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Page 1 Name ___________solutions___________

Physics 321 Electromagnetism I Second midterm 8.20 am Wednesday 21 November

Autumn 2012 Instructor: David Cobden A102

Do not turn this page until the buzzer goes at 8.20.

Hand your exam to me before I leave the room at 9.25.

Attempt all the questions.

Please write your name on every page and your SID on the first page.

Write all your working on these question sheets. Use this front page for extra working. It is importantto show your calculation or derivation. Some marks are given for showing clear and accurate working

and reasoning.

Watch the blackboard for corrections and clarifications during the exam.

This is a closed book exam. No books, notes, or calculators allowed.

The dipole potential is.

.

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Page2

Name

otuti

b

r\5:

,

l.

[6]

State

Laplace's

equation

and

also state carefully

what

must

be known

in

order

to be able

to

solve

it

in

a

specified

region.

v'

v(r)

=

o

";:J

":

rffi#

fu:;*"

2.

t

I 0] An

"infinite slot" is

the interior

of

a set of three

perpendicular

planes.

Two of them,

the

upper

and lower,

atz

=

aandz

=0,

are

grounded

conductors.

The

third,

the

bottom of

the slot,

at

x:0,

has

a

varying

potential

V

=VosilYapplied

over

it.

We

want

to

find

the

potential

inside

the

slot.

First, find

by

separation

of

variables

the

general

eigenfunction

expansion

of the

potential

7(r) in

Cartesian coordinates.

[x@Y(Oz({

=o

k,.'*ki+k}o

7X

"

=

k,]

X

Y"=ko-Y

Z"

=

k;Z

L

Sn^i\"{y

6. f q..l

L

,o

v=r6*

k*, k:

I

x>O

V=o

a,

w:

L,O

)j

-

k:=O

:.

kJ*k.Lo

4.

[5]

Apply

the

boundary

condition

at

infinity.

V,

O

o.5.

x-)

e

.'.

i.

V=

VrO

a.\^

Z=O

,4

{".

q\(

rc

Z

n

*=O

nT(

o\

r+

P.-LX.k,3o

c-

e"r(*E

"y

or

1*Va.\.-.h.

A,

9,,

c,

D

J"r"

on

k^,

ka'

Ve

wq^b

."^(

Y6orrcr."h

(.

2c-

-fo

1

v

=

A

@

*'aB

e

*)6"s

k*e+

D't

,r

el\^\V

c\(

2

k,.

3.

[5]

Now

simplifu

for

this

case,

taking account

of translation

symmetry

in

y.

"'

a,\\

D'

=

O

(

koqs^T

;.

k.=

rr

1(d

A

:

ATl+

A".c

Stn

q-

n-'*1[

o,

k-t

+

b'"ink^)

ar e1*I

5.

[6]

Apply

the boundary

conditions

at the conducting

planes.

;.

It

7/24/2019 321_MT2_2012_solutions

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-+

A^=O,

n{

5

;

hzrV

z1lt

sln

{i

Ii

ti

v

(

(

I

\

(

I

(

1

%

/

4r@4T|rfl

r^aq

Page

3

Name

5ol.tH

c/Yu

6.

[8]

Finally,

apply

the boundary

condition

at x

=

0

to find

the complete

solution

7(r).

v,s\n

7.nt

:

+

A,^

e r,\l

ry

(

t\

6v{rL',o1o'h\ib

i

siqr.r

;.V:

Vr{ry

7.

[3]

How

do

you

know

this

is the

only

solution?

tA^iq,^=

.*s,

,IlN.-orq.F

\

{o.

t-q\^Js

1*^t'\on

8.

[6]

The image

charge system

for

a

point

charge

q

atr1

in

the

neutral

conducting sphere whose

surface isr

=

R

(in

spherical

polar

coordinates) consists of two charges:

q'

=

-turr=

(i)'.r,

and

q

at

the origin. What

is

the value

of

q ,

and

why?

1 =*1

Ne-qhra\

+

no

/. $

fl.^x

o*h

#

yh*rc*

Fi

Zay;-o.?Fs

=

O

9.

[0]

A

point

dipole

of moment

p

is

located

at

a

position

11

fl&r a

neutral

conducting

sphere of

radius R.

Construct

an

appropriate image

system

and thereby

write

down an

expression for

the

potential

I/(r)

everywhere

outside the sphere.

-

-

dnr.[i-r-,

l]

4l

r.,f

i-el'

'*,

'

\-{

\

f,T>.

nfY

DipL

u

*1*f ,, 1J

bb

s-^\\

Ai \a'rc*

d=

Ehd.

a(

K.so-

hot

o ..

Po\\r.

10.

[4]

A

positive

charge

Q

is now

transferred to the sphere. How

should the image

system be changed

to

give

the

correct V

(r)?

Ad.r[

G

a.F

H^o

oAgi^..

11.

[6]

Sketch

on the above

diagram the electric

field lines

(solid)

and

equipotentials

(dashed)

around

the

sphere and

dipole now with

this charge on

the

sphere.

t',

o

i'

Qr

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Page 4

Name

S\un\i t t

12.

L4)

Write

down

the leading

order term in

the multipole

expansion

of

the

potential

far

away

from

the dipole

+

charged

sphere

system.

13.

Uzl

A

thin

circular ring

of

radius

R lies inthe

x-y

plane

centered

on the

origin.

It has

a line

charge

density

7

=

A

cos

@

distributed along its

circumference, where

@

is

the azimuthal

angle

relative to

the

i direction.

Find the monopole

and dipole moments

of the ring

(in

Cartesian

coordinates.)

-I

r\s

cl qeJ.

so

V

=

_L

-F

(h)

flar

^=

Ac

t /

(:

f

h.o,

@,1

:

Ktt:'cc,s6

aq

-O

z

J

ni

A

cosl&Aq

c

fi=

[t t'

'=u{J'b

=

trA*

5.'

a

TrR'A

*

s/Ao

os

fi

co

coig

Jy

14.

[6]

Hence

find

coordinates.

the

potential

7(r) far

from

the

ring

(r

>>

R).

Express

it in

spherical

polar

'-51r\'tJ

v(n)=*?.ea*

##

+

O(b\

Ld:o

[-+

oo

FJ

7c

/s

f/

w

\,

,J

J*O

6

ot p--a

:

?r

KA*,

+

R}4 cor

@

*'llt r

t

zor*

I

5 .

[

1 0] A

parallel-plate

capacitor,

with

plate

separation

d and area A

>> d'

,

is floating in

outer

space

with

a

potential

difference

76

between the

plates.

Find the

(dominant

term

in the multipole

expansion

ofthe)

potential

far from

the

capacitor.

a

-,

r=

V

=

c%i+.

ed$=e AV i

no

+-

$-+

-

raorgffu '

4fi-rp

v

[i) =

z

Auc.i

--

1V,

6.

''-l

GP

V, A

cosO

*71

ct