electricity magnetism lecture 2: electric fields lecture 02... · the electric field e at a point...

Electricity & Magnetism Lecture 2: Electric Fields

Today’sConcepts: A)TheElectricField

B)Con9nuousChargeDistribu9ons

Electricity&Magne/smLecture2,Slide1

★ Couldyoupleasereschedulethefinalexamtooneweeklater?Asit'sawfullyscheduledbetweenmyotherexams,anditsalsothesameweekschoolends,whichmakesit's9mingevenmorehec9c.

★ Andalsothesecondmidterm,couldyoupleasemakeitinsteadofJuly18thhaveitbeeitherWednesdayorFridayofthatveryweekthatthesecondmidtermison?(thelaterthebeNer:D)

★ Thanks!

There’sawriNenhomeworkassignmentforUnit19now.

Suddenly,terriblehaiku:Posi9vetestchargeRepelledbyop9mistsCuphalfelectric

I had problems understanding the concept and calculation of charge on and infinite line

aliNlebitconcentra9onitrequires.

defineElectricfieldandelectricforceandchargeinwordsplease

Posi9vesandnega9ves,whatdotheyreallymean?Itseemslikewe,ashumans,justdecidedtonamethemthis.

??Infinitelinesofchargee???itwouldhelpalottogooverthose,everythingelseissimmplle

Your Comments

Electricity&Magne/smLecture2,Slide2

I almost forgot how exciting smartphysics was, yay

Irony?

Inthediagramsbelow,themagnitudeanddirec9onoftheelectricfieldisrepresentedbythelengthanddirec9onofthebluearrows.Whichofthediagramsbestrepresentstheelectricfieldfromanega%vecharge?

3equal–chargesareatthecornersofanequilateraltriangle

WhereistheEfield0?Theblacklineis1/2waybetweenthebaseandthetopcharge.A.Abovetheblackline

B. BelowtheblacklineC. ontheblackline

D.nowhereexceptinfinity

E. somewhereelse

!

Electric Field PhET

Electric Field Line Applet

TheelectricfieldEatapointinspaceissimplytheforceperunitchargeatthatpoint.

Electricfieldduetoapointchargedpar/cle

Superposi/on

E2

E3

E4

EFieldpointstowardnega/veandAwayfromposi/vecharges.

“Whatexactlydoestheelectricfieldthatwecalculatemean/represent?““Whatistheessenceofanelectricfield?“

Electric Field

q4

q2

q3

Electricity&Magne/smLecture2,Slide4

Twoequal,butoppositechargesareplacedonthexaxis.Theposi/vechargeisplacedtotheleLoftheoriginandthenega/vechargeisplacedtotheright,asshowninthefigureabove.

Whatisthedirec/onoftheelectricfieldatpointA?

oUpoDownoLeLoRightoZero

CheckPoint: Electric Fields1

Electricity&Magne/smLecture2,Slide5

A

Bx+Q −Q

ABCDE

Twoequal,butoppositechargesareplacedonthexaxis.Theposi/vechargeisplacedtotheleLoftheoriginandthenega/vechargeisplacedtotheright,asshowninthefigureabove.

Whatisthedirec/onoftheelectricfieldatpointA?

oUpoDownoLeLoRightoZero

CheckPoint Results: Electric Fields1

Electricity&Magne/smLecture2,Slide6

A

Bx+Q −Q

ABCDE

Twoequal,butoppositechargesareplacedonthexaxis.Theposi/vechargeisplacedtotheleLoftheoriginandthenega/vechargeisplacedtotheright,asshowninthefigureabove.

Whatisthedirec/onoftheelectricfieldatpointB?

oUpoDownoLeLoRightoZero

CheckPoint: Electric Fields2

Electricity&Magne/smLecture2,Slide7

A

Bx+Q −Q

ABCDE

Twoequal,butoppositechargesareplacedonthexaxis.Theposi/vechargeisplacedtotheleLoftheoriginandthenega/vechargeisplacedtotheright,asshowninthefigureabove.

