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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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
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
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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
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
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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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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
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