Maxwell’s Equations and Electromagnetic Waves

Chapter 29

James Clerk Maxwell1831-1879

Maxwell’s Theory• Electricity and magnetism were originally thought to be

unrelated

• Maxwell’s theory showed a close relationship between all electric and magnetic phenomena and proved that electric and magnetic fields play symmetric roles in nature

• Maxwell hypothesized that a changing electric field would produce a magnetic field

• He calculated the speed of light – 3x108 m/s – and concluded that light and other electromagnetic waves consist of fluctuating electric and magnetic fields

James Clerk Maxwell1831-1879

Maxwell’s Theory• Stationary charges produce only electric fields

• Charges in uniform motion (constant velocity) produce electric and magnetic fields

• Charges that are accelerated produce electric and magnetic fields and electromagnetic waves

• A changing magnetic field produces an electric field

• A changing electric field produces a magnetic field

• These fields are in phase and, at any point, they both reach their maximum value at the same time

Modifications to Ampère’s Law• Ampère’s Law is used to analyze magnetic fields

created by currents

• But this form is valid only if any electric fields present are constant in time

• Applying Ampère’s law to a circuit with a changing current results in an ambiguity

• The result depends on which surface is used to determine the encircled current.

IsdB 0

Modifications to Ampère’s Law• Maxwell used this ambiguity, along with symmetry

considerations, to conclude that a changing electric field, in addition to current, should be a source of magnetic field

• Maxwell modified the equation to include time-varying electric fields and added another term, called the displacement current, Id

• This showed that magnetic fields are produced both by conduction currents and by time-varying electric fields

IsdB 0

dt

d E 00

dt

dI Ed

0

Maxwell’s Equations• In his unified theory of electromagnetism, Maxwell

showed that the fundamental laws are expressed in these four equations:

0q

AdE

0 AdB

dt

dsdE B

IsdB 0

dt

d E 00

Maxwell’s Equations• Gauss’ Law relates an electric field to the charge

distribution that creates it

• The total electric flux through any closed surface equals the net charge inside that surface divided by o

0q

AdE

0 AdB

dt

dsdE B

IsdB 0

dt

d E 00

Maxwell’s Equations• Gauss’ Law in magnetism: the net magnetic flux

through a closed surface is zero

• The number of magnetic field lines that enter a closed volume must equal the number that leave that volume

• If this wasn’t true, there would be magnetic monopoles found in nature

0q

AdE

0 AdB

dt

dsdE B

IsdB 0

dt

d E 00

Maxwell’s Equations• Faraday’s Law of Induction describes the creation of an

electric field by a time-varying magnetic field

• The emf (the line integral of the electric field around any closed path) equals the rate of change of the magnetic flux through any surface bounded by that path

0q

AdE

0 AdB

dt

dsdE B

IsdB 0

dt

d E 00

Maxwell’s Equations• Ampère-Maxwell Law describes the creation of a

magnetic field by a changing electric field and by electric current

• The line integral of the magnetic field around any closed path is the sum of o times the net current through that path and o times the rate of change of electric flux through any surface bounded by that path

0 AdB

0q

AdE

IsdB 0

dt

d E 00

dt

dsdE B

Maxwell’s Equations• Once the electric and magnetic fields are known at

some point in space, the force acting on a particle of charge q can be found

• Maxwell’s equations with the Lorentz Force Law completely describe all classical electromagnetic interactions

0 AdB

IsdB 0

dt

d E 00

0q

AdE

dt

dsdE B

BvqEqF

Maxwell’s Equations• In empty space, q = 0 and I = 0

• The equations can be solved with wave-like solutions (electromagnetic waves), which are traveling at the speed of light

• This result led Maxwell to predict that light waves were a form of electromagnetic radiation

0 AdB

dt

dsdE B

IsdB 0

dt

d E 00

0q

AdE

Electromagnetic Waves• From Maxwell’s equations applied to empty space, the

following relationships can be found:

• The simplest solutions to these partial differential equations are sinusoidal waves – electromagnetic waves:

• The speed of the electromagnetic wave is:

2

2

002

2

t

E

x

E

2

2

002

2

t

B

x

B

00

1

k

v m /s 1 0 2 .9 9 7 9 2 8 c

)co s();co s( maxmax tk xBBtk xEE

max

max

B

E

B

E

Plane Electromagnetic Waves• The vectors for the electric and magnetic

fields in an em wave have a specific space-time behavior consistent with Maxwell’s equations

• Assume an em wave that travels in the x direction

• We also assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only

• The electric field is assumed to be in the y direction and the magnetic field in the z direction

