Transcript PPT

Physics 2113
Jonathan Dowling
James Clerk Maxwell
(1831-1879)
Lecture 37: WED 22 APR CH32:
Maxwell’s Equations I
Maxwell I: Gauss’ Law for E-Fields:
charges produce electric fields,
field lines start and end in charges
S
S
S
Maxwell II: Gauss’ law for B-Fields:
field lines are closed
or, there are no magnetic monopoles
S
F = å BA = 0
d >b>c>a=0
FTa &B = 2 ´ 6 - 4 ´ 3 = 12 - 12 = 0
FTc &B = -2 ´ 6 + 2 ´ 8 = -12 +16 = +4
T &B
F Side
=
-F
=0
a
a
F Side
= -FTc &B = -4
c
F Side
=0
a
F Side
=4
c
T &B
FTb &B = -2 ´ 1- 4 ´ 2 = -2 - 8 = -10 F d = +2 ´ 3 + 3 ´ 2 = +6 + 6 = +12
F bSide = -FTb &B = 10
T &B
F Side
=
-F
= -12
d
d
F bSide = 10
F Side
= 12
d
Maxwell III: Ampere’s law:
electric currents produce magnetic fields
C
Maxwell IV: Faraday’s law:
changing magnetic fields produce (“induce”)
electric fields
Maxwell Equations I – IV:
In Empty Space with No Charge or Current
q=0
?
i=0
…very suspicious…
NO SYMMETRY!
Maxwell’s Displacement Current
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 Maxwell reasoned this without any
experiment!
The missing
Maxwell
Equation!
But there is no current producing it! ?
E
id=0d E/dt
Maxwell’s Fix
We calculate the magnetic field produced by the
currents at left and at right using Ampere’s law :
We can write the current as:
dq d(CV )
dV e 0 A d(Ed)
d(EA)
dF E
i=
=
=C
=
= e0
= e0
dt
dt
dt
d
dt
dt
dt
q = VC C = e 0 A / d V = Ed
Displacement “Current”
Maxwell proposed it based on
symmetry and math — no experiment!
B
B!
B
i
i
E
Changing E-field Gives Rise to B-Field!
Maxwell’s Equations I – V:
I
E
·
dA
=
q
/
e
0
ò
S
II
B
·
dA
=
0
ò
S
IV
V
d
B
·
ds
=
m
e
E
·
dA
+
m
i
III
0
0
0
òC
ò
dt S
d
E
·
ds
=
B
·
dA
òC
ò
dt S
Maxwell Equations in Empty Space:
Changing E gives B.
Changing B gives E.
Fields without
sources?
Positive
Feedback
Loop!
32.3: Induced Magnetic Fields:
Here B is the magnetic field induced along a
closed loop by the changing electric flux FE
in the region encircled by that loop.
Fig. 32-5 (a) A circular parallel-plate capacitor, shown in side view, is being charged by
a constant current i. (b) A view from within the capacitor, looking toward the plate at
the right in (a).The electric field is uniform, is directed into the page (toward the plate),
and grows in magnitude as the charge on the capacitor increases. The magnetic field
induced by this changing electric field is shown at four points on a circle with a radius r
less than the plate radius R.
Ba > Bc > Bb > Bd = 0
dF E d
dE
= EA = A
µ slope
dt
dt
dt
The displacement current id = i is
distributed evenly over grey area.
So rank by i
enc
d
= amount
of grey area enclosed by each loop.
d =c>b>a
32.3: Induced Magnetic Fields: Ampere Maxwell Law:
Here ienc is the current encircled by the closed
loop.
In a more complete form,
When there is a current but no change in electric
flux (such as with a wire carrying a constant
current), the first term on the right side of the
second equation is zero, and so it reduces to the
first equation, Ampere’s law.
Example, Magnetic Field Induced by
Changing Electric Field:
Example, Magnetic Field Induced by Changing Electric
Field, cont.:
32.4: Displacement Current:
Comparing the last two terms on the right side of the above equation shows that the
term
must have the dimension of a current. This product is usually treated as
being a fictitious current called the displacement current id:
in which id,enc is the displacement current that is encircled by the integration loop.
The charge q on the plates of a parallel plate capacitor at any time is related to the
magnitude E of the field between the plates at that time by
in which A is
the plate area.
The associated magnetic field are:
AND
Example, Treating a Changing Electric Field as a Displacement Current:
Example, Treating a Changing Electric Field as a Displacement Current:
id
id
32.5: Maxwell’s Equations: