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