Maxwell’s equationsthe dawn of 20th century physics

James Clerk Maxwell (1831-1879)

Physics 2102Gabriela González

Gauss’ Law:charges produce electric fields,

field lines start and end in charges

S

qdAE 0/

S

S S S

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:electric currents produce magnetic fields

C

idsB 0

C

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

electric fields

SC

dABdt

ddsE

All together:

S

qdAE 0/

S

dAB 0

C

idsB 0

SC

dABdt

ddsE

S

dAE 0

S

dAB 0

C

dsB 0

SC

dABdt

ddsE

No charges or currents:

…very suspicious…!

?

q=0

i=0

S

qdAE 0/

idsBC 0

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.

There is an electric field inside the capacitor.But no magnetic field there?

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

But there is no current producing it! ?

Maybe we can make it right…

dt

d

dt

EAd

dt

Edd

d

A

dt

dVC

dt

CVd

dt

dqi E

000 )()()(

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 0

q=CV V=EdC=e0A/d FE=EA

SC

dAEdt

ddsB 00

B !

E

i

B

i

B

Displacement current

Maxwell proposed it, and it was confirmed.

C

dsB 0

“Maxwell” equations:

S

qdAE 0/

S

dAB 0

idAEdt

ddsB

SC

000

SC

dABdt

ddsE

S

dAE 0

S

dAB 0

SC

dAEdt

ddsB 00

SC

dABdt

ddsE

Maxwell equations in free space:

Fields withoutsources?

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

00

1

v

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

Light itself is a wave of electricity and magnetism!?

Maxwell, waves and light

=3 108 m/s

electric and magnetic “forces” can travel!

The velocity of the “Maxwell” waves is…

m0 = 1.256 10-6 Tm/Ae0 = 8.854 10-12 C2/Nm2

[m0e0 ]=(Tm/A)(C2/Nm2) =TsC/Nm =(Ns/Cm)(sC/Nm) =s2/m2

SC

dAEdt

ddsB 00

SC

dABdt

ddsE

A physics t-shirt!

S

qdAE 0/

S

dAB 0

idAEdt

ddsB

SC

000

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 wavesFirst person to prove that electromagnetic waves existed:

Heinrich Hertz (1875-1894)

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.