Mon. 10.1 - .2.1 Potential Formulation...
Transcript of Mon. 10.1 - .2.1 Potential Formulation...
Fri. 7.3.1-.3.3 Maxwell’s Equations
Mon. 10.1 - .2.1 Potential Formulation HW8
Exercise: ‘very long’ solenoid, with radius a and n turns per unit length, carries time varying current, I(t). What’s an expression for the electric field a distance s from axis? Recall that inside a solenoid .
z
I
Using Faraday’s Law
t
BE
dt
d BEmfor area
B
tldE
or
zInB oˆ
Example: A slowly varying alternating current, , flows down a long, straight, thin wire and returns along a coaxial conducting tube of radius a. In what direction must the electric field point?
Using Faraday’s Law
t
BE
dt
d BEmfor area
B
tldE
or
a
z
I t I0 cos t
I I
E
E
B
Lenz’ law says in the direction to drive current that would oppose changing flux, so down and up as the current varies up and down. z
What’s the electric field?
adBt
ldE
where B
0I
2 sˆ s a,
0 s a.
Calls for an Amperian loop
bottomouttopin
ldEldEldEldEldE
and I t I0 cos t
zsEzsEldE outin
a
sin
zsBdt
adBt
a
s
o
in
zsds
I
t 2z
s
aI
t in
o ln2
Example: A slowly varying alternating current, , flows down a long, straight, thin wire and returns along a coaxial conducting tube of radius a.
Using Faraday’s Law
t
BE
dt
d BEmfor area
B
tldE
or
a
z
I t I0 cos t
I I
E
E
B
Lenz’ law says in the direction to drive current that would oppose changing flux, so down and up as the current varies up and down. z
What’s the electric field?
adBt
ldE
zsEsE outinz
s
aI
t in
o ln2
In what direction must the electric field point?
Right-hand-side is independent of how far out of loop sout is, so E is constant outside. But it should be 0 quite far away, so must be 0 everywhere outside.
insEin
o
s
aI
tln
2 in
oo
s
atI
tln
2
cos
insEin
oo
s
atIln
2
sin
Inductance
What is flux through path 2 due to current following path 1?
2
111
4 r
r
ldIB o
1
2 212 adB
22
112
4ad
ldIo
r
r22
11
4ad
ldI o
r
r
Equivilantly, can rephrase using product rules, or use A to get same result
212 adA11 AB
212 adB
Purely geometric factor
21 ldA
r11
14
ldIA o
r21
124
ldldI o
rr
r 2122
12,1
44
ldldad
ldM oo
Symmetric between two loops
2,11MI
12,12 IM
22,11 IM
Inductance
1
2 As with Resistance, sometimes it’s easiest to do the geometric integral, sometimes it’s easiest to find flux, factor out current, and thus find M.
rr
r 2122
1
44
ldldad
ld oo
2,1
2
1
1
2 MII
Example: Coaxial solenoids of radii a1 > a2 and windings per length n1 and n2.
a
z
a1 a2
B2 0n2I2
1
2
221 NaB
111 lnN
11
2
2221 lnaIno 21
2
212 Ilanno
M1,2
Overlapping volumes
Demo!
Faraday’s Law: time varying current in one solenoid induces Emf and drives current in other
22,1 IMdt
d
dt
d B 1.1Emf
Self Inductance
Current passing along the loop is itself responsible for flux through the loop
rr
r 1112
1
44
ldldad
ld oo
LI1
1
Time varying current along one segment of the loop produces field and Emf felt by other segments of the same loop.. z
a
nIB 0
NaB 2
nLN
nlanIo
2
1Ilano
22
L
volume
dt
d BEmf LIdt
d
Example: single solenoid
Consider driving charges around a solenoid. How much work would you have to do to get it going?
z
a
As you accelerate it up to speed, self inductance means a counter force is generated, so you must at least provide equal and opposite force.
q
ldF
Emfq
W
So per unit charge,
WqEmf
Then the rate at which work is done by the inductance’s emf is
PI
Pdt
dq
Emf
Emf
Or using the self-inductance relationship
PLIdt
dI
PLIdt
d 2
21
So bringing the current up to speed, you must oppose this, and invest energy at
2
21 LI
dt
dPyou
2
21 LIWyou
Energy to Generate Current (physical and special case derivation)
Energy to Generate Current (physical and special case derivation)
Consider driving charges around a solenoid. How much work would you have to do to get it going?
z
a
2
21 LIWyou
Rephrasing in terms of the corresponding field that’s generated, n
BI
0
2nL o
2
2
21
n
BnW
o
oyou
2
21 BW
oyou
Extrapolating to more general cases,
dBWo
2
21
(Griffiths does a more general derivation much like he did for the work of generating E field.)
“Where is the energy stored, field or current?” Neither / both – energy isn’t a substance (no “caloric fluid”) to be stored some where. It’s kinetic and potential energy, it’s “stored in” the motion of charges and their interactions situation of current flowing and field generated.
