electricity & magnetism lecture 6: electric potential lecture 06...checkpoint: electric field...

Electricity & Magnetism Lecture 6: Electric Potential

Today’sConcept:ElectricPoten4al

(DefinedintermsofPathIntegralofElectricField)

Electricity&Magne4smLecture6,Slide1

Stuff you asked about:

!  “ExplainmorewhyEisnega4veofdeltaV”

!  “Idon'tlikethatweweretoldtousegradientswhenwehaven'tevendonetheminmathyet.”

!  “Sowejustneedtosuperimposetheradialfieldlineswhicharefoundbytakingthenega4veofthegradientoftheelectricpoten4alin3dcartesian/spherical/cylindricalcoordinatesystemandareperpendiculartoequipoten4als,thelocuspointofallpointwiththesamepoten4aldifference.Simpleenough...We'lljustdothat!....Ohhhhhwait...WHAT?.”

!  “soelectricpoten4alisthean4deriva4veofelectricfieldisthatcorrect?inotherwordstheareaunderelectricfieldfunc4ongivestheelectricpoten4al?

!  Whatisthedifferencebetweenanintegralofadotproductandaintegralofasimpleproduct?

!  How are the equipotentials spaced so far apart? doesn't the Voltage change continually with r?


Big Idea

Last4mewedefinedtheelectricpoten4alenergyofchargeqinanelectricfield:

€

ΔUa→b = −r F ⋅d

r l = − q

r E ⋅d

r l

a

b∫

a

b∫

Theonlymen4onofthepar4clewasthroughitschargeq.Wecanobtainanewquan4ty,theelectricpoten4al,whichisaPROPERTYOFTHESPACE,asthepoten4alenergyperunitcharge.

Notethesimilaritytothedefini4onofanotherquan4tywhichisalsoaPROPERTYOFTHESPACE,theelectricfield.

ΔVa→b ≡ΔUa→b

q= −

rE ⋅ d

rl

a

b

∫

vE ≡

rFq


Electric Potential from E field

ConsiderthethreepointsA,B,andClocatedinaregionofconstantelectricfieldasshown.

WhatisthesignofΔVAC = VC-VA?

A)ΔVAC < 0B)ΔVAC = 0C)ΔVAC > 0 Rememberthedefini4on:

ΔVA→C = −rE ⋅ d

rl

A

C

∫

Chooseapath(anywilldo!)

D

Δx

ΔVA→C = −rE ⋅ d

rl

A

D

∫ −rE ⋅d

rl

D

C

∫ ΔVA→C = 0 −rE ⋅ d

rl

D

C

∫ = −EΔx < 0


CheckPoint: Zero Electric Field

Supposetheelectricfieldiszeroinacertainregionofspace.Whichofthefollowingstatementsbestdescribestheelectricpoten4alinthisregion?

A.  Theelectricpoten4aliszeroeverywhereinthisregion.B.  Theelectricpoten4aliszeroatleastonepointinthisregion.C.  Theelectricpoten4alisconstanteverywhereinthisregion.D.  Thereisnotenoughinforma4ongiventodis4nguishwhichoftheabove

answersiscorrect.

ΔVA→B = −rE ⋅ d

rl

A

B

∫


Rememberthedefini4on

0=Er

0=Δ →BAV Visconstant!

E from V Ifwecangetthepoten4albyintegra4ngtheelectricfield:

Weshouldbeabletogettheelectricfieldbydifferen4a4ngthepoten4al?

ΔVa→b = −rE ⋅ d

rl

a

b

∫

VE ∇−=rv

InCartesiancoordinates:

Ex = −

dVdx

Ey = −dVdy

Ez = −dVdz


CheckPoint: Spatial Dependence of Potential 1


Theelectricpoten4alinacertainregionisplodedinthefollowinggraph

AtwhichpointisthemagnitudeoftheE-FIELDgreatest?

o Ao Bo Co D

VE ∇−=rv




AtwhichpointisthemagnitudeoftheE-FIELDgreatest?


HowdowegetEfrom V ?

VE ∇−=rv Ex = −

∂Vdx Lookatslopes!




Atwhichpointisthedirec4onoftheE-fieldalongthenega4vex-axis?


“At B, the slope is decreasing (-) so the direction of the E field is negative “ “E is negative when the slope of V is positive (E=-dV/dx). Therefore E is directed along the x-axis at point C. “

HowdowegetEfrom V ?

VE ∇−=rv

Ex = −dVdx

Lookatslopes!

Equipotentials Equipoten4alsarethelocusofpointshavingthesamepoten4al.

Equipotentials produced by a point charge

Equipoten4alsareALWAYS

perpendiculartotheelectricfieldlines.

TheSPACINGoftheequipoten4alsindicatesTheSTRENGTHoftheelectricfield.


Contour Lines on Topographic Maps



Visualizing the Potential of a Point Charge

Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 2 / 19

CheckPoint: Electric Field Lines 1


Thefield-linerepresenta4onoftheE-fieldinacertainregioninspaceisshownbelow.Thedashedlinesrepresentequipoten4allines.

AtwhichpointinspaceistheE-fieldtheweakest?


“The electric field lines are the least dense at D” “ From what I know, the answer should be D” “ D is where the electric field lines are the least dense “ “ I’m pretty sure the electric field lines are the least dense at D” “ I’d guess D “



Comparetheworkdonemovinganega4vechargefromAtoBandfromCtoD.Whichonerequiresmorework?

