16 Overview

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16 Overview

• work, energy, voltage

• relation between field and voltage

• capacitance

• homework:

• 4, 8, 9, 13, 19, 40, 41, 55, 69, 82, 95, 97

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Electrostatic Potential Energy, UE

& Electric Potential, V• Charge-charge interaction stores energy

• Ex. two + + close have high UE

• Electric Potential V is energy per test charge in (J/C = V) (volts)

• Two steps to find V at a point of interest “P”:

• 1) Measure UE when q is moved to P (from far away)

• 2) Calculate V = UE/q

• /

Work-Energy Theorem

• Relates change in energy stored in a system to work done by that system.

• UE = -WE

• If positive work is done by an electric system, then the change in the stored energy is negative.

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Example V calculation

• q = +1.0 C moved close to another + charge (from far away).

• If UE = +3.0 J,

• Then V = UE/q = (+3.0 J)/(+1.0 C)

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VV 0.3

Point Charge Potential, VQ

• VQ = kQ/r

• Ex. Potential 2.0m from Q = +4.0nC is VQ = kQ/r = (9E9)(+4E-9)/(2) = +18V.

• Electric Potential is + near +charges

• Ex. Potential 4.0m from Q = -4.0nC is VQ = kQ/r = (9E9)(-4E-9)/(4) = -9V.

• Electric Potential is - near -charges

• /5

Potential Due to Several Charges

• Point charge potentials add algebraically

• VP = VQ1 + VQ2 + …

• Ex. If “P” is 2.0m from Q1 = +4nC and 4.0m from Q2 = -4nC, Then

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2

2

1

1

r

kQ

r

kQVP

0.4

)104(109

0.2

)104(109 9999

VVP 99180.4

36

0.2

36

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Potential Difference & Average Electric Field

• Let + test charge q move in the direction of the field E (°)

• UE = -WE

• UE = -FEd

• UE = -qEavd

d

VEav

qd

UE Eav

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Ex. Average Electric Fieldd

VEav

X(m) V(volts)

0 100

2 90

10 80

30 70

50 65

Interval

0 to 2

2 to 10

10 to 30

30 to 50

mVm

VEav /5

)02(

)10090(

mVm

VEav /25.1

)210(

)9080(

mVm

VEav /50.0

)1030(

)8070(

mVm

VEav /25.0

)3050(

)7065(

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Equipotential Surfaces

• surfaces which have the same potential at all points.

• Ex. A sphere surrounding an isolated point charge is an equipotential surface.

• Ex. A charged conductor in electrostatic equilibrium is an equipotential surface. (this also implies E near surface is perpendicular to the surface)

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Capacitance: Charge Stored per Volt AppliedThe capacitance is defined as C = Q/VThe capacitance is defined as C = Q/V Units: C/V = farad = FUnits: C/V = farad = F

CVQ

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Capacitors

• store energy… and give it back fast, e.g. flash unit

Permittivity

• Relates to ability of material to store electrostatic potential energy

• Empty space value:

• Material values are:

• … is the dielectric constant

• Exs. = 1.0 air, 3.5 paper 12

21212 C 1085.84

1 mNke

o

o

Parallel Plate Capacitance

• Ex. Area A = 100 square-cm, d =1mm

13

d

AC o

(empty) 1085.8101

)101()1( 113

22

Fm

mC o

filled)(paper 1010.3101

)101()5.3( 103

22

Fm

mC o

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Energy Stored in a Capacitor

q

UV E

qVUE

Charge Q added to Capacitor over average potential of V/2

QVVQUE 21)2/(

QVUE 21

Capacitor Energy

QVUE 21 CVQ

221

21 )( CVVCVUE

CQCQQUE /)/( 221

21

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Supercapacitors

• Porous structure with high internal surface area (A) and small spacing (d) resulting in very large capacitance

• Have capacitances greater than 1 farad

Capacitor Circuits

• Parallel: each gets potential V, so capacitance increases

• Series: each gets potential less than V, so capacitance decreases

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Capacitors in “Parallel” Arrangement

CVQ

""12 VVVV BA

eqBA QQQ

BA QQ VCVCVC eqBA

eqBA CCC

Ex. FFFCeq 18126

19

eqBABA C

Q

C

Q

C

QVV

Capacitors in “Series” Arrangement

C

QV eqBA QQQ

Q = 0eqBA CCC

111

12

1

6

11

eqCEx.

FCeq 4

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Summary

• Welectric = qEd = -EPE

• V = EPE/q

• V = V1 + V2 +…

• Eavg = -ΔV/d

• C = q/V = KoA/d

• Capacitor Energy = ½CV2

• Capcitors in series & parallel