Energy to Generate Current
Exercise: Work to turn on co-axial solenoids of different wire density, n, and opposite current, I.
dBWo
2
21
a
z
a1 a2
For an individual solenoid
outside 0
inside 110
1
InB
Energy to Generate Current (mathematical and general case derivation)
Self-inductance should be a real phenomenon for any current path; the work to establish a current along any path should be
LIdt
d
dt
dEmf
fIP Em
PdtW
2
21 LI
dt
dLI
dt
dIP
dtLIdt
dW 2
21
2
21 LIW
LI adB
AB
adA
LI ldA
ldAIW
21 lIdA
21
dqv
dJAW
21
BJo
1
dBAWo
2
1
BAABBA
AB
dBABBWo
2
1
dBAdBWo
2
21
daBAdBWo
2
21
Rephrasing as sum over a volume containing current
Sending volume to cover all space, surface to infinity, where B is presumed to be 0
Correcting Ampere’s law Physical Motivation
J
J
1adJo
Io
a1 a2
2adJo
0
ldB
Mathematical Motivation It’s a mathematical fact that, the divergence of a curl of a vector field is 0
0B
JB
0So,
JB
0
Continuity Equation:
tJ Note: in the scenario above
this isn’t zero So,
tB 0
Correcting Ampere’s law The Fix
J
J
a1 a2
It’s a mathematical fact that, the divergence of a curl of a vector field is 0
0B
tsomethingB 00
So need
E
0
tE
too
0
tE
tB o 00 0
tt
EB o 00
Or rephrasing in terms of J again
Jt
EB o
00
Jt
EB o
00
In the scenario above, E is changing as the plates charge
Unfortunately permanent historically mistaken name: “Displacement Current”
t
EJ D
0
Conceptually, a stand-in for the effect of currents elsewhere
Corrected Maxwell-Ampere’s law
J
J
Jt
EB o
00
a) Find the electric field between the plates as a function of time t.
Example: Thin wires connect to the centers of thin, round capacitor plates. Suppose that the current I is constant, the radius of the capacitor is a, and the separation of the plates is w (<< a). Assume that the current flows out over the plates in such a way that the surface charge is uniform at any given time and is zero at t = 0.
zta
ItE
o
wire ˆ2
z
zt
tEo
ˆ
Approximating at infinite sheets, recall from Gauss’s law
or zAreatq
tEo
ˆ/)(
or since
a
tItqdt
tdqI wirewire )(
)(
Constant current and q(0)=0
and 2aArea
s
b) Using this as an Amperian Loop, find the magnetic field between the capacitor plate.
adJadt
EdB
000
2
2s
a
Iwireo
adJadt
EdB
000
None across this surface
adt
EdB
00
2
2s
a
Iwireo
sB 2Symmetry, as always, tells us B parallels our loop 2
2
0a
sIwire
ˆ2 2
0
a
sIB wire
so Just like field inside the wire!
Corrected Maxwell-Ampere’s law
J
J
Jt
EB o
00
a) Find the electric field between the plates as a function of time t.
Example: Thin wires connect to the centers of thin, round capacitor plates. Suppose that the current I is constant, the radius of the capacitor is a, and the separation of the plates is w (<< a). Assume that the current flows out over the plates in such a way that the surface charge is uniform at any given time and is zero at t = 0.
zta
ItE
o
wire ˆ2
z
a
c) Find the current along the surface of the capacitor plate.
adJadt
EdB
000
ˆ2 2
0
a
sIB wireb) Using this as an Amperian Loop, find the
magnetic field between the capacitor plate.
Compare Maxwell-Ampere’s Law for two, wisely-chosen surfaces bound by our Amperian loop.
Surface 1 Can lid
2.
0
2.
0
1.
0
1.
0
surfacesurfacesurfacesurface
adt
EadJdBad
t
EadJ
Surface 2 Can body
s
2.1.
0
surfacesurface
adJadt
E
platewire II
walllcylindricacapend
adJadJ..
2
2s
a
Iwire
21
as
wireplate II
Iwire
s
Iplate
ad
parallel
anti-parallel
Corrected Maxwell-Ampere’s law
Jt
EB o
00
Execise: Current flows down a long, straight, thin wire and returns along a thin, coaxial conducting tube of radius a. The electric field for the region s < a is
adJadt
EdB
000
a) Find an expression for the “displacement current” density.
I t I0 cos t
zts
aItsE ˆsinln
2, 00
t
E
0
a
z
I I
E
B
b) Integrate over a cross-section it pierces to find the “displacement current”.
adt
E
0
Fri. 7.3.1-.3.3 Maxwell’s Equations
Mon. 10.1 - .2.1 Potential Formulation HW8
E 0
E da Qenc
0
B 0 B da 0
E B
tad
t
B
tdE
a
B
Jt
EB
000
adJadt
EdB
000
Gauss’s Law
Gauss’s Law for Magnetism
Faraday’s Law
Maxwell – Ampere’s Law