Post on 22-Sep-2020
transcript
Fri. 2.4.3-4.4 Work & Energy in Electrostatics
Mon.
Wed.
Thurs
2.5 Conductors
3.1-.2 Laplace & Images Summer Science Research Poster Session:
Hedco7pm~9pm
HW3
VQldEQldFbaW
b
a
b
a
you
Work to construct charge distribution Source charges
a
sensing charge, Q
b
q1
q2
12r
q3
23r
13r
q1
14r
24r
34r
i ij ij
jin
i
i
j ij
ji qqqqW
oo rr4
1
2
1
1
41
Work to construct a discrete charge distribution n
i
ii
n
i ji ij
j
i
n
i ji ij
jin
i ij ij
ji
assemble PVqq
qqqqq
W11 01010 2
1
4
1
2
1
4
1
2
1
4
1
rrr
If we sum twice over all charges (but self interaction) correct by dividing by 2
Avoiding self energy
ji ij
j
i
qPV
r04
1 Potential at location of qi due to all other charges but qi
dVPVdPVdqWn
i
id
n
i
iidq
assemble 2
1
2
1
2
1
10
10
limlim
Work to construct a continuous charge distribution
where
dVo
4
1
r
Self energy issues vanish for continuous distribution since
00
drrdqd
)(lim)(
Concrete example, universal results – Charging a Capacitor
First, E-field expression
E
o
o
o
lr
lr
llrr
o
encl
EEE
EE
dadaEdaE
QadE
2
1
E
One plate
E
doubles
cancels
oinE
Another plate
Concrete example, universal results – Charging a Capacitor
xA
QxE
oo
ˆˆ 1
sA
QldEV
o
l
r
lr1
x̂s
dQsA
QVdQdW
o
Work to move one morsel of charge, dQ
2
21 Q
A
sdQ
A
QsdQ
A
QsdWW
ooo
A
sQsA
A
QsAW
adsadVW
o
assemble
assemble
oo
o
22
121
1
2
1
2
1
2
1
2
1
2
1
Or using what we’d just derived:
Rephrased in terms of field
VolEAsEVQW oo
create
2
2
2
221
Same result, more mathematically
dVW 2
1
E
0
W 0
2E V d
VEVEVE
Gauss’s Law
Product Rule
V E
W 0
2E V d E 2d
SV
adVEdVE
Divergence Theorem
0S
adVE Sending Volume & Area out to
infinity To contain all of charge distribution’s field
spaceall
dEW.
20
2
Not for point charges
dVW 2
1
spaceall
dEW.
20
2
Recall in derivation of
that we explicitly required
00
drrdqd
)(lim)(
to avoid accidentally including
0
2
41
2
41 i
ii
i q
r
qoo
in our sum.
So,
doesn’t apply for a point charge.
Example: Work of assembling a charged, solid sphere