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

Physics 2102

Jonathan Dowling

Maxwell’s equationsMaxwell’s equations

the dawn of the 20th century revolution in physics

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

field lines start and end in charges

∫ =•S

qdAE 0/ ε

S

S S S

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

or, there are no magnetic monopoles

∫ =•S

dAB 0

S

S

S

S

S

S

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

∫ =•C

idsB 0μ

C

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

electric fields

∫∫ •−=•SC

dABdt

ddsE

All together:All together:

∫ =•S

qdAE 0/ ε

∫ =•S

dAB 0

∫ =•C

idsB 0μ

∫∫ •−=•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 right…Something is not right…If we are charging a capacitor, there is a current left and right of the capacitor.

Thus, there is the same magnetic field right and left of the capacitor, with circular lines around the wires.

But no magnetic field inside the capacitor?

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

But there is no current producing it! ?

EEBB BB

Maybe we can make it right…

dt

d

dt

EAd

dt

Edd

d

A

dt

dVC

dt

CVd

dt

dqi EΦ

====== 000 )()()( εεε

We can write the current as:

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

∫ =•C

idsB 0μ

q=CV V=EdC=ε0A/d ΦE=∫E•A=EA

EE

id=ε0dΦ/dt

∫∫ •=•SC

dAEdt

ddsB 00εμ

B !

E

i

B

i

B

Displacement currentDisplacement currentMaxwell proposed it, and it was confirmed.∫ ≠•

C

dsB 0

““Maxwell” equations:Maxwell” equations:

∫ =•S

qdAE 0/ ε

∫ =•S

dAB 0

idAEdt

ddsB

SC

000 μεμ +•=• ∫∫

∫∫ •−=•SC

dABdt

ddsE

∫ =•S

dAE 0

∫ =•S

dAB 0

∫∫ •=•SC

dAEdt

ddsB 00εμ

∫∫ •−=•SC

dABdt

ddsE

Maxwell equations in free Maxwell equations in free space:space:

Fields withoutsources?

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

00

1

εμ=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−∝⇒−= εμ

electric and magnetic “forces” can travel!

∫∫ •=•SC

dAEdt

ddsB 00εμ ∫∫ •−=•

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 and Morley looked and looked, and decided it wasn’t there. How do waves travel???

Electricity and magnetism are “relative”: Whether charges move or not depends on which frame we use…

This was how Einstein began thinking about his “theory of special relativity”…

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 and magnetic fields everywhere in space if we are given the sources: electric charges and currents.

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