0 )3( ,0 (2) ,0 )1(
Wave equation; The Telegraph Equations !
This leads to the telegraph equation in terms of E field, (5). We apply the same treatment on equation (4) and obtain the telegraph equation for the B field, (6).
With the step mentioned in last slide we obtain (3’);
Et
EB
t
BEE
)'4( ,0
)(-)( )'3( 2
The terms on the right hand side of telegraph equations are dissipative terms.
t
B
t
BB
t
E
t
EE
2
22
2
22 )6( ,)5( Telegraph Equations
Vector Potential and Scalar Potential
In Lecture-1 we saw every vector field is associated with a scalar field, the gradient of the scalar is the vector and we called the scalar field as the potential. We saw that for many vectors, there are potentials associated with each of them.
These vectors were always a space gradient (or any space derivative) of the scalar potentials. Hence the name vector field and (scalar) potential.
When it’s a scalar whose gradient creates a vector field it’s a scalar potential and when it’s a vector whose curl (another way to create space derivative of a vector) creates a vector field the former vector is called a vector potential.
Magnetic Vector Potential
In Lecture-1 we saw for every vector whose divergence vanishes the vector is necessarily a curl of another vector and for every vector that is a curl of another vector the divergence of the former vector vanishes.
Since the divergence of the magnetic field is zero it implies from the above, magnetic field vector B is curl of another vector A, the magnetic vector potential.
FG then 0G
0G then FG if
0;)(
if
FSince
potential vector magneticA
0)Cf( as CfAA
AB 0;B
Since
Electro-Magnetic Scalar Potential
We take the 3rd of Maxwell equations (check slide 15), eqn (1) below, use the fact that the magnetic field is the curl of the vector potential A we just defined, in last slide. Interchange the order of the curl and the time derivative;
! ,t
-
- t
0;)(
,0)t
( ,AB with 0,t
E (1)
potentialscalarA
E
AE
fSince
AE
B
We already saw
in Lecture-1 that curl of any gradient is always 0.
Phi, the scalar potential, is partly electric and partly magnetic potential.
Gauge Transformations, Lorentz and Coulomb’s Gauge !
We saw that vector potentials and scalar potentials that we just discussed are arbitrary, that is, there is not a single one of them.
0; ,1
; ..
. , ;
),( , , ;
2
AGaugeCoulombtc
AGaugeLorentzge
ConditionGaugecalleddconstraineAandGauged
trfft
ffAAtionsTransformaGauge
Gauge Conditions are therefore a particularly chosen definitions of vector or scalar potentials. The Gauge Transformations are a change of the definition of these potentials, so that the E and B field stand unchanged, thus Physics stays unique and not arbitrary;
Wave Equation in terms of Scalar Potential
We saw that vector potential and scalar potential, that we just discussed are now unique, given to a particular Gauge Condition.
011
;
0)(
; 0)(
.0 ; , 'x
2
2
2
2
tctcAGaugeLorentzApply
t
Aor
t
AE
ELawGaussEqnswellMaFreespace
Using this, we can write the wave equation in terms of the scalar potential, instead of just in E and B fields.
01
2
2
tc
wave equation in terms of the scalar potential
Wave Equation in terms of Vector Potential
We saw that scalar potential has its own wave equation just as E and B fields did. Lets find the wave equation for the vector potential.
0][][
0)(
)(
0)()(,0 (1)
2
2
00
2
00
2
2
0000
2
0000
t
AA
tA
t
A
tAA
t
A
tA
t
EB
Using Freespace Ampere-Maxwell Law (1), we can write the wave equation in terms of the vector potential.
2
2
2
2 1
t
A
cA
wave equation in terms of the vector potential
1st [term] is 0; Lorentz Gauge
Transverse nature of electromagnetic waves
In the beginning we claimed that electromagnetic waves are transverse in nature, that is the propagation of the energy and wave as such, occurs in a direction perpendicular to the oscillations of the vector fields of E and B. Lets see this.
We see that the wave equations (1) are differential equations, 2nd order in space and 2nd order in time. Plane waves (2) and (3) given above are their solutions.
Lets prove that E and B fields are perpendicular to each other as well to the wave propagation direction; k vector. Note that E is along e and B along b, the unit vectors.
)(
0
)(
02
2
00
2 ˆ),( )3( ,ˆ),( )2( ,0),(
),( )1( trkitrki eBbtrBeEetrEt
BEBE
Transverse nature of electromagnetic waves
e and b are thus unit vectors, constant in space and time. k is vector along which wave propagates, called wave vector and its magnitude is called as wave number and gives wavelength as well as momentum of the plane wave. Omega is the angular frequency and gives the time period and frequency of the wave, as well as the energy. e, b, k form a right hand trio. E_zero and B_zero are the amplitudes or maximum value of the fields. Below we see e and k are perpendicular.
0ˆ0))(iEˆ(or 0)(ˆ
. ̂;0ˆ),(ˆ)(ˆ0
)()( Use0;)ˆ( ,0
)(
0
)(
0
)(
0
)(
0
)(
0
keekeeEe
constiseeeEeeEe
VAVAVAeEeE
trkitrki
trkitrki
trki
0ˆ ke
e and k are perpendicular
Homework
Homework; (1) prove the result we used in last slide;
)(
0
)(
0 )(iE)( trkitrki ekeE
Homework; (2) prove that b and k are also perpendicualar to each other just like e and k.