A.  Moreworkisrequiredtomoveanega4vechargefromAtoBthanfromCtoD

B.  Moreworkisrequiredtomoveanega4vechargefromCtoDthanfromAtoB

C.  Thesameamountofworkisrequiredtomoveanega4vechargefromAtoBastomoveitfromCtoD

D.  Cannotdeterminewithoutperformingthecalcula4on


Whatarethese?

ELECTRICFIELDLINES!

Whatarethese?

EQUIPOTENTIALS!

WhatisthesignofWAC = workdonebyEfieldtomovenega4vechargefromAtoC?

A)WAC<0B)WAC=0C)WAC>0

AandCareonthesameequipoten4al

Equipoten4alsareperpendiculartotheEfield:Noworkisdonealonganequipoten4al

Clicker Question: Electronic Field 2

WAC = 0


CheckPoint Results: Electric Field Lines 2


Comparetheworkdonemovinganega4vechargefromAtoBandfromCtoD.Whichonerequiresmorework?

A.  Moreworkisrequiredtomoveanega4vechargefromAtoBthanfromCtoD

B.  Moreworkisrequiredtomoveanega4vechargefromCtoDthanfromAtoB

C.  Thesameamountofworkisrequiredtomoveanega4vechargefromAtoBastomoveitfromCtoD



! AandCareonthesameequipoten4al! BandDareonthesameequipoten4al! Thereforethepoten4aldifferencebetweenAandBistheSAMEasthepoten4albetweenCandD



Comparetheworkdonemovinganega4vechargefromAtoBandfromAtoD.Whichonerequiresmorework?

A.  Moreworkisrequiredtomoveanega4vechargefromAtoBthanfromAtoD

B.  Moreworkisrequiredtomoveanega4vechargefromAtoDthanfromAtoB

C.  Thesameamountofworkisrequiredtomoveanega4vechargefromAtoBastomoveitfromAtoD



Insulating charged sphere


Prelecture Question


Which of the following equipotential diagrams best describes the spherical insulator in the previous example? (The colored

circle in the center represents the insulator.)

Prelecture Question


The separation of the equipotentials is smallest at the outer radius of the pink sphere since the electric field is strongest

there. The separation of the equipotentials increases where the electric field is weaker; both toward the center of the sphere and

far away from the sphere.

Calculation for Potential

Pointchargeqatcenterofconcentricconduc4ngsphericalshellsofradiia1,a2,a3,anda4.Theinnershellisuncharged,buttheoutershellcarrieschargeQ.WhatisVasafunc4onof r?

StrategicAnalysis:!  Sphericalsymmetry:UseGauss’LawtocalculateEeverywhere!  Integrate E togetV

metal

metal

+Q

a1

a2

a3 a4

+q

cross-sec4on

ConceptualAnalysis:!  ChargesqandQwillcreateanEfieldthroughoutspace

! 

V (r) = −rE

r0

r

∫ ⋅ drl


Calculation: Quantitative Analysis

r > a4 : WhatisE(r)?A)0B)C)

50

metal

metal

+Q

a1

a2

a3 a4

D)E)

Why?

Gauss’law:

r

rE∫ ⋅ d

rA = Qenclosed

ε0

204

1rqQE +

=πε

+q

cross-sec4on


204

1rQ

πε rqQ +

021πε

204

1rqQ −

πε204

1rqQ +

πε

0

24ε

πqQrE +

=

metal

metal

+Q

a1

a2

a3 a4

+q

cross-sec4on


a3< r<a4:WhatisE(r)?

A)0B)C)

D)E)

Howisthispossible?-qmustbeinducedatr=a3surface

r

ApplyingGauss’law,whatisQenclosedforredsphereshown?

A)qB)-qC)0

chargeatr=a4surface=Q+q


204

1rq

πε rq

021πε

204

1rqQ −

πε204

1rq−

πε

E4πr2 = Q+ qε0

24

4 4 aqQ

πσ

+=

metal

metal

+Q

a1

a2

a3 a4

+q

cross-sec4on


Con4nueonin…

a2 < r < a3 :

TofindV:1) Chooser0suchthatV(r0)=0(usual:r0=infinity)2) Integrate!


E = 14πε0

qr2

r r < a1 : 204

1rqE

πε=

a1 < r < a2 : 0=E

r > a4 : V =14πε0

Q+ qr

a3 < r < a4 : A)

B)

C)

( ) 041

440

+→∞Δ=+

= aVaqQV

επ

3041

aqQV +

=επ

0=V

r > a4 :

a3 < r < a4 :


metal

metal

+Q

a1

a2

a3 a4

+q

cross-sec4on


V =14πε0

Q+ qr

V =14πε0

Q+ qa4

a2 < r < a3 : V (r) = ΔV (∞→ a4 )+ 0+ΔV (a3→ r)

V (r) = Q+ q4πε0a4

+ 0+ q4πε0

1r−1a3

⎛

⎝⎜

⎞

⎠⎟ V (r) = 1

4πε0Q+ qa4

+qr−qa3

⎛

⎝⎜

⎞

⎠⎟

a1 < r < a2 : V (r) = 14πε0

Q+ qa4

+qa2−qa3

⎛

⎝⎜

⎞

⎠⎟

0 < r < a1 : V (r) = 1

4πε0Q+ qa4

+qa2−qa3+qr−qa1

⎛

⎝⎜

⎞

⎠⎟

electricity & magnetism lecture 6: electric potential lecture 06...checkpoint: electric field...

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