ELECTROMAGNETIC WAVES

Electromagnetic Theory

• Theoretical understanding of electricity

and magnetism

– Seemed complete by around 1850

– Coulomb’s Law and Gauss’ Law explained electric fields and forces

– Ampère’s Law and Faraday’s Law explained magnetic fields and forces

Unanswered Questions

• What was the nature of electric and

magnetic fields?

• What is the idea of action at a distance?

• How fast do the field lines associated with

a charge react to a movement in the

charge?

• Maxwell’s work led to the discovery of

electromagnetic waves

Discovery of EM Waves

• A time-varying magnetic field gives rise to an electric field

– A magnetic field can produce an electric field

• Maxwell proposed a modification to Ampère’s Law

– A time-varying electric field produces a magnetic field

– This gives a new way to create a magnetic field

– Also gives equations of electromagnetism a symmetry

• James Clerk Maxwell studied some of these questions in the mid-1800’s

Symmetry of E and B

• The correct form of Ampère’s Law (due to Maxwell) says that a changing electric flux produced a magnetic field.

– Since a changing electric flux can be caused by a changing E, there was an indication that a changing

electric field produces a magnetic field

• Faraday’s Law says that a changing magnetic flux produces an induced emf, and an emf is always

associated with an electric field

– Since a changing magnetic flux can be caused by a changing B, we can also say that a changing

Section 23.1

Symmetry of E and B, cont.

Electromagnetic Waves

• Self-sustaining oscillations involving E and B are possible

• The oscillations are an electromagnetic wave

– Electromagnetic waves are also referred to as electromagnetic radiation

• Both the electric and magnetic fields must be changing with time

• Maxwell worked out the details of em waves in great mathematical detail in ~1850

• Experimental proof of the existence of the waves was not found until 1887

Perpendicular Fields

• According to Faraday’s Law, a changing

magnetic flux through a given area produces an electric field

– The direction of the electric field is perpendicular to the magnetic field that

produced it

• Similarly, the magnetic field induced by a

changing electric field is perpendicular to the

electric field that produced it

Properties of EM Waves

• An electromagnetic wave involves both an

electric field and a magnetic field

• These fields are perpendicular to each

other

• The propagation direction of the wave is

perpendicular to both the electric field and

the magnetic field

DEMO 7B‐20

EM Waves are Transverse Waves

• Imagine a snapshot of the electromagnetic wave • The electric field is along the x-axis

• The wave travels in the z-direction

– Determined by the right-hand rule #2

• The magnetic field is along the y-direction

• Because both fields are perpendicular to the direction of propagation, the wave is a transverse wave

Right handed  coordinate system

QUIZ 1, OCTOBER 14, 2013 An electromagnetic wave is traveling in +y direction and the magnetic field at a particular  point on the y‐axis points in the +z direction at a  certain instant in time.  At this same point and  instant, what is the direction of the electric field? Z B Y a) ‐z b) ‐x c)  ‐y d)  +x e)  None of the above _X

Light is an EM Wave

• Maxwell found the speed of an em wave can be expressed in terms of two universal constants

– Permittivity of free space, ε_o (electrostatic value) – Magnetic permeability of free space, μ_o

– (magnetostatic value) –

• The speed of an em wave is denoted by c

• Inserting the values, c = 3.00 x 108 _m/s

– The value of the speed of an electromagnetic wave is

o o

c

 

Light as an EM Wave, cont.

• Maxwell answered the question of the

nature of light – it is an electromagnetic

wave

• He also showed that the equations of

electricity and magnetism provide the

theory of light

EM Waves in a Vacuum

• Remember that mechanical waves need a medium to travel through

• Many physicists searched for a medium for em waves to travel through (The ether)

• EM waves can travel through many materials, but they can also travel through a vacuum

• All em waves travel with speed c through a vacuum

• The frequency f and wavelength λ are

EM Waves in Material Substances

• When an em wave travels through a

material substance, its speed depends on

the properties of the substance

• The speed of the wave is always less than

• The speed of the wave depends on the

wave’s frequency

Section 23.2 o o c    1

EM Waves Carry Energy

• An em wave carries

energy in the electric and magnetic fields associated with the wave

• Assume a wave interacts with a charged particle • The particle will

EM Waves Carry Energy, cont.