Plane Electromagnetic Waves• The components of the electric and

magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation

• Thus, electromagnetic waves are transverse waves

• Waves in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves

John Henry Poynting 1852 – 1914

Poynting Vector• Electromagnetic waves carry energy

• As they propagate through space, they can transfer that energy to objects in their path

• The rate of flow of energy in an em wave is described by a vector, S, called the Poynting vector defined as:

• Its direction is the direction of propagation and its magnitude varies in time

• The SI units: J/(s.m2) = W/m2

• Those are units of power per unit area

1

oμ S E B

Poynting Vector• Energy carried by em waves is shared equally by the

electric and magnetic fields

• The wave intensity, I, is the time average of S (the Poynting vector) over one or more cycles

• When the average is taken, the time average of cos2(kx - ωt) = ½ is involved

2 2max max max max

avg 2 2 2o o o

E B E c BI S

μ μ c μ

Chapter 29Problem 29

What would be the average intensity of a laser beam so strongthat its electric field produced dielectric breakdown of air (whichrequires Ep = 3 MV/m)?

Polarization of Light• An unpolarized wave: each atom

produces a wave with its own orientation of E, so all directions of the electric field vector are equally possible and lie in a plane perpendicular to the direction of propagation

• A wave is said to be linearly polarized if the resultant electric field vibrates in the same direction at all times at a particular point

• Polarization can be obtained from an unpolarized beam by selective absorption, reflection, or scattering

Polarization by Selective Absorption• The most common technique for polarizing light

• Uses a material that transmits waves whose electric field vectors in the plane are parallel to a certain direction and absorbs waves whose electric field vectors are perpendicular to that direction

Polarization by Selective Absorption• The intensity of the polarized beam transmitted

through the second polarizing sheet (the analyzer) varies as S = So cos2 θ, where So is the intensity of the polarized wave incident on the analyzer

• This is known as Malus’ Law and applies to any two polarizing materials whose transmission axes are at an angle of θ to each other

Étienne-Louis Malus1775 – 1812

Chapter 29Problem 40

A polarizer blocks 75% of a polarized light beam. What’s the angle between the beam’s polarization and the polarizer’s axis?

Electromagnetic Waves Produced by an Antenna

• Neither stationary charges nor steady currents can produce electromagnetic waves

• The fundamental mechanism responsible for this radiation: when a charged particle undergoes an acceleration, it must radiate energy in the form of electromagnetic waves

• Electromagnetic waves are radiated by any circuit carrying alternating current

• An alternating voltage applied to the wires of an antenna forces the electric charge in the antenna to oscillate

Electromagnetic Waves Produced by an Antenna

• Half-wave antenna: two rods are connected to an ac source, charges oscillate between the rods (a)

• As oscillations continue, the rods become less charged, the field near the charges decreases and the field produced at t = 0 moves away from the rod (b)

• The charges and field reverse (c) and the oscillations continue (d)

Electromagnetic Waves Produced by an Antenna

• Because the oscillating charges in the rod produce a current, there is also a magnetic field generated

• As the current changes, the magnetic field spreads out from the antenna

• The magnetic field lines form concentric circles around the antenna and are perpendicular to the electric field lines at all points

• The antenna can be approximated by an oscillating electric dipole

The Spectrum of EM Waves• Types of electromagnetic

waves are distinguished by their frequencies (wavelengths): c = ƒ λ

• There is no sharp division between one kind of em wave and the next – note the overlap between types of waves

The Spectrum of EM Waves• Radio waves are used in

radio and television communication systems

• Microwaves (1 mm to 30 cm) are well suited for radar systems + microwave ovens are an application

• Infrared waves are produced by hot objects and molecules and are readily absorbed by most materials

The Spectrum of EM Waves• Visible light (a small

range of the spectrum from 400 nm to 700 nm) – part of the spectrum detected by the human eye

• Ultraviolet light (400 nm to 0.6 nm): Sun is an important source of uv light, however most uv light from the sun is absorbed in the stratosphere by ozone

The Spectrum of EM Waves• X-rays – most common

source is acceleration of high-energy electrons striking a metal target, also used as a diagnostic tool in medicine

• Gamma rays: emitted by radioactive nuclei, are highly penetrating and cause serious damage when absorbed by living tissue

Answers to Even Numbered Problems

Chapter 29:

Problem 14

3.9 μA

Answers to Even Numbered Problems

Chapter 29:

Problem 22

(a) 3 m(b) 6 cm(c) 500 nm(d) 3 Å

Answers to Even Numbered Problems

Chapter 29:

Problem 32

(a) 160 W/m2

(b) 350 V/m(c) 1.2 μT

Answers to Even Numbered Problems

Chapter 29:

Problem 36

3.1 cm