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Maxwells Equations

- Chapter 32, Sections 9, 10, 11
- Maxwells Equations

Electromagnetic Waves

Chapter 34, Sections 1,2,3

The Equations of Electromagnetism (at this point

)

Gauss Law for Electrostatics

Gauss Law for Magnetism

Faradays Law of Induction

Amperes Law

The Equations of Electromagnetism

..monopole..

Gausss Laws

1

?

2

...theres no magnetic monopole....!!

The Equations of Electromagnetism

.. if you change a magnetic field you induce an

electric field.........

Faradays Law

3

Amperes Law

4

.......is the reverse true..?

Look at charge flowing into a capacitor

E

B

Amperes Law

Here I is the current piercing the flat surface

spanning the loop.

Look at charge flowing into a capacitor

E

B

Amperes Law

Here I is the current piercing the flat surface

spanning the loop.

E

For an infinite wire you can deform the surface

and I still pierces it. But something goes wrong

here if the loop encloses one plate of the

capacitor in this case the piercing current is

zero.

B

Side view (Surface is now like a bag)

Look at charge flowing into a capacitor

E

It must still be the case that B around the

little loop satisfies

B

E

where I is the current in the wire. But that

current does not pierce the surface.

B

What does pierce the surface? Electric flux - and

that flux is increasing in time.

Look at charge flowing into a capacitor

E

B

E

B

Thus the steady current in the wire produces a

steadily increasing electric flux. For the

sac-like surface we can write Amperes law

equivalently as

Look at charge flowing into a capacitor

E

B

The best way to write this result is

E

B

Then whether the capping surface is the flat

(pierced by I) or the sac (pierced by electric

flux) you get the same answer for B around the

circular loop.

Maxwell-Ampere Law

E

B

This result is Maxwells modification of Amperes

law

Can rewrite this by defining the displacement

current (not really a current) as

Then

Maxwell-Ampere Law

E

B

This turns out to be more than a careful way to

take care of a strange choice of capping surface.

It predicts a new result

A changing electric field induces a magnetic field

This is easy to see just apply the new version

of Amperes law to a loop between the capacitor

plates with a flat capping surface

B

x

x x x x

x x x x x

x x

Maxwells Equations of Electromagnetism

Gausss Law for Electrostatics

Gausss Law for Magnetism

Faradays Law of Induction

Amperes Law

Maxwells Equations of Electromagnetism

Gausss Law for Electrostatics

Gausss Law for Magnetism

Faradays Law of Induction

Amperes Law

These are as symmetric as can be between electric

and magnetic fields given that there are no

magnetic charges.

Maxwells Equations in a Vacuum

Consider these equations in a vacuum no charges

or currents

Maxwells Equations in a Vacuum

Consider these equations in a vacuum no charges

or currents

These integral equations have a remarkable

property a wave solution

Plane Electromagnetic Waves

Ey

Bz

This pair of equations is solved simultaneously

by

c

x

as long as

Static wave F(x) FP sin (kx ?) k 2?

? ? k wavenumber ? wavelength

Moving wave F(x, t) FP sin (kx - ?t) ?

2? ? f ? angular frequency f frequency v ?

/ k

F

v

Moving wave F(x, t) FP sin (kx - ?t )

x

At time zero this is F(x,0)Fpsin(kx).

F

v

Moving wave F(x, t) FP sin (kx - ?t )

x

At time zero this is F(x,0)Fpsin(kx). Now

consider a snapshot of F(x,t) at a later fixed

time t.

F

v

Moving wave F(x, t) FP sin (kx - ?t )

x

At time zero this is F(x,0)Fpsin(kx). Now

consider a snapshot of F(x,t) at a later fixed

time t. Then

F(x, t) FP sinkx-(?/k)t

This is the same as the time-zero function, slide

to the right a distance (?/k)t.

F

v

Moving wave F(x, t) FP sin (kx - ?t )

x

At time zero this is F(x,0)Fpsin(kx). Now

consider a snapshot of F(x,t) at a later fixed

time t. Then

F(x, t) FP sinkx-(?/k)t

This is the same as the time-zero function, slide

to the right a distance (?/k)t. The distance it

slides to the right changes linearly with time

that is, it moves with a speed v ?/k. The wave

moves to the right with speed ?/k

Plane Electromagnetic Waves

These are both waves, and both have wave speed

?/k.

Plane Electromagnetic Waves

These are both waves, and both have wave speed

?/k. But these expressions for E and B solve

Maxwells equations only if

Hence the speed of electromagnetic

waves is

Plane Electromagnetic Waves

These are both waves, and both have wave speed

?/k. But these expressions for E and B solve

Maxwells equations only if

Hence the speed of electromagnetic

waves is Maxwell plugged in the values of the

constants and found

Plane Electromagnetic Waves

These are both waves, and both have wave speed

?/k. But these expressions for E and B solve

Maxwells equations only if

Hence the speed of electromagnetic

waves is Maxwell plugged in the values of the

constants and found

Plane Electromagnetic Waves

These are both waves, and both have wave speed

?/k. But these expressions for E and B solve

Maxwells equations only if

Hence the speed of electromagnetic

waves is Maxwell plugged in the values of the

constants and found

Thus Maxwell discovered that light is

electromagnetic radiation.

Plane Electromagnetic Waves

Ey

Bz

c

- Waves are in phase.
- Fields are oriented at 900 to one another and to

the direction of propagation (i.e., are

transverse). - Wave speed is c
- At all times EcB.

x

The Electromagnetic Spectrum

infra -red

ultra -violet

Radio waves

g-rays

m-wave

x-rays

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