Chapter 24

Electromagnetic waves

So far you have learned

1. Coulomb’s Law – Ch. 19

2. There are no Magnetic Monopoles – Ch. 22.1

3. Faraday’s Law of Induction – Ch. 23

4. Ampere’s Circuital Law – Ch. 22.9

Each law is empirical and Faraday’s is the most remarkable of these.

Faraday’s Law of Induction is:

The EMF in a circuit is equal to the

1.Line integral of the magnetic field.

2.Surface integral of the curl of the magnetic field.

3.Rate of change of the magnetic flux enclosed by the circuit

4.Square of the hypotenuse of the current.

5.Inverse of the distance squared.

Faraday’s Law of Induction (Chapter 23)

implies an electric field is produced by a changing magnetic field.

SC

adBdt

dldE

E B

Long solenoid with changing B.

Imagine a highly resistive loop of wire in a changing magnetic field….

Ampere’s circuital law (Chapter 22.9)

is not correct for rapidly varying currents.

SC

adJIldB

00

Something is rotten… Remember my

“law”?

IdP

0 sB

For S1 : get I on the right

But for S2: get zero on the right

02

1

S ofboundary

0

S ofboundary

P

P

d

Id

sB

sB

Apply it to this case…

I is the current that crosses a surface bound by curve P

Something is rotten…

IdP

0 sB

02

1

S ofboundary

0

S ofboundary

P

P

d

Id

sB

sB 0 I

I can fix this!

No current goes through S2, but an electric field does!

For S1 : get I on the right and

dt

dQI

E

E

SS

dQ

210

AE

Gauss’s law

Almost all electric flux goes

through S2

E

210 SS

dQ

AE

Take d/dt of this equation

dt

dI

dt

dQ E 0

Gives dQ/dt across S1

Gives dQ/dt across S2

S1

S2

E

dt

dId E

P

000 sB

I propose to replace I in

Ampere’s law with:

dt

d

dt

dQI E

0

The Ampere-Maxwell Law:

The ordinary electrical current The displacement current

Origin of the term “Displacement Current”

Dielectric material

-Q +Q

I I

The dielectric material is polarized by the displacement of bound charges, and there is a “displacement current” associated with the movement of the bound charges.

Origin of the term “Displacement Current”

Vacuum

-Q +Q

I I

The amazing experimental fact is that there is a “displacement current” associated with the [polarization of the] vacuum!

The conclusion is that

A magnetic field is produced by a changing electric field.

Direct observation of the “Displacement Current” is not so easy.

In fact only conduction currents contribute to the magnetic field at low frequencies, so the first test by Hertz was really the best test of the Ampere-Maxwell law: the existence of Hertzian waves.

My equations:

0

QdAE

0AB d

dt

dd BsE

dt

dId E

000 sB

BvEF q

And the Lorentz force from E and B on A charges q:

0

QdAE

0AB d

dt

dd BsE

Id 0sB

BvEF q

describe all electric and magnetic phenomena

dt

d E 00

and LIGHT

Maxwell’s equations

1.May be derived from pure thought.

2.Are mathematical descriptions of empirical fact.

3.Are irrelevant to modern physics.

Electromagnetic waves If we look at Maxwell’s eqns where there are no charges or currents -

after a bit of math we will get…

Units of 1/speed2

constant2

2

constant2

2

00

tx

dx

d

dt

d EE

)'sin(

)'cos(11

)'cos(1

')'sin()(

)sin(

000

000

0000

0

000

0

tkzk

E

tkzkdt

dE

tkzkdt

dE

dztkzdt

dEzB

tkzEE

z

z

x

Suppose we start with a sine wave electric field in the x direction traveling in the z direction:

The Ampere-Maxwell law says

)sin(

1)cos(1

)'cos(

')'()(

2

000

2000

02000

0

tkzk

E

tkzkdt

dE

tkzk

Edt

d

dzzBdt

dzE

z

z

yx

Then the Faraday law of induction says

12

00

k

This can only be true if

The speed of the waves isUnits of

1/speed2

m/s1099792.21 8

00

c

• Light is an electromagentic wave• It is described by Maxwell’s equations• Electricity, magnetism and optics are different aspects of the same theory

constant2

2

constant2

2

00

tx

dx

d

dt

d EE

This is in the vacuum.

In a medium of dielectric constant and magnetic permeability the speed of light is

m/s1099792.21 8

medium

c