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Physics 2102
Jonathan Dowling
Lecture 22
James Clerk Maxwell (1831-1879)
Maxwell’s equations
the dawn of the 20th century
revolution in physics
Gauss’ Law:
charges produce electric fields,
field lines start and end in charges
E

dA

q
/

0

S
S
S
S
Gauss’ law for magnetism:
field lines are closed
or, there are no magnetic monopoles
B

dA

0

S
S
Ampere’s law:
electric currents produce magnetic fields
B

ds


i
0

C
C
Faraday’s law:
changing magnetic fields produce (“induce”)
electric fields
d
E

ds


B

dA
C

dt S
All together:
E

dA

q
/

0

S
B

ds


i
0

B

dA

0

S
C
d
E

ds


B

dA
C

dt S
No charges or currents:
 E  dA  0
q=0
S
 B  dA  0
?
i=0
S
B

ds

0

C
…very suspicious…!
d
E

ds


B

dA
C

dt S
Something is not right…
B
E
B
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! ?
E
id=0d/dt
Maybe we can make it
right…
We calculate the magnetic field produced by the
currents at left and at right using Ampere’s law :
 B  ds   i
0
C
We can write the current as:
dq d (CV )
dV  0 A d ( Ed )
d ( EA)
d E
i

C

 0
 0
dt
dt
dt
d
dt
dt
dt
q=CV
C=0A/d
V=Ed
E=E•A=EA
Displacement current
B

ds

0

C
Maxwell proposed it, and
it was confirmed.
d
C B  ds  0 0 dt S E  dA
B
B!
B
i
i
E
“Maxwell” equations:
E

dA

q
/

0

S
 B  dA  0
S
d
B

ds



E

dA


i
0
0
0
C

dt S
d
E

ds


B

dA
C

dt S
Maxwell equations in free
space:
E

dA

0

S
 B  dA  0
S
d
B

ds



E

dA
0
0
C

dt S
d
E

ds


B

dA
C

dt S
Fields without
sources?
Maxwell, waves and light
A solution to the Maxwell equations in free space is
a “traveling wave”…
d
C B  ds 0 0 dt S E  dA
d
C E  ds   dt S B  dA
electric and magnetic “forces” can travel!
d 2E
d 2E
   0 0 2  E  sin k ( x  vt)
2
dx
dt
v
1
 0 0
=3 108 m/s
The “electric” waves travel
at the speed of light!?
Light itself is a wave of
electricity and magnetism!?
Electromagnetic waves
First person to prove that electromagnetic waves existed:
Heinrich Hertz (1875-1894)
First person to use electromagnetic waves for communications:
Guglielmo Marconi (1874-1937), 1909 Nobel Prize
(first transatlantic
commercial wireless
service, Nova Scotia,
1909)
Electromagnetic waves:
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?
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.
Summary
• 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.