0ˆ0 kbB
In the following slide we will prove from Maxwell’s relations, e, b, k are all mutually perpendicular in a right handed way.
Transverse nature of electromagnetic waves
Lets show that E, B and k are mutually perpendicular in a right handed way, that is e, b and k are;
eeeEeEeeEe
AVAVVA
eBbeEet
BE
rkititrkitrki
trkitrki
ˆ)()ˆ()ˆ(
)()( use
,0)ˆ(t
)ˆ(0
0
)(
0
)(
0
)(
0
)(
0
From homework (1) we saw, rkirki ekie
See eg Lecture-1
This leads to; 0ˆ)ˆk( ,)ˆk()ˆ( )(
0
)(
0
)(
0
)(
0 trkitrkitrkitrki eBbieEeieEeieEe
So we have the transverse condition;
bE
Be ˆ)ˆk(
0
0
e, b and k are mutually RH-perpendicular
Lets prove this
Transverse nature of electromagnetic waves
k has a magnitude equaling to product of angular frequency and ratio of amplitude of magnetic field to that of electric field, also note that for waves, angular frequency divided by wave number is the speed of wave, hence the ratio of amplitude of electric field to that of magnetic field is also the speed of the wave;
Z_zero is called impedance of vacuum. It has dimension of electric resistance.
0
0
0
00
00
0
00000
00
. , ,1
zcH
EcBEHBcAs
E, B are in phase and their magnitude and directions are correlated.
0
0
0
0
0
0 , )()(kB
Ecor
E
Bck
E
B
Electromagnetic Energy.
Electric Energy per unit volume or electric energy density and Magnetic Energy per unit volume or magnetic energy density are given by;
So the total energy density called the electromagnetic energy density or energy per unit volume is given by;
22
2
1
2
1 ,
2
1
2
1HHBuEDEu BE
)(2
1)(
2
1 22 HEHBDEuEM
Poynting vector and Poynting Theorem.
Poynting Vector measures the rate of energy flow of electromagnetic waves per unit time, per unit area, normal to the direction of wave propagation (SI unit, );
BEHES
2m
watt
Differential form (2) and integral form (1) of Poynting’s Theorem. Intergral form is obtained by taking volume integral and using Gauss Divergence Theorem. see Lecture-1
jEt
uS EM
)2(
VV
EM
A
dVjEdVt
uAdS
)1(
VV
EM
A
VV
EM
A
dVjEdVt
uAdSLHSontheoremDivGaussUse
dVjEdVt
udVSofIntegralVolumeTake
)1( ; .
; )2(
Proof of Poynting Theorem.
This proves the differential form of Poynting’s Theorem from Maxwell’s equations.
jEt
uS
jEHE
tHE
E
tt
EE
t
DERHS
H
tt
HH
t
BHRHS
HEHEEHLHS
jEt
DE
t
BHHEEHiiEiH
iijt
DHi
t
BE
EM
)22
()(
)2
()(
;
)2
()(
;
)()()( ;
)()( )()(
)( ),(
22
2
2
Maxwell’s equations.
Significance of Poynting Theorem.
We already saw the significance of Poynting’s Theorem as the quantitative relation depicting the flow of electro-magnetic energy across regions in given time, this can be readily seen as an equation of continuity;
If there are no sources of currents j, that the fields are interacting with, it means no energy is dissipated that way, we set j = 0 and we have the equation of continuity of energy flow, which represents the conservation of electromagnetic energy.
0 ,0
jwhen
t
uS EM
equation of continuity
Poynting vector and intensity of em-wave
Above we took the maximum value of Poynting’s vector, S = EH. (cross product maximum) Realize that; the maximum comes when the amplitudes are attained. For a plane wave E is given as a time varying sine function, whose integral average is ½ and thus given as rms value of the field.
2
22
0
2
0
22
00
2
0
0
22
sin ,sin ,
,
rms
rms
EcSI
EcEc
c
E
c
tEStEE
c
ES
cH
E
H
EEBEHS
Intensity of electro-magnetic wave
Electro-magnetic waves in conducting media
What happens when electromagnetic waves pass through conducting media, such as metallic conductors and ionized gases or electrolytes. For wave propagation along z–axis, electric field decays with distance by an exponetial factor f. E reduces to 1/e of its value after a distance of z which is called skin-depth or penetration depth of conducting media.
)1( 02
22
t
E
t
EE
Electromagnetic-waves passing through conducting media, the 3rd term in (1) gives dissipation. High frequency e.m. waves cannot propagate through conducting media like metals and ionized gases as seen in factor f.
2
z ,z
2 efatt
Thank you
We discussed electromagnetic waves at the undergraduate engineering level. Its also useful for the Physics ‘honors’ and ‘pass’ students.
This was a course I delivered to engineering first years, around
12th November 2009. But I have added contents, added the diagrams and many explanations now; 28th Aug - 2nd Sept 2015.
More soon. Eg Quantum Mechanics and Applications (for
engineering degree and for grad and undergrad of Physics Major students) Reach me @ [email protected]
