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