1 16 overview work, energy, voltage relation between field and voltage capacitance homework: 4, 8,...

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1 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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1

16 Overview

• work, energy, voltage

• relation between field and voltage

• capacitance

• homework:

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

2

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.

3

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)

4

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

6

2

2

1

1

r

kQ

r

kQVP

0.4

)104(109

0.2

)104(109 9999

VVP 99180.4

36

0.2

36

7

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

8

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(

9

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)

10

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

11

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

14

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

16

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

17

18

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

20

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