ELECTROMAGNETIC WAVES

• Types of electromagnetic waves• Electromagnetic spectrum• Propagation of electromagnetic wave• Electric field and magnetic field• Qualitative treatment of electromagnetic waves

• Electromagnetic (EM) waves were first postulated by James Clerk Maxwell and subsequently confirmed by Heinrich Hertz

• Maxwell derived a wave form of the electric and magnetic equations, revealing the wave-like nature of electric and magnetic fields, and their symmetry

• Because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave

• According to Maxwell’s equations, a spatially-varying electric field generates a time-varying magnetic field and vice versa

• Therefore, as an oscillating electric field generates an oscillating magnetic field, the magnetic field in turn generates an oscillating electric field, and so on

• These oscillating fields together form an electromagnetic wave

Introduction

• In the studies of electricity and magnetism, experimental physicists had determined two physical constants - the electric (o) and magnetic (o) constant in vacuum

• These two constants appeared in the EM wave equations, and Maxwell was able to calculate the velocity of the wave (i.e. the speed of light) in terms of the two constants:

• Therefore the three experimental constants, o, o and c previously thought to be independent are now related in a fixed and determined way

Speed of EM waves

m/s100.31 8

oo

c 0 = 8.8542 10-12 C2 s2/kgm3 (permittivity of vacuum)

0 = 4 10-7 kgm/A2s2 (permeability of vacuum)

Name Differential form Integral form

Gauss's law

Gauss's law for magnetism

Maxwell–Faraday equation (Faraday's law of induction)

Ampère's circuital law(with Maxwell's correction)

 

Formulation in terms of free charge and current

Maxwell’s Equations

fD

tBE

0 B 0 AdB

V

)(VQAdD fV

tldE SB

S

,

tDJH

tIldH SD

fSS

,

,

zz

yy

xx

ˆˆˆ

vzv

yv

xv

vdiv zyx

zyx

zyx

vvv

zyxv

ˆˆˆ

Maxwell’s EquationsFormulation in terms of total charge and current

0

E

tBE

0 B

tEJB

000

0

)(

VQAdE

V

tldE SB

S

,

0 AdBV

tIldB SE

SS

,

000

Differential form Integral form

Gauss's law

Gauss's law for magnetism

Maxwell–Faraday equation (Faraday's law of induction)

Ampère's circuital law(with Maxwell's correction)

line integral of the electric field along the boundary ∂S of a surface S (∂S is always a closed curve)line integral of the magnetic field over the closed boundary ∂S of the surface S

The electric flux (surface integral of the electric field) through the (closed) surface  (the boundary of the volume V )The magnetic flux (surface integral of the magnetic B-field) through the (closed) surface  (the boundary of the volume V )

Maxwell’s Equations

ldES

ldBS

AdEV

AdBV

(1) Gauss’s law for the electric fieldGauss’s law is a consequence of the inverse-square nature of Coulomb’s law for the electrical force interaction between point like charges

(2) Gauss’s law for the magnetic fieldThis statement about the non existence of magnetic monopole; magnets are dipolar. Magnetic field lines form closed contours

(4) The Ampere-Maxwell lawThis law is a statement that magnetic fields are caused by electric conduction currents and or by a changing electric flux (via the displacement current)

(3) Faraday’s law of electromagnetic inductionThis is a statement about how charges in magnetic flux produce (non-conservative) electric fields

Maxwell’s Equations

Electromagnetic Spectrum

Generating an Electromagnetic Waves

An arrangement for generating a traveling electromagnetic wave in the shortwave radio region of the spectrum: an LC oscillator produces a sinusoidal current in the antenna, which generate the wave. P is a distant point at which a detector can monitor the wave traveling past it

Generating an Electromagnetic Waves

Variation in the electric field E and the magnetic field B at the distant point P as one wavelength of the electromagnetic wave travels past it.

The wave is traveling directly out of the page

The two fields vary sinusoidally in magnitude and direction

The electric and magnetic fields are always perpendicular to each other and to the direction of travel of the wave

• Close switch and current flows briefly. Sets up electric field

• Current flow sets up magnetic field as little circles around the wires

• Fields not instantaneous, but form in time

• Energy is stored in fields and cannot move infinitely fast

Generating an Electromagnetic Waves

• Figure (a) shows first half cycle

• When current reverses in Figure (b), the fields reverse

• See the first disturbance moving outward

• These are the electromagnetic waves

Generating an Electromagnetic Waves

• Notice that the electric and magnetic fields are at right angles to one another

• They are also perpendicular to the direction of motion of the wave

Generating an Electromagnetic Waves

Electromagnetic Waves• The cross product always gives the direction of travel

of the wave• Assume that the EM wave is traveling toward P in the positive

direction of an x-axis, that the electric field is oscillating parallel to the y-axis, and that the magnetic filed is the oscillating parallel to the z-axis:

