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Mon. Wed. Fri.,
10.3 Point Charges (C 14) 4.1 Polarization (C 14) 4.2 Field of Polarized Object
HW9
Mon. (C14) 4.3 Electric Displacement
Continuous Source Distribution c
ttr
rwhere d
trJtrA ro
r
),(,
4
http://web.mit.edu/viz/spin/ choose slow spin up – time evolving magnetic field for a sphere of charge spinning up
Charged sphere spinning up from rest
Continuous Source Distribution
Exercise: find the Vector potential for a wire that momentarily had a burst of current. c
ttr
rwhere d
trJtrA ro
r
),(,
4
bo ttqtI )(
ssr ˆ
zzr ˆ
r
ldtrI
trA ro
r
),(,
4
22 szr
22 sz rso
Defined piecewise through time
So, at some time, tb, the current will blink on and off again. The observer will first notice the middle blink, then just either side of the middle, then a little further out,…
zzdttq
trA broo ˆ)(
,4
r
So, we get contribution to our integral only when
b
rb
ttc
cttt
r
r
Which is true at two locations at any moment t:
22sttcz b
We could rephrase the delta function as being a spike at these two locations, or we could observe the integral is ‘even’ and then wave our hands
0
4ˆ
)(2, zzd
ttqtrA broo
r
z
ttc
q
b
oo ˆ2
Continuous Source Distribution
HW Exercise: A neutral current loop made of two concentric arcs. The current rises with time as I(t) = kt (presumably just since t=0, but we’ll assume we’re long enough out.) What are A, and E at the origin?
cttr
rwhere d
trJtrA ro
r
),(,
4
a b
lldtrI
trA roL
ˆ),(
4),(
r
b
a
rrr
a
b
roL bd
b
ktxxd
x
ktad
a
ktxxd
x
kttA
0
0
ˆˆˆˆ4
),0(
Observe: since integrating over negative values, |x|= -x
b
a
a
b
oL yxd
c
btkxxd
x
c
xtk
yxdc
atkxxd
x
c
xtk
tA0
0
ˆcosˆsinˆˆcosˆsinˆ4
),0(
b
a
a
b
oL yxd
c
at
c
btkxxd
x
c
xtk
xxdx
c
xtk
tA0
ˆcosˆsinˆˆ4
),0(
b
a
oL yxd
c
bakxxd
x
c
xtk
tA0
ˆcosˆsinˆ24
),0(
xc
bakx
c
ab
a
btktA o
Lˆ2ˆln2
4),0(
x
a
bkto ˆln2
4
Continuous Source Distribution
HW Exercise: A neutral current loop made of two concentric arcs. The current rises with time as I(t) = kt (presumably just since t=0, but we’ll assume we’re long enough out.) What are A, and E at the origin?
cttr
rwhere d
trJtrA ro
r
),(,
4
a b
lldtrI
trA roL
ˆ),(
4),(
r
t
AVE
Neutral, so no charge density, no V
xa
bko ˆln
2
xa
bkto ˆln
2
Time-Dependent Coulomb’s & Biot-Savart General Solutions to Maxwell’s Equations
Jefimenko’s Equations
t
AVE
AB
dtr
trV r
o
Lr
),(
4
1),(
dtrJ
trA roL
r
),(
4),(
and
where
While proving, we’d found
Meanwhile
dc
trJ r
o r2
),(
4
1
dtrJ
tt
A roL
r
),(
4
dtrJ ro
r
),(
4
so
t
AVE
2
1
co
o
dc
trJtrtr rrr
o rr
r
cr
r2 2
),(ˆ),(ˆ),(
4
1
Time-Dependent Coulomb’s & Biot-Savart General Solutions to Maxwell’s Equations
Jefimenko’s Equations
AB
dtrJ
trA roL
r
),(
4),(
where
dc
trJtrtrtrE rrr
o rr
r
cr
r2 2
),(ˆ),(ˆ),(
4
1),(
Product rule 7 fAAfAf
= -
Time-Dependent Coulomb’s & Biot-Savart General Solutions to Maxwell’s Equations
Jefimenko’s Equations
dc
trJtrtrtrE rrr
o rr
r
cr
r2 2
),(ˆ),(ˆ),(
4
1),(
=
The x,y, and z dependence is locked up in
where
so, for example
=
=
=
=
Time-Dependent Coulomb’s & Biot-Savart General Solutions to Maxwell’s Equations
Jefimenko’s Equations
dc
trJtrtrtrE rrr
o rr
r
cr
r2 2
),(ˆ),(ˆ),(
4
1),(
dc
trJ
c
trJtrv
c
trJtrtrqtrF rrrrr
o
rrrrr
r
cr
r22
ˆ),(),(
,),(ˆ),(ˆ),(
4
1),(
232
Force experienced by a moving charge in presence of changing charge and current densities.
