Chapter 35. Electromagnetic Fields and Waves · • All electromagnetic waves travel in a vacuum with the same speed, ... This paradox is resolved by Einstein’s Special Theory of

Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Chapter 35. Electromagnetic Fields and Waves

To understand a laser beam, we need to know how electric and magnetic fields change with time. Examples of time-dependent electromagnetic phenomena include high-speed circuits, transmission lines, radar, and optical communications. Chapter Goal: To study the properties of electromagnetic fields and waves.


E or B? It Depends on Your Perspective

Whether a field is seen as “electric” or “magnetic” depends on the motion of the reference frame relative to the sources of the field.


Transformations

The Galilean field transformation equations are

where V is the velocity of frame S' relative to frame S and where the fields are measured at the same point in space by experimenters at rest in each reference frame.

NOTE: These equations are only valid if V << c.


Example transformation

Consider a charge at rest at the origin in S where B=0 and E is given by Coulomb’s Law.

In S’, the charge is moving and there is both an electric field E’=E and a magnetic field B’.

Biot Savart Law


Maxwell’s Equations and Electromagnetic Waves

Maxwell’s equations provide a unified description of the electromagnetic field and predict that

•  Electromagnetic waves can exist at any frequency, not just at the frequencies of visible light. This prediction was the harbinger of radio waves. •  All electromagnetic waves travel in a vacuum with the same speed, a speed that we now call the speed of light.


Light speed Maxwell’s equations predict EM waves move at a unique (light) speed relative to ANY frame of reference.

That is impossible if the Galilean velocity addition rule v’ = v+vrel holds.

This paradox is resolved by Einstein’s Special Theory of Relativity.


Discovery of artificial EM waves by Hertz

Magnified view of the spark gap and dipole transmitting ("feed") antenna at the focal point of the reflector. The high voltage spark jumped the gap between the spherical electrodes. The electrical impulse produced by the spark generated damped oscillations in the dipole antenna.

Magnified view of the spark gap and dipole receiving antenna at the focal point of a receiving reflector similar to the transmitting one. The width of the small spark gap on the right is controlled by the screw below it. The vertical dipole antenna at the left was about 40 centimeters long.

EM wave

Transmitter spark gap

Receiver spark gap


Schematic of operation of antenna

An oscillating electric dipole is surrounded by an oscillating electric (and magnetic) field. Field disturbances propagate away at light speed perpendicular to the dipole. Operating in reverse, an EM wave excites charges oscillation in a dipole.


Marconi’s transatlantic signal experiment Left to right: Kemp, Marconi, and Paget pose in front of a kite that

was used to keep aloft the receiving aerial wire used in the transatlantic radio experiment.


Marconi’s transatlantic signal generator Capacitor

banks Induction coils

Spark gap


Radio Like two identical tuning forks coupled resonantly through a sound wave, two tuned LC or other resonant circuits can be coupled through an EM wave. A high frequency EM can carry audio frequency analog information by warbling the frequency (FM) or amplitude (AM).

An optimal antenna has a size of order a wavelength. More compact loop antennas work through induction.


Harmonic Waves

A harmonic plane wave is generated by a single frequency source current distribution and is characterized by frequency f and wavelength c/f.

E, B, v form a right handed coordinate system.

A “linearly polarized” wave has the orientation of E fixed.


Generation of electromagnetic waves

EM waves are emitted in general by accelerated charges which shed free force fields.

Synchrotron light source Dipole antenna


EM wave spectrum

EM waves are observed over a wide range of frequencies.

Megahertz natural and artificial sources produce radio waves.

Ultra high frequency motions inside nuclei source gamma ray radiation.


Properties of Electromagnetic Waves

1. The fields E and B and are perpendicular to the direction of propagation vem.Thus an electromagnetic wave is a transverse wave.

2.  E and B are perpendicular to each other in a manner such that E × B is in the direction of vem.

3.  The wave travels in vacuum at speed vem = c 4.  E = cB at any point on the wave.

Any electromagnetic wave must satisfy four basic conditions:


Energy of Electromagnetic Waves

Energy density in E-field

Energy density in B-field

�

uE = εoE2 r,t( ) /2

�

uB = B2 r,t( ) /2µo

Total

�

uTot = εoE2 /2 + B2 /2µo

= εoE2 /2 + E 2 /2c 2µo = εoE

2 r,t( ) = B2 r,t( ) /µo

�

uTot = εoE2 = εoEo

2 cos2 kz −ωt( ) moves w/ EM wave at speed c

EM waves carry energy density u and momentum density u/c.


The energy flow of an electromagnetic wave is described by the Poynting vector defined as

The magnitude of the Poynting vector is

Poynting vector

The intensity of an electromagnetic wave whose electric field amplitude is E0 is


EXAMPLE 35.4 The electric field of a laser beam

Laser light is monochromatic and sourced by coherent high frequency motion of atomic electrons.


EXAMPLE 35.4 The electric field of a laser beam


Radiation Pressure A electromagnetic wave carries momentum density U/c and if the momentum is absorbed or reflected a pressure is exerted called the radiation pressure prad. The radiation pressure on an object that absorbs all the light is

where I is the intensity of the light wave. Note reflection implies twice the momentum transfer and twice the pressure.


EXAMPLE 35.5 Solar sailing

QUESTION:


EXAMPLE 35.5 Solar sailing


Polarization

Unpolarized

Plane Polarized x

y z

�

E = Eo cos kz −ωt( ) ˆ x B = Bo cos kz −ωt( ) ˆ y

Superposition of plane polarized waves


Polarization filters

An array of linear conductors absorbs energy only from the component of electric field along the conducting direction transmitting the orthogonal component.


Polarization filters

x

y

transmission

�

E inc =

E o cos kx −ωt( )

Plane-polarized incident wave

polarizer

�

Einc cosθ( ) ˆ x + Einc sinθ( ) ˆ y

absorbed transmitted

  Transmitted wave =

�

E trans = Eo cosθ cos kx −ωt( ) ˆ x


Malus’s Law Suppose a polarized light wave of intensity I0 approaches a polarizing filter. θ is the angle between the incident plane of polarization and the polarizer axis. The transmitted intensity is given by Malus’s Law:

If the light incident on a polarizing filter is unpolarized, the transmitted intensity is

In other words, a polarizing filter passes 50% of unpolarized light and blocks 50%.


Malus’s Law Here we see the effect of two filters. When parallel, light transmitted by the first is transmitted by the second. When orthogonal, light transmitted by the first is absorbed by the second.


Polarization by reflection

Unpolarized Incident light

Reflection polarized with E-field parallel to surface

Refracted light

  Unpolarized light reflected from a surface becomes partially polarized

  Degree of polarization depends on angle of incidence

n


Glare reduction

Transmission axis

  Reflected sunlight partially polarized.   Horizontal reflective surface ->the E-

field vector of reflected light has strong horizontal component.

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Chapter 35. Electromagnetic Fields and Waves · • All electromagnetic waves travel in a vacuum with the same speed, ... This paradox is resolved by Einstein’s Special Theory of