• As the electric field oscillates, so will the force • The electric force will do work on the charge • The charge’s kinetic energy will increase

• Energy is transferred from the wave to the particle

• The wave has energy

• The total energy per unit volume is the sum of its electric and magnetic energies

• u_total = u_elec + u_mag

Section 23.3 elec o mag o u  E and u B   1 2  1 2 2 2

EM Waves Carry Energy, final

• As the wave propagates, the energies per unit volume oscillate (energy densities)

• The electric and magnetic energies are equal and this leads to the proportionality between the peak electric and magnetic fields

o o o o o o E B E c B     2 2 1 1 2 2

Energy Propagation in Electromagnetic Waves Energy transmitted through unit time per unit area 0 0 1 E_rms B_rms EB P u c A        • Intensity I = Average energy flux density (W/m2) Define Poynting vector 0 1 S E B        Direction is that of wave propagation   average magnitude is the intensity 2 0 2 2 0 0 0 1 1 2 2 2 m m m m S I c E B E B c E c         

Intensity of an EM Wave

• The strength of an em wave is usually measured in terms of its intensity

– SI unit is W/m2

• Intensity is the amount of energy transported per unit time across a surface of unit area

• Intensity also equals the energy density multiplied by the speed of the wave

• I = u_total × _{c = (½ εo c Eo}2 _{+ ½ μ}

o B2)

the intensity is proportional to the square of the electric and magnetic field amplitudes

TRANSVERSE ELECTROMAGNET WAVE 

TEM  wave

in y‐z plane In x‐z plane X Y Z

Solar Cells

• The intensity of sunlight on a typical sunny day is about 1000 W/m²

• A solar cell converts the energy from sunlight into electrical energy

• Current photovoltaic cells convert only about 15% of the energy that strikes them

• Also must account for nights and cloudy days • Making better and more practical solar cells

EM Waves Carry Momentum

• An electromagnetic w

carries momentum

• Consider the collision

• between w and electron q • The momentum is carried

by the wave before the collision and by the

electron in the wall after the collision

EM Waves Carry Momentum, cont.

• The absorption of the wave occurs through the electric and magnetic forces on charges in the object

• When the charge absorbs an electromagnetic wave, there is a force on the charge in the

direction of propagation of the original wave

• The force on the charge is related to the charge’s

change in momentum: F_B = ∆p / ∆t

• According to conservation of momentum, the final momentum on the charge must equal the initial momentum of the electromagnetic wave

Radiation Pressure

• When an electromagnetic wave is absorbed by an object, it exerts a force on the object

• The total force on the object is proportional to its exposed area

• Radiation pressure is the force of the electromagnetic force divided by the area

• This can also be expressed in terms of the intensity

radiation F I P A c   Section 23.3

Electromagnetic Spectrum

• All em waves travel through a vacuum at the speed c

– c = 2.99792458 x 108 _{m/s ~ 3.00 x 10}8 _m/s

– c is defined to have this value and the value of a meter is derived from this speed

• Electromagnetic waves are classified according to their frequency and wavelength

• The wave equation is true for em waves: c = ƒ λ

• The range of all possible electromagnetic waves is called the electromagnetic spectrum

Section 23.4

EM Spectrum, Diagram

QUIZ 2 October 14, 2013

Light is an electromagnetic wave  where the  wavelength λ in meters times the frequency f in  Hertz (H_z) is equal to the velocity of light  c = 3 x 108 _{meters/second in vacuum. Which of}  the following statements is correct?  (a) λ = 100 meters,    f = 1x106 _Hz (b) λ = 105 _{meters,    f = 3x10}4 _Hz (c) λ  = 10‐6 _{meters,   f = 3x10}14 _Hz (d) λ = 1 meter,         f = 3/2 x 108 _Hz