FIZ102E Lecture4CapacitanceandDielectrics

AlexandrJonasDepartmentofPhysicsEngineering

T

Whatdidwecoverlastweek?

Electricpotentialenergyinauniformfield

Uniformelectricfieldisconservative workWabdonebyelectricforcesonchargeq0 movingbetweenpointsa andb doesnotdepend onthepath

ElectricpotentialenergyU=q0 Ey Wab=Ua Ub =q0 E(ya yb)

Workofelectricfieldtomovechargeq0 betweenpointsa andb

pathcharge

0

pathcharge

ldEqldFW ba

..

Electricpotentialenergyoftwopointcharges

Workofelectricfieldtomovechargeq0 froma tobalonganarbitrarypath

cosdldr

baba

r

r

r

r

r

rba UUrr

qqdrrqqdrFdlFW

b

a

b

a

b

a

1144 1cos 00200

20

041

rqqFF

rqqU 0

041Workispathindependent potentialenergy

Electricpotentialenergyofasystemofpointcharges

Totalworkdonebythefieldonchargeq0 betweenpointsa andb

b

a

b

a

b

a

b

a

b

aba ldFldFldFldFldFW ..... 4321total

Potentialenergy: i i

i

rqqUUUUU

0

04321 4

Totalforceactingonchargeq0

iN

iF

FFFFF

1

4321total

Totalpotentialenergy=algebraicsum ofindividualpotentialenergies

ElectricpotentialElectricpotentialV =potentialenergyU perunitchargeassociatedwithatestchargeq0 placedattheselectedpointinanelectricfield

0qWVVV babaab

pointa potentialVa

pointb potentialVb

0qUV Units:1Volt[V]=1J/C

Voltage:

VoltageVab =workperunitchargedonebytheelectricfieldtomoveachargeq0 frompointa topointb

chargeq0

+

Calculatingelectricpotential

Singlepointcharge:rqV

041

Multiplepointcharges: i i

i

rqV

041

Continuouschargedistribution:

ondistributi

charge rdqV

041

Relationbetweenelectricpotentialandelectricfield

Potentialdifference=lineintegralofelectricfield

b

a

b

a

b

ababa ldEq

ldEq

q

ldF

qWVV

...

0

0

00

(a)Inthedirectionoftheelectricfield potentialdecreasesforanysignofthesourcecharge(b)Againstthedirectionoftheelectricfield potentialincreasesforanysignofthesourcecharge

Movingfrompointa topointb

Equipotentialsurfaces

Equipotentialsurface =threedimensionalsurfaceonwhichtheelectricpotentialV isthesameateverypoint

00

b

a

baba ldEq

WVV

.

foranytwopointsa,b onequipotentialsurfacesinceVa =Vb

0ldE . electricfieldisperpendicular toequipotentialsurfaces

Potentialgradient

Electricfield atanypointinspace=negativeofthepotentialgradient

Theelectricfieldpointsinthedirectioninwhichthepotentialdecreasesmostrapidly

b

aba ldEVV

. Vk

zVj

yVi

xVE

constzydxdV

xV

,

...partialderivativeofV withrespecttox(similarlyfory andz)

kz

jy

ix

...gradientoperator

Whatwillwecovertoday?

Capacitorsandcapacitance

Capacitorsinseriesandparallel

Energystorageincapacitorsandelectricfieldenergy

Dielectrics

MolecularmodelofinducedchargeandGaussslawindielectrics