2
21 )()( tgtvrtr oo
Apparently Maxwell’s Laws require time separation, but don’t dictate whether field precedes or follows charge behavior. In addition to imposing Maxwell’s equations, we also imposed causality – effect follows cause.
BvqEqtrF
),(
Note: we did that all the time in classical mechanics too – think of the quadratic
Continuous Source Distribution
Exercise: If we have a slowly varying current density, then what does the general magnetic field expression reduced to?
cttr
rwhere d
trJ
c
trJtrB rro r
rr 2ˆ
),(),(
4),(
),(),(),( rrr trJtttrJtrJ
Taylor series
Remarkably, for constant current, even though the charge configuration is changing, the field here and now depends on the charge configuration right now, not back when it was emitted.
Continuous Source Distribution
HW Exercise: If we have a constant current density, then what does the general electric field expression reduced to?
cttr
rwhere d
c
trJtrtrtrE rrr
o rr
r
cr
r2 2
),(ˆ),(ˆ),(
4
1),(
)0,(),(
0),(
rJtrJ
trJ
r
r
),(),( rr trJtr
)0,(),(),()0,( rJtrJtrr rr
rr trrtr )0,()0,(),(
Constant
By continuity equation
But if J is constant in time, its divergence is constant in time, so r varies at a constant rate
and trrtr )0,()0,(),(
so ttrtrtr rr )0,(),(),(
or
crtrtr r
r)0,(),(),(
dtr
trEo
2r
r̂),(
4
1),(
Point Source d
trJtrA ro
r
),(,
4
dtr
trV r
o r
),(,
41
Continuous Source Distribution d
trJ
c
trJtrB rro r
rr 2ˆ
),(),(
4),(
dc
trJtrtrtrE rrr
o rr
r
cr
r2 2
),(ˆ),(ˆ),(
4
1),(
Differentially small volume of charge
zyx
qtr ),(
v
observer
r
r
),(,
41 rtr
trVo
But appears to occupy wider volume x
x
tvxx
Apparent Extra length
tv
xx
Extra distance light travels from back vs. from front
tc cosx
c
xt
cos
c
vx
x
cos1
1
c
v r̂1
1
zyxzyx
q
o r4
1
x
xqtrV
o r4
1,
c
v
qtrV
o rr ˆ1
1,
41
rr
vc
qctrV
o41,
Point Source d
trJtrA ro
r
),(,
4
dtr
trV ro
r
),(,
4
Continuous Source Distribution d
trJ
c
trJtrB rro r
rr 2ˆ
),(),(
4),(
dc
trJtrtrtrE rrr
o rr
r
cr
r2 2
),(ˆ),(ˆ),(
4
1),(
Differentially small volume of charge
zyx
qtr ),(
v
observer
r
r
vtrtrA ro
),(,
4
But appears to occupy wider volume x
x
tvxx
Apparent Extra length
tv
xx
Extra distance light travels from back vs. from front
tc cosx
c
xt
cos
c
vx
x
cos1
1
c
v r̂1
1
rr
vc
qctrV
o41,
zyx
vzyx
q
o
r
4
x
xvqtrA o
r
4,
rr
vc
vqctrA o
4,
rr
vc
vqctrA
c
co
2
2
4,
rr
vc
vqctrA
co2
14
1,
trVc
vtrA ,,
2
Point Source
How about Fields rr
vc
qctrV
o41, trV
c
v
vc
vqctrA o ,,
24
rr
AB
At
VE
rrc
v
qctrV r
o
r
1
4),(
rrcrrc
v
v
qcr
o
2
1
4
rcrrrcrrc
vvv rrrrˆ
Product Rule 4
rrrrr
rrrrr vvvvv
)( rrr twr
r
Notational note: to reinforce that our r now points to a moving source, Griffiths replaces “r’”, that’s stationary in time, with “w”, that tracks the moving source.