)sin()sin(

0

0

tkxBBtkxEE

E0 = amplitude of the electric fieldB0 = amplitude of the magnetic field = angular frequency of the wavek = angular wave number of the waveAt any specified time and place: E/B = c

cBE 00 /(speed of electromagnetic wave)

BE

Electromagnetic wave represents the transmission of energy

The energy density associated with the electric field in free space:

202

1 EuE

The energy density associated with the magnetic field in free space:

2

0

121 BuB

Electromagnetic Waves

BEBE uuuuu 22 Total energy density:

2

0

20

1 BEu

ExampleImagine an electromagnetic plane wave in vacuum whose electric field (in SI units) is given by

0,0),109103(sin10 1462 zyx EEtzE

Determine (i) the speed, frequency, wavelength, period, initial phase and electric field amplitude and polarization, (ii) the magnetic field.

Solution(i) The wave function has the form: )(sin),( 0 vtzkEtzE xx

)]103(103sin[10Here, 862 tzEx

1816 ms103,m103 vk

Hz105.4,nm7.6662 14

vf

k

Solution (continued)

Period , and the initial phase = 0s102.2/1 15 fT

Electric field amplitude V/m1020 xE

The wave is linearly polarized in the x-direction and propagates along the z-axis

(ii) The wave is propagating in the z-direction whereas the electric field oscillates along the x-axis, i.e. resides in the xz-plane.

Now, is normal to both and z-axis, so it resides in the yz-plane. Thus,

E

B

E

),(ˆand,0,0 tzBjBBB yzx

Since, cBE

T)109103(sin1033.0),( 1466 tztzBy

refer to the fields of a wave at a particular point in space and

indicates the Poynting vector at that point

Energy Transport and the Poynting Vector

S

• Like any form of wave, an EM wave can transport from one location to another, e.g. light from a bulb and radiant heat from a fire

• The energy flow in an EM is measured in terms of the rate of energy flow per unit area

• The magnitude and direction of the energy flow is described in terms of a vector called the Poynting vector: S

BES

0

1

B ,E

S

is perpendicular to the plane formed by , the direction is determined by the right-hand rule.

S

B E

and

Energy Transport and the Poynting Vector

S

Because are perpendicular to each other in an EM wave, the magnitude of is:

B E

andS

EBS0

1

2Ec

S0

1

E/cB Instantaneous energy flow rate

Intensity I of the wave = time average of S, taken over one or more cycles of the wave

)(sin11 22

00

tkxEc

Ec

SI m2

rmsrmsrms BEEc

SI0

2

0

12

1

In terms of rms :

rmsm EE 2mmm BEE

cI

0

2

0 21

21

Example[source: Halliday, Resnick, Walker, Fundamentals of Physics 6th Edition, Sample Problem 34-1

An observer is 1.8 m from a light source whose power Ps is 250 W. Calculate the rms values of the electric and magnetic fields due to the source at the position of the observer.

Energy Transport and the Poynting Vector

S

0

2

24

cE

rPI rms

V/m48)m8.1(4

H/m)10m/s)(4π10(250W)(34 2

78

20

rPc

Erms

T106.1m/s103

V/m48 78

c

EB rms

rms

Polarization of Electromagnetic Wave

Polarization of Electromagnetic WaveThe transverse EM wave is said to be polarized (more specifically, plane polarized) if the electric field vectors are parallel to a particular direction for all points in the wave

direction of the electric field vector E = direction of polarization

xtkzEE ˆ)sin(0

Example, consider an electric field propagating in the positive z-direction and polarized in the x-direction

ytkzEc

B ˆ)sin(10

ztkzEcS ˆ)sin(200

BES

0

1

oo

1

c

ExampleA plane electromagnetic harmonic wave of frequency 6001012 Hz, propagating in the positive x-direction in vacuum, has an electric field amplitude of 42.42 V/m. The wave is linearly polarized such that the plane of vibration of the electric field is at 45o to the xz-plane. Obtain the vector BE

and

Solution:bygiven isvectorelectricThe E

here 2/120

200,0 zyx EEEE

8

120 103

106002sin xtEE

102

100 Vm30 EEE zy

x

y

z

Solution (continued)

So

8

12

103106002sin30,0 xtEEE zyx

8127

103106002sin10,0 xtBBB yzx

cBE

)ˆˆ(ˆˆ kjEkEjEE yzy

)ˆˆ(ˆˆ kjBkBjBB yzy

BEBE

tonormalis,0Then

required.as,ˆ2)ˆˆ(and

2

icE

iiBEBES yzy

Harmonic Waves

)](sin[ txkAy v

)sin( tkxAy

A = amplitude k = 2/ (propagation constant)

)](cos[ txkAy vor

v = f = f (2/k) k v = 2f = (angular frequency)

)cos( tkxAy or

Phase : = k(x + vt) = kx + t moving in the – x-direction = k(x - vt) = kx - t moving in the + x-

direction

Harmonic Waves

)sin( 0 tkxAy

In general, to accommodate any arbitrary initial displacement, some angle 0 must be added to the phase, e.g.