Whatisthedirec/onoftheelectricfieldatpointB?

oUpoDownoLeLoRightoZero

CheckPoint Results: Electric Fields2

Electricity&Magne/smLecture2,Slide8

A

Bx+Q −Q

Polarization Demo

ABCDE

CheckPoint: Magnitude of Field (2 Charges)

InwhichofthetwocasesshownbelowisthemagnitudeoftheelectricfieldatthepointlabeledAthelargest?

oCase1oCase2oEqual

+Q

+Q

+Q

−Q

AA

Case1 Case2

Electricity&Magne/smLecture2,Slide9

ABC

InwhichofthetwocasesshownbelowisthemagnitudeoftheelectricfieldatthepointlabeledAthelargest?

oCase1oCase2oEqual

“Theupperle^+Qonlyaffectsthexdirec9oninbothandthelowerright(+/-)Qonlyaffectstheydirec9onsoinboth,nothingcancelsout,sothey'llhavethesamemagnitude.”

CheckPoint Results: Magnitude of Field (2 Chrg)

E

E

Electricity&Magne/smLecture2,Slide10

+Q

+Q

+Q

−Q

AA

Case1 Case2

ABC

CheckPoint: Magnitude of Field (2 Charges)

InwhichofthetwocasesshownbelowisthemagnitudeoftheelectricfieldatthepointlabeledBthelargest?

oCase1oCase2oEqual

+Q

+Q

+Q

−Q

AA

Case1 Case2

Electricity&Magne/smLecture2,Slide9

B B

CheckPoint: Magnitude of Field (2 Charges)

InwhichofthetwocasesshownbelowisthemagnitudeoftheelectricfieldatthepointlabeledBthelargest?

oCase1oCase2oEqual

+Q

+Q

+Q

−Q

AA

Case1 Case2

Electricity&Magne/smLecture2,Slide9

B B

CheckPoint: Magnitude of Field (2 Charges)

InwhichofthetwocasesshownbelowisthemagnitudeoftheelectricfieldatthepointlabeledBthelargest?

oCase1oCase2oEqual

+Q

+Q

+Q

−Q

AA

Case1 Case2

Electricity&Magne/smLecture2,Slide9

B B

Twochargesq1andq2arefixedatpoints(-a,0) and(a,0) asshown.Togethertheyproduceanelectricfieldatpoint(0,d) whichisdirectedalongthenega/vey-axis.

x

y

q1 q2(−a,0) (a,0)

(0,d)

Whichofthefollowingstatementsistrue:

A)Bothchargesarenega/veB)Bothchargesareposi/veC)ThechargesareoppositeD)Thereisnotenoughinforma/ontotellhowthechargesare

related

Clicker Question: Two Charges

Electricity&Magne/smLecture2,Slide11

_ _+ +

_+

Electricity&Magne/smLecture2,Slide12

CheckPoint Results: Motion of Test Charge

Electricity&Magne/smLecture2,Slide13

Aposi/vetestchargeqisreleasedfromrestatdistancerawayfromachargeof+Qandadistance2rawayfromachargeof+2Q.HowwillthetestchargemoveimmediatelyaLerbeingreleased?

oTotheleLoTotherightoStays/lloOther

“Theforceispropor9onaltothechargedividedbythesquareofthedistance.Therefore,theforceofthe2Qchargeis1/2asmuchastheforceoftheQcharge.“

“Eventhoughthechargeontherightislarger,itistwiceasfaraway,whichmakestheforceitexhertsonthetestchargehalfthatasthechargeonthele^,causingthechargetomovetotheright.”

Thera9obetweentheRandQonbothsidesis1:1meaningtheywillresultinthesamemagnitudeofelectricfieldac9nginoppositedirec9ons,causingqtoremains9ll.

Ex

= k

q

d2� q�p

2d�2 cos

⇡

4

!

Ey = k

0BBBBBBBB@

qd2 �

q⇣p

2d⌘2 sin ⇡4

1CCCCCCCCA

Electric Field Example

CalculateEatpointP.

d

P

d

A) B) C) D)Needtoknowd

Needtoknowd&qE)

Whatisthedirec/onoftheelectricfieldatpointP,theunoccupiedcornerofthesquare? −q +q

+q

Electricity&Magne/smLecture2,Slide14

About r

~r12θ1

2

r12

y

x

j

i ~r12

=r12

cos ✓ ˆi + r12

sin ✓ ˆj

r12

=r

12

cos ✓ ˆi + r12

sin ✓ ˆjr

12

r12

= cos ✓ ˆi + sin ✓ ˆj

For example

~r12(1,2)

(5,5)

r12

θ

~r12

=4

ˆi + 3

ˆj

r12

=p

4

2 + 3

2 = 5

cos ✓ = 4

5

sin ✓ = 3

5

r12

=r

12

cos ✓ ˆi + r12

sin ✓ ˆjr

12

r12

= 4

5

ˆi + 3

5

ˆj

λ = Q/L

Summa/onbecomesanintegral(becarefulwithvectornature)

“Idon'tunderstandthewholedqthingandlambda.”