rtwrrr
r
)(0 rtw
z
t
t
w
y
t
t
w r
r
yr
r
z
z
tw
y
twtw
ryrz
xrr
)()()(
Focus on just one component
zy
tv
x
tvy
z
tv
z
tvx
z
tv
y
tvtw r
xr
yx
zr
xr
yr
zrˆˆˆ)(
z
tv
y
tv r
yr
z
rtv
vtr
rr tv
rSo far:
Point Source
How about Fields rr
vc
qctrV
o41, trV
c
v
vc
vqctrA o ,,
24
rr
AB
At
VE
rrc
v
qctrV r
o
r
1
4),(
rrcrrc
v
v
qcr
o
2
1
4
rcrrrcrrc
vvv rrrrˆ
Product Rule 4
rrrrr
rrrrr vvvvv
)( rrr twr
r )( rtw
cttr
rr
1
c
2r1
crr
1
c
rrrr
1
121
c
rrrrrr
2
Product Rule 4
vttv rr
rr
rr tv
r
rtv
rrr
Product Rule 2
So far:
quoting
rrrr
vttv rr2
rrrrr
vttv
ct rrr
1
Point Source
How about Fields rr
vc
qctrV
o41, trV
c
v
vc
vqctrA o ,,
24
rr
AB
At
VE
rrc
v
qctrV r
o
r
1
4),(
rrcrrc
v
v
qcr
o
2
1
4
rcrrrcrrc
vvv rrrrˆ
Product Rule 4
rrrrr
rrrrr vvvvv
)( rrr twr
r )( rtw
rr tv
r
rrrrr
vttv
ct rrr
1
rtw-rzyx
zyx rrrrr
r
rr
r
rr
r
rr
t
tw
z
t
t
tw
y
t
t
tw
x
t
z
r
y
r
x
r
zyxzyx rrrrrrrr
z
t
y
t
x
t
t
twzyx rrr
r
rzyxzyx rrrrrrrr
ˆˆˆ
rtv rrrr
So rrrr tvvttv
ct rrrr
r
1rr
r
vt
cr
1
vc
rr
r
Point Source
How about Fields rr
vc
qctrV
o41, trV
c
v
vc
vqctrA o ,,
24
rr
AB
At
VE
More of the same…
auuvcu
qtrE
o
r
r
r 22
34),(
vcu
-r̂
),(),(1
),(22
trVvvtrVc
trVc
vB
Ec
B
auuvcu
q
cB
o
r
rrr
1
ˆ1
4
1 22
3
Force between moving charges (Eq’n 10.74)
auuvcc
Vauuvc
u
qQF
o
rrr
r
r 2222
3ˆ
4
vcu
r̂q Q r
v
a
V
“The entire theory of classical electrodynamics is contained in that equation…but you see why I preferred to start out with Coulomb’s law.” - Griffiths
Mon. Wed. Fri.,
10.3 Point Charges (C 14) 4.1 Polarization (C 14) 4.2 Field of Polarized Object
HW9
Mon. (C14) 4.3 Electric Displacement