Suppose the initial boundary conditions are such that y = y0 when x = 0 and t = 0 , then

y = A sin 0 = y0

0 = sin-1 (y0/A)

Plane WavesThe wave “displacement” or disturbance y at spatial coordinates (x, y, z): )sin( tkxAy

Traveling wave moving along the +x-direction

At fixed time, let take at t = 0: kxAy sin

When x = constant, the phase = kx = constant

the surface of constant phase are a family of planes perpendicular to the x-axis

these surfaces of constant phase are called the wavefronts

Plane Waves

Plane wave along x-axis. The waves penetrate the planes x = a, x = b, and , x = c at the points shown

Plane Waves

Generalization of the plane wave to an arbitrary direction. The wave direction is given by the vector k along the x-axis in (a) and an arbitrary direction in (b)

x= r cos

)cossin( krAy

)sin( tAy rk

zyx xkxkxk rk

)( zyx k,k,k

are the components of the propagation direction

)( tiAey rk

Spherical & Cylindrical Waves

Spherical Waves:

Cylindrical Waves:

)( tkrierAy

)( tkieAy

r = radial distance from the point source to a given point on the waveform

A/ r = amplitude

= perpendicular distance from the line of symmetry to a point on the waveform

e.g. of the z-axis is the line of symmetry, then 22 yx

Mathematical Representation of Polarized Light

yExEE yx ˆˆ

Consider an EM wave propagating along the z-direction of the coordinate system shown in figure.

The electric field of this wave at the origin of the axis system is given by:

z

x

y

E

xE

yE

0

Propagation direction

Complex field components for waves traveling in the +z-direction with amplitude E0x and E0y and phases x and y :

)(0

~ xtkzixx eEE )(

0~ xtkzi

yy eEE

xx EE ~Re yy EE ~Re

yxE ˆˆ~ )(0

)(0

yx tkziy

tkzix eEeE

)(0

)(00

~~ˆˆ[~

tkzi

tkziiy

ix

e

eeEeE yx

EE

]yxE

]ˆˆ[~000 yxE yx i

yi

x eEeE = complex amplitude vector for the polarized wave

Since the state of polarization of the light is completely determined by the relative amplitudes and phases of these components, we just concentrate only on the complex amplitude, written as a two-element matrix – called Jones vector:

y

x

iy

ix

y

x

eEeE

EE

0

0

0

00 ~

~~E

Mathematical Representation of Polarized Light

Linear PolarizationFigures representation of -vectors of linearly polarized light with various special orientations. The direction of the light is along the z-axis

oscillations along the y-axis between +A and A

Vertically polarized Horizontally polarized Linearly polarized

+A

A

linear polarization along y

100~

0

00 A

AeEeE

y

x

iy

ix

E

E

E

Linear Polarization

10

= Jones vector for vertically linearly polarized light

ba

= vector expression in normalized from for 122 ba

In general:

Linear Polarization

AEE xxy 00 ,0,0

01

0~

0

00 A

AeEeE

y

x

iy

ixE

linear polarization along x

Horizontally polarized

+A-A

Linear Polarization

0

cos,sin 00

yx

yx AEAE

sincos

sincos~

0

00 A

AA

eEeE

y

x

iy

ixE

linear polarization at

oscillations along the a line making angle with respect to the x-axis

E

Linearly polarized

31

21

2/32/1

60sin60cos~

0E

Linear Polarization

For example = 60o :

ba

0~EGiven a vector a, b = real numbers

the inclination of the corresponding linearly polarized light is given by

ox

oy

EE

ab 11 tantan

Suppose = negative angle

E0y = negative number

Since the sine is an odd function, thus E0x remain positive

The negative sign ensures that the two vibrations are out of phase, as needed to produce linearly polarized light with -vectors lying in the second and fourth quadrants

E

The resultant vibration takes places place along a line with negative slope

ba

Jones vector with both a and b real numbers, not both zero,

represents linearly polarized light at inclination angle

ab1tan

Linear Polarization

• In determining the resultant vibration due to two perpendicular components, we are in fact determining the appropriate Lissajous figure

• If other than 0 or , the resultant E-vector traces out an ellipse

Lissajous Figures

Lissajous figures as a function of relative phase for orthogonal vibrations of unequal amplitude. An angle lead greater than 180o may also be represented as an angle lag of less that 180o . For all figures we have adopted the phase lag convention y x

Lissajous Figures

Linear Polarization ( = m)

Circular Polarization ( = /2)

Elliptical Polarization