Maxwell’s Equations

• We have been examining a variety of electrical and magnetic phenomena

• James Clerk Maxwell summarized all of electricity and magnetism in just four equations

• Remarkably, the equations predict the existence of electromagnetic waves

Gauss

• The electric field is due to electric charges.– Related to electric flux

• Electric field lines start or stop on a charge or make a closed loop.

0

)(

qAEE n

No Monopoles

• There are no magnetic charges.

• Magnetic field lines can only make a closed loops.

0)( ABB n

Faraday • An emf is induced by a varying magnetic field within a closed path.– Magnetic forces

will move charges.

• This implies that a changing magnetic field creates an electric field.

dt

BdlEECircfem

)()( ||

Ampere

• Ampere’s Law: IlBBCirc 0||)( Earlier, we just went on a closed path enclosing surface 1. But according to Ampere’s Law, we could have considered surface 2. The current enclosed is the same as for surface 1. We can say that the current flowing into any volume must equal that coming out.

IV Maxwell’s Equation

• Suppose we have a charged capacitor and it begins to discharge

Surface 1 works but surface 2 has no current passing through the surface yet there is a magnetic field inside the surface.

Problems with Ampere’s Law

oB 2 r I

oIB2 r

IlBBCirc 0||)(

But if we use Surface 2 ...

oB 2 r 0

B 0

?????

IlBBCirc 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

Ampere’s Extended law-IV Maxwell equation:

• A magnetic field is induced by an electric current.

• There is an electric flux as well.– Changes create magnetic

fields

• This implies that a changing magnetic field creates an electric field.

B field surrounds electricfield, although there is no“current” flowing here

t

IBCirc E)(

Maxwell’s Equations:• Maxwell noted the

symmetry between electric and magnetic fields.

• Changing magnetic fields create electric fields

– Current and changing electric fields create magnetic fields

– Electric field lines originate from charges or form closed loops

– Magnetic field lines form closed loops

0

)(

qSEEFlusso n

0)( BFlusso

tIlBBoneCircuitazi E||)(

t

BlEEoneCircuitazi

)()( ||

Electromagnetic Waves• So, a magnetic field will be

produced in space if there is a changing electric field

• But, this magnetic field is changing since the electric field is changing

• A changing magnetic field produces an electric field that is also changing

• We have a self-perpetuating system

A capacitor being charged by a current ic has a displacement current equal to iC between the

plates, with displacement current iD = e A dE/dt. This changing E field can be regarded as the source

of the magnetic field between the plates.

Electromagnetic Waves

Notice that the electric and magnetic fields are at right angles to one another! They are also perpendicular to the direction of motion of the wave.

This picture defines the coordinate system we will use in our discussion. Wave propagates along the x-axis. The electric field varies in the y-direction and the magnetic field in the z-direction.

Wave Motion• Changing electric and magnetic fields

create a wave.– Electric field creates a magnetic field– Magnetic field creates an electric field

Electromagnetic Waves

Close switch and current flows briefly. Sets up electric field. Current flow sets up magnetic field as little circles around the wires. Fields not instantaneous, but form in time. Energy is stored in fields and cannot move infinitely fast.

Speed of EM Waves

We are going to apply Faraday’s Law to the imaginary moving rectangle abcd. Compute the magnetic flux change

vByt

tvBy

t

AB

tfem B

00)1

Speed of EM Waves

• We can say the emf around the loop is the sum of the individual emfs going along each straight line segment in the loop

• We look at the work done in moving a test charge around the loop

• 2) fem = W/q = Fd/q = Ed = Ey0 = By0v

3) E = Bv

Speed of EM Waves

Now we are going to look at the change in electric flux. Set a new imaginary rectangle and play the same game as before.

B||l 00

E

t 00

Ez0vt

t00Ez0v Ampere’s Law:

vEzBz 0000)4

Speed of EM Waves:

smv

v

v

v

vBvB

BvE

EvBvEzBz

/103

1041085.8

1

1

1

)()4

)3

)4

8

712

00

200

00

000000

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 equation’s 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

fk vf

SPETTRO ELETTROMAGNETICO

generazione di un’onda e.m.

B

E

Generatore di onde em

Rivelatore di onde em Generatore di differenza di potenziale

Hertz nel 1886 riuscì per la prima volta a produrre e a rivelare le onde elettromagnetiche di cui Maxwell aveva previsto l’esistenza.

Le onde elettromagnetiche furono generate da oscillazioni di cariche elettriche lungo un circuito.

La trasmissione delle onde era rilevata da un cerchio di grosso filo di rame interrotto da uno spazio di lunghezza regolabile tra due sferette.

Il passaggio di una corrente oscillante nel cerchio di rame si manifestava attraverso una scintilla che illuminava le due sferette

Le onde generate con questo apparato avevano una frequenza di 109 Hz

ESPERIMENTO DI HERTZ