WHATDOESTHISMEAN?

Integrateoverallcharges(dq)

risvectorfromdqtothepointatwhichEisdefined

r

dE

Continuous Charge Distributions

LinearExample:

charges

ptforE

dq = λ dx

Electricity&Magne/smLecture2,Slide15

Clicker Question: Charge Density

Linear(λ =Q/L)Coulombs/meter

Surface(σ = Q/A)Coulombs/meter2

Volume(ρ=Q/V) Coulombs/meter3

Whathasmorenetcharge?.A)Aspherew/radius2metersandvolumechargedensityρ=2C/m3

B)Aspherew/radius2metersandsurfacechargedensityσ=2C/m2

C)BothA)andB)havethesamenetcharge.

“Iwouldliketoknowmoreaboutthechargedensity.”

SomeGeometry

Electricity&Magne/smLecture2,Slide16

Prelecture Question

A)

B)

C)

D)

E)

What is the electric field at point a?

“Howistheintegra/onofdEoverLworkedout,stepbystep?”

Clicker Question: Calculation

A) B) C) D) E)

Whatis?

Chargeisuniformlydistributedalongthex-axisfromtheorigintox=a.ThechargedensityisλC/m.Whatisthex-componentoftheelectricfieldatpointP:(x,y)=(a,h)?

x

y

a

h

P

x

r

dq = λ dx

Electricity&Magne/smLecture2,Slide19

Weknow:

Clicker Question: Calculation

Weknow:

Whatis?

A) B) C) D)

Chargeisuniformlydistributedalongthex-axisfromtheorigintox=a.ThechargedensityisλC/m.Whatisthex-componentoftheelectricfieldatpointP:(x,y)=(a,h)?

xa

P

x

r

θ1 θ2

θ2

dq = λ dx

h

y

Electricity&Magne/smLecture2,Slide20

k� cos ✓2

Z 1

�1

dx

(a � x)

2 + h

2

k� cos ✓2

Za

0

dx

(a � x)

2 + h

2

Weknow:

cosθ2 DEPENDSONx!

Clicker Question: Calculation

Whatis?

A) B)

C) noneoftheabove

Chargeisuniformlydistributedalongthex-axisfromtheorigintox=a.ThechargedensityisλC/m.Whatisthex-componentoftheelectricfieldatpointP:(x,y)=(a,h)?

xa

P

x

r

θ1 θ2

θ2

dq = λ dx

h

y

Electricity&Magne/smLecture2,Slide21

Weknow:

Clicker Question: CalculationChargeisuniformlydistributedalongthex-axisfromtheorigintox=a.ThechargedensityisλC/m.Whatisthex-componentoftheelectricfieldatpointP:(x,y)=(a,h)?

xa

P

x

r

θ1 θ2

θ2

dq = λ dx

h

y

Electricity&Magne/smLecture2,Slide22

Whatis?

A) B) C) D)

E

x

(P) = k�

1 � hp

a

2 + h

2

!

Weknow:

Calculation

Whatis?

xa

P

x

r

θ1 θ2

θ2

dq = λ dx

h

y

Electricity&Magne/smLecture2,Slide23

Chargeisuniformlydistributedalongthex-axisfromtheorigintox=a.ThechargedensityisλC/m.Whatisthex-componentoftheelectricfieldatpointP:(x,y)=(a,h)?

E

x

(P) = k�

Za

0dx

a � x

p(a � x)2 + h

2

E

x

(P) =k�

h

1 � hp

h

2 + a

2

!E

x

(P) =k�

h

(1 � sin ✓1)

E

x

(P) =k⇥

h

Z ⇤/2

�1

d� cos �

Exerciseforstudent:

Changevariables:writexintermsofθ

Result:obtainsimpleintegralinθ

Observation

NotethatourresultcanberewriNenmoresimplyintermsofθ1.

Chargeisuniformlydistributedalongthex-axisfromtheorigintox=a.ThechargedensityisλC/m.Whatisthex-componentoftheelectricfieldatpointP:(x,y)=(a,h)?

xa

P

x

r

θ1 θ2

θ2

dq = λ dx

h

y

Electricity&Magne/smLecture2,Slide24