General form of Faraday’s Lawb

b aba b a

a

U UV V V E ds

q

So the electromotive force around a closed path is:

E ds

And Faraday’s Law becomes:

BdE ds

dt

A changing magnetic flux produces an electric field.

This electric field is necessarily non-conservative.

E produced by changing B

BdE d

dt

2d B

E 2 r rdt

d BrE

2 dt

How about outside ro ?

Problems with Ampere’s Law

o enclCB d I ?

oB 2 r I

oIB2 r

But what if…..

o enclCB d I ?

oB 2 r 0

B 0

?????

Maxwell’s correction to Ampere’s Law

Q CV

oACd

V Ed

oo o E

AQ Ed AE

d

Eo

dQ dI

dt dt

Called “displacement current”, Id

Maxwell’s Equations

Gauss's law electric

0 Gauss's law in magnetism

Faraday's law

Ampere-Maxwell lawI

oS

S

B

Eo o o

qd

ε

d

dd

dtd

d μ ε μdt

E A

B A

E s

B s

•The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gauss’s law for magnetism•Faraday’s law and the Ampere-Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the respective fluxes

• Gauss’s law (electrical):• The total electric flux through any

closed surface equals the net charge inside that surface divided by o

• This relates an electric field to the charge distribution that creates it

• Gauss’s law (magnetism): • The total magnetic flux through any

closed surface is zero• This says the number of field lines

that enter a closed volume must equal the number that leave that volume

• This implies the magnetic field lines cannot begin or end at any point

• Isolated magnetic monopoles have not been observed in nature

oS

qd

ε E A

0S

d B A

• Faraday’s law of Induction:• This describes the creation of an electric field by a

changing magnetic flux• The law states that the emf, which is 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

• One consequence is the current induced in a conducting loop placed in a time-varying B

• The Ampere-Maxwell law is a generalization of Ampere’s law

• It describes the creation of a magnetic field by an electric field and electric currents

• The line integral of the magnetic field around any closed path is the given sum

Bdd

dt

E s

I Eo o o

dd μ ε μ

dt

B s

The Lorentz Force Law

• Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be calculated

• F = qE + qv x B• This relationship is called the Lorentz force law• Maxwell’s equations, together with this force law,

completely describe all classical electromagnetic interactions

Maxwell’s Equation’s in integral form

A Vo o

Q 1E dA dV

AB dA 0

Gauss’s Law

Gauss’s Law for Magnetism

B

C A

d dE d B dA

dt dt

Eo encl o o o oC A

d dEB d I J dA

dt dt

Faraday’s Law

Ampere’s Law

Maxwell’s Equation’s in free space (no charge or current)

AE dA 0

AB dA 0

Gauss’s Law

Gauss’s Law for Magnetism

B

C A

d dE d B dA

dt dt

Eo o o oC A

d dB d E dA

dt dt

Faraday’s Law

Ampere’s Law

Hertz’s Experiment• An induction coil is connected to a

transmitter• The transmitter consists of two spherical

electrodes separated by a narrow gap• The discharge between the electrodes

exhibits an oscillatory behavior at a very high frequency

• Sparks were induced across the gap of the receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitter

• In a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave properties

– Interference, diffraction, reflection, refraction and polarization

• He also measured the speed of the radiation

Implication • A magnetic field will be produced in empty space if there

is a changing electric field. (correction to Ampere)

• This magnetic field will be changing. (originally there was none!)

• The changing magnetic field will produce an electric field. (Faraday)

• This changes the electric field.

• This produces a new magnetic field.

• This is a change in the magnetic field.

An antenna

We have changed the magnetic field near the antenna

Hook up an AC source

An electric field results! This is the start of a “radiation field.”

Look at the cross section

E and B are perpendicular (transverse) We say that the waves are “polarized.”E and B are in phase (peaks and zeros align)

Called:“Electromagnetic Waves”

Accelerating electric charges give rise to electromagnetic waves

Angular Dependence of Intensity

• This shows the angular dependence of the radiation intensity produced by a dipole antenna

• The intensity and power radiated are a maximum in a plane that is perpendicular to the antenna and passing through its midpoint

• The intensity varies as (sin2 θ / r2

Active Figure 34.3

(SLIDESHOW MODE ONLY)

Harmonic Plane Waves

x

At t = 0

At x = 0

spatial period or wavelength

temporal period

phase velocity

2v f

T T 2 k

t

E

E

Applying Faraday to radiation

B

C

dE d

dt

C

E d E dE y E y dE y

Bd dBdx y

dt dt

dBdE y dx y

dt

dE dB

dx dt

Applying Ampere to radiation

Eo oC

dB d

dt

C

B d B z B dB z dB z

Ed dEdx z

dt dt

o o

dEdB z dx z

dt

o o

dB dE

dx dt

Fields are functions of both position (x) and time (t)

o o

dB dE

dx dt

dE dB

dx dt

E B

x t

o o

B E

x t

2

2

E B

x x t

2

o o 2

B E

t x t

Partial derivatives are appropriate

2 2

o o2 2

E E

x t

This is a wave equation!

