Lecture 22Lecture 22James Clerk Maxwell (1831-1879)

Physics 2102

Jonathan Dowling

MaxwellMaxwell’’s equationss equationsthe dawn of the 20th century

revolution in physics

GaussGauss’’ Law: Law:charges produce electric fields,

field lines start and end in charges

! =•S

qdAE0

/"

S

S S S

GaussGauss’’ law for magnetism: law for magnetism:field lines are closed

or, there are no magnetic monopoles

! =•S

dAB 0

S

S

S

S

S

S

AmpereAmpere’’s law:s law:electric currents produce magnetic fields

! =•C

idsB0

µ

C

FaradayFaraday’’s law:s law:changing magnetic fields produce (“induce”)

electric fields

!! •"=•SC

dABdt

ddsE

All together:All together:

! =•S

qdAE0

/"

! =•S

dAB 0

! =•C

idsB0

µ

!! •"=•SC

dABdt

ddsE

! =•S

dAB 0

! =•C

dsB 0

!! •"=•SC

dABdt

ddsE

No charges or currentsNo charges or currents::

…very suspicious…!

?

qq=0=0

ii=0=0

!

E • dA = 0S

"

Something is not rightSomething is not right……If we are charging a capacitor, there is acurrent left and right of the capacitor.

Thus, there is the same magnetic field rightand left of the capacitor, with circular linesaround the wires.

But no magnetic field inside the capacitor?

With a compass, we can verify there is indeeda magnetic field, equal to the field elsewhere.

But there is no current producing it! ?

EEBB BB

Maybe we can make itright…

dt

d

dt

EAd

dt

Edd

d

A

dt

dVC

dt

CVd

dt

dqi E!====== 00

0 )()()(""

"

We can write the current as:

We calculate the magnetic field produced by thecurrents at left and at right using Ampere’s law :

! =•C

idsB0

µ

q=CV V=EdC=ε0A/d ΦE=∫E•A=EA

EE

id=ε0dΦ/dt

!! •=•SC

dAEdt

ddsB

00"µ

B !

E

i

B

i

B

Displacement currentDisplacement currentMaxwell proposed it, andit was confirmed.! "•

C

dsB 0

““MaxwellMaxwell”” equations: equations:

! =•S

qdAE0

/"

! =•S

dAB 0

idAEdt

ddsB

SC

000µ!µ +•=• ""

!! •"=•SC

dABdt

ddsE

! =•S

dAE 0

! =•S

dAB 0

!! •=•SC

dAEdt

ddsB

00"µ

!! •"=•SC

dABdt

ddsE

Maxwell equations in freeMaxwell equations in freespace:space:

Fields withoutsources?

A solution to the Maxwell equations in free space is a “traveling wave”…

00

1

!µ=v

The “electric” waves travelat the speed of light!?

Light itself is a wave of electricity and magnetism!?

Maxwell, waves and lightMaxwell, waves and light

=3 108 m/s)(sin2

2

002

2

vtxkEdt

Ed

dx

Ed!"#!= $µ

electric and magnetic “forces” can travel!

!! •=•SC

dAEdt

ddsB

00"µ !! •"=•

SC

dABdt

ddsE

First person to use electromagnetic waves for communications:Guglielmo Marconi (1874-1937), 1909 Nobel Prize

(first transatlantic commercial wirelessservice, Nova Scotia, 1909)

Electromagnetic wavesElectromagnetic wavesFirst person to prove that electromagnetic waves existed:

Heinrich Hertz (1875-1894)

Electromagnetic waves: Electromagnetic waves: one velocity, many frequencies!one velocity, many frequencies!

with frequencies measured in “Hertz” (cycles per second)and wavelength in meters.

http://imagers.gsfc.nasa.gov/ems/http://www.astro.uiuc.edu/~kaler/sow/spectra.html

How do waves travel?How do waves travel?Is there an ether they ride on? Michelson andMorley looked and looked, and decided it wasn’tthere. How do waves travel???

Electricity and magnetism are “relative”:Whether charges move or not depends on whichframe we use…This was how Einstein began thinkingabout his “theory of specialrelativity”…

We’ll leave that theory for later…maybe.

SummarySummary• Changing electric fields produce (induce)magnetic fields: displacement currents.

•Maxwell’s laws allow us to calculate electric andmagnetic fields everywhere in space if we are giventhe sources: electric charges and currents.

• If there are no sources, we can still have electricand magnetic fields as electromagnetic waves,which travel at the speed of light.