Name ___________________________________Student ID ____________ Score_______ last first

VECTOR DERTVATIVES

Cunesian. i +. / r !+. / . i i l r =dr. l \ .1.

Name ___________________________________Student ID ____________ Score_______ last first

, l \ t t ( s rL-T.DA \ I ENT-{L CO \ST{ \TS

.:N8ar lorrCr, \ r r r

/ i : l r IO \ Al

( =.1 l | r . l l l 'nr \

. = 1.6{) , l (J L"C

' , : ( r l l , l ( l lg

iPennr l l i \ i l \ of l rcc \ t r ! ( |

rpcrnrr ihr | r \ ( i f i i .c \ f ! ( ( l

rchrrg. o l the elecrron)

rmi\ \ o l the ele.rro l

SPHERICAL AND CYLI\DRICAL COORI)INATESSphc cal

t '= ' i i l 'n \oIr=/nn"!od,

l ' : , ' , r , , r - :t " : ' , " ,1, , , ; .1Ilo=lun i \ t r

( rlindrical

i : . in i r .o.d i - .os ' .*ar i .1" o ii : \ in, \ ,n @i-.o\ , \ ;n r , r i - corodt : c,r , i . rn" P.f .

ll i : s in r ! {^or i -sLn ! \urdi +co\di{ : e," ' , .^o i - .o." ' ! r ( ' , i \ in

p i0 = \u 'o i - .o 'o i

i : cosdG nn OCr" : \ rn p\ + .osp9i :2

s^ : e,"oi - ' 'n o iC: \ rndi-eo.dii : i

l ,=. \ , - \ ;I

F = 140qQr2 r Coulombs Law

F =QE Force on charge Q due to electric field

E = 140 !r( )r2 r d ! Electric field from continuous charge distribution

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Eda =10Qencl Gauss Law (integral), andE =

10

Gauss Law (differential)

E dl = 0 and E = 0

V r( ) = Ed l

r

Electrostatic potential, and E = V

2V = 0

Poissons equation, and 2V = 0 Laplaces equation (regions of no charge)

V (r) = 140qr and V r( ) =

140

!r( )r d ! (setting reference point at infinity)

Eabove Ebelow =0n and

Vaboven

Vbelown =

10 boundary conditions

V b( )V a( ) =WQ

W = 12 qii=1

n

V ri( ) and W = 12 V d and W =02 E

2 d

P = 02 E2

Electrostatic pressure

Q =CV Capacitor