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Lecture 18

Electromagnetic Waves Propagation and Generation

From Maxwell to DAlembert

Wavefronts and rays

Polarization

The e.m. spectrum

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From Maxwells equations to e.m. waves

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MAXWELLs EQUATIONS

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Maxwell equations:

NO charges

NO current

REMIND the external product of 3 vectors (a, b and c):

a X b X c = b (a c) - c (a b) AND APPLY this rule to the external product of Nabla and electric field:

2.

3. APPLY the curl operator to equation III:

From Maxwells equations to e.m. waves

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4. APPLY the time derivation to equation IV and combine:

5. A similar result can be obtained with the magnetic field

6. FINALLY, from Maxwell equations we have obtained the DAlembert equations for electric and magnetic field:

From Maxwells equations to e.m. waves

IN VACUUM:

NO charges

NO current

Supposing that E and B are transeverse waves

orthogonal to the x-axis, which is the propagation direction

From Maxwells equations to e.m. waves (alternative procedure)

x

E

t

B

x

E

t

B

t

B

yz

zy

x 0

!!!!zero! are derivative z andy The

x

B

t

E

x

B

t

E

t

E

yz

zy

x

00

00

1

1

0

2

2

002

2

dt

Bd

dx

Bd yy

2

2

002

2

dt

Bd

dx

Bd zz

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2

2

002

2

dt

Ed

dx

Ed yy

2

2

002

2

dt

Ed

dx

Ed zz

2

2

002

2

dt

Bd

dx

Bd yy

2

2

002

2

dt

Bd

dx

Bd zz

Both magnetic and electric fields satisfy DAlembert equation

From Maxwells equations to e.m. waves (alternative procedure)

x

y

z

Campo

elettrico

Campo

magnetico

Electric field

Magnetic field

1. Ex=Bx=0 that is: e.m. waves are transverse-wave

2. The e.m. wave propagation velocity in vacuum is constant and given by:

3. Ez, Bz , Ey, By components satisfy the wave-equation, and harmonic

solutions can be given by:

4. The simple solution for E and B fields propagation in vacuum along x-axis

is to have E//y-axis and B//z-axis:

smcv

v /1031

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1 8

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200

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FIRST IMPORTANT RESULTS:

2. And making the derivatives::

OTHER VERY IMPORTANT RESULTS:

c

EB Demonstration of

1. Assuming a plane wave with E//y and B//z:

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ELECTROMAGNETIC WAVES:

http://web.mit.edu/viz/EM/visualizations/light/EBlight/EBlight.htm

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E.M. WAVES PROPAGATION: Rays and wavefronts

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e.m. WAVES PROPAGATION

PHASE of THE Harmonic WAVE: tkx

WAVE-FRONT: is composed by all the points where the field has the same value. A wavefront is the locus of points having the same phase: a line (or a

curve) in 2-D, or a surface in 3-D.

PLANE WAVE

It propagates along the x axis, the electric field E(x,t) has to have nul x components: Ex(x,t)=0

E has the same value in a plane that is orthogonal to the x axis, that is: in the planes // (y,z)

Plane

wave-front

The simplest form of a wavefront is the plane wave, where the rays are parallel to one another. The light from this type of wave is referred to as collimated light. The plane wavefront is a good model for a surface-section of a very large spherical wavefront; for instance, sunlight strikes the earth with a spherical wavefront that has a radius of about 150 million kilometers (1 Angstrom). For many purposes, such a wavefront can be considered planar.

00

1

c

e.m. WAVES PROPAGATION

The wave velocity propagation in vacuum is:

The propagation direction is given by the direction of :

Wave front

Huygens' principle provides a quick method to predict the propagation of a wavefront: for

example, a spherical wavefront will remain spherical as the energy of the wave is carried

away equally in all directions. Such directions of energy flow, which are always

perpendicular to the wavefront, are called rays creating multiple wavefronts.

HUYGENS PRINCIPLE: any point on a wave front may be regarded as the source of

secondary waves and that the surface that is tangent to the secondary waves can be used

to determine the future position of the wave front.

RAYS and WAVEFRONTS

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E.M. WAVES : the e.m. spectrum

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THE ELECTRO-MAGNETIC SPECTRUM:

c = /k = /T =

Maxwell 1864: e.m. wave idea Hertz 1887: first experimental proof The wave equation for e.m. wave

admit solutions for ANY FREQUENCY

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Images taken of the Whirlpool galaxy recordiung radiation in

different frequency ranges (and a s consequensce different details

are revealed)

ELECTROMAGNETIC WAVES ARE REAL

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E.M. WAVES GENERATION

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Electromagnetic Waves generation

Electric and magnetic fields are coupled through Ampres and Faradays laws

Once created they can continue to propagate without further input

Only accelerating charges will create electromagnetic waves

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Electromagnetic Waves generation

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Electromagnetic Waves

With the changing current restricted to a line, the fields propagate with cylindrical symmetry outward from the current line.

The electric field is aligned parallel to the current and the

magnetic filed is aligned perpendicular to both the electric field

and to the direction of propagation. These are general features

of electromagnetic waves.

The current must change in time if it is to give rise to propagating fields (as a steady current merely produces a static magnetic

field). We can translate this into a statement about the charges

whose flow gives rise to the current: The charges that give rise

to the propagating electric and magnetic fields must be

accelerating. Harmonically varying currents will give rise to

harmonically varying electric and magnetic fields.

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DIPOLE RADIATION

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DIPOLE ANTENNA

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DIPOLE RADIATION

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DIPOLE RADIATION

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DIPOLE RADIATION: angular distribution

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ACCELERATED CHARGE

http://www.tapir.caltech.edu/~teviet/Waves/empulse.html

http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html

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