The Trial Solution

• The simplest solution to the partial differential equations is a sinusoidal wave:– E = Emax cos (kx – ωt)

– B = Bmax cos (kx – ωt)

• The angular wave number is k = 2π/λ– λ is the wavelength

• The angular frequency is ω = 2πƒ– ƒ is the wave frequency

The trial solution

y oE E E sin kx t 2 2

o o2 2

E E

x t

2

2o2

EE sin kx t

t

22

o2

Ek E sin kx t

x

2 2o o o ok E sin kx t E sin kx t

2

2o o

1

k

The speed of light (or any other electromagnetic radiation)

o o

1v c

k

2v f

T T 2 k

3. The speed of an electromagnetic wave traveling in a transparent nonmagnetic substance is , where κ is the dielectric constant of the substance. Determine the speed of light in water, which has a dielectric constant at optical frequencies of 1.78.

5. Figure 34.3 shows a plane electromagnetic sinusoidal wave propagating in the x direction. Suppose that the wavelength is 50.0 m, and the electric field vibrates in the xy plane with an amplitude of 22.0 V/m. Calculate (a) the frequency of the wave and (b) the magnitude and direction of B when the electric field has its maximum value in the negative y direction. (c) Write an expression for B with the correct unit vector, with numerical values for Bmax, k, and ω, and with its magnitude in the form

6. Write down expressions for the electric and magnetic fields of a sinusoidal plane electromagnetic wave having a frequency of 3.00 GHz and traveling in the positive x direction. The amplitude of the electric field is 300 V/m.

The electromagnetic spectrum

2v f

T T 2 k

Another lookdE dB

dx dt

y oE E E sin kx t

o o

d dE sin kx t B sin kx t

dx dt

o oE k cos kx t B cos kx t

o

o o o

E 1c

B k

z oB B B sin kx t

Energy in Waves

2 20

0

1 1u E B

2 2

o

o o o

E 1c

B k

20u E

2

0

1u B

0

0

u EB

Poynting Vector

• Poynting vector points in the direction the wave moves• Poynting vector gives the energy passing through a unit

area in 1 sec.• Units are Watts/m2

0

1S E B

S cu

2 2

o o o

E cBEBS

μ μ c μ

Intensity

• 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 2

max max max maxav 2 2 2

I aveo o o

E B E cBS cu

μ μ c μ

11. How much electromagnetic energy per cubic meter is contained in sunlight, if the intensity of sunlight at the Earth’s surface under a fairly clear sky is 1 000 W/m2?

16. Assuming that the antenna of a 10.0-kW radio station radiates spherical electromagnetic waves, compute the maximum value of the magnetic field 5.00 km from the antenna, and compare this value with the surface magnetic field of the Earth.

21. A lightbulb filament has a resistance of 110 Ω. The bulb is plugged into a standard 120-V (rms) outlet, and emits 1.00% of the electric power delivered to it by electromagnetic radiation of frequency f. Assuming that the bulb is covered with a filter that absorbs all other frequencies, find the amplitude of the magnetic field 1.00 m from the bulb.

Radiation Pressure

Up

c

F 1 dpP

A A dt

aveS1 dUP

Ac dt c

Maxwell showed: (Absorption of radiation by an object)

What if the radiation reflects off an object?

Pressure and Momentum

• For a perfectly reflecting surface, p = 2U/c and P = 2S/c

• For a surface with a reflectivity somewhere between a perfect reflector and a perfect absorber, the momentum delivered to the surface will be somewhere in between U/c and 2U/c

• For direct sunlight, the radiation pressure is about 5 x 10-6 N/m2

26. A 100-mW laser beam is reflected back upon itself by a mirror. Calculate the force on the mirror.

27. A radio wave transmits 25.0 W/m2 of power per unit area. A flat surface of area A is perpendicular to the direction of propagation of the wave. Calculate the radiation pressure on it, assuming the surface is a perfect absorber.

29. A 15.0-mW helium–neon laser (λ = 632.8 nm) emits a beam of circular cross section with a diameter of 2.00 mm. (a) Find the maximum electric field in the beam. (b) What total energy is contained in a 1.00-m length of the beam? (c) Find the momentum carried by a 1.00-m length of the beam.

Background for the superior mathematics student!

Harmonic Plane Waves

In general, We will only be concerned with the real part of the complex phasor representation of a plane wave.

Using Euler’s formula:

Re cos E E kx todi b g

E E e E kx t i kx to

i kx to b g b g b gcos sin

k 2 = propagation number

2

= angular frequency

(kx-t) = phase

Phase Velocity - Another View

kx t

d

dtk

dx

dtkv

0

Since the plane waves remain plane waves, the phase on a plane does not change with time

vk

Vector Calculus Theorems

F dl F dS

C Az zz

A VF dA FdV

And an Important Identity

Gauss’ Divergence Theorem

Stokes Theorem

F F Fe j e j 2

Maxwell’s Equation’s In Differential Form

D

B 0

Gauss’s Law

Gauss’s Law for Magnetism

EB

t

H JD

t

Faraday’s Law

Ampere’s Law