Energy of electric force
Electrostatic potential
Electrostatic potential difference
Potential energy
B
Z
Z
B
∆UAB = −WAB = −
~ · d~r = −q
F
Z
A
Only electrostatic potential difference is physical meaningful.
B
~ · d~l
E
It is the potential energy difference per charge of two points.
A
A
~ · d~l
E
∆VAB = −
The electrostatic potential energy difference moving charge q when a
charge is moved from point A to piont B
∆VAB a property of two points:
Potential energy difference is negative of worked doned by electric field.
Sign of the potential energy difference depends on sign of charge.
Line integral of a conservative force do not depends on the path: recall
gravity.
Do not depends on any path that connects them, so choose wisely.
Measured in volt (V): joule per coulomb.
A natual unit for energy of quantized charge: electron volt (eV).
e = 1.6 × 10−19 C
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The potential
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1eV = 1.6 × 10−19 J
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Equipotential and electric field
EPD of a point charge
Equipotential surface is collection of all points with the same value for φ.
∆VAB = kq
1
1
−
rB
rA
Electric field is along the direction of steepest descend:
Electric field
Fix the value at a some point ~r0 define a single point function:
φ(~r) = ∆V (~r,~r0 ) + φ(~r0 )
~ = −∇φ
~
E
For point charge, a particular convenient choice is limr→∞ φ(r) = 0:
Density of equipotential line along certain direction ' Electric field
strength along that direction
Streamline of electric field is everywhere perpendicular to equipotential
surfaces.
Electrostatic potential of a point charge
φ(r) =
kq
r
Where there is no electric field, potential is constant
Use the principle of superposition to compute electrostatic potential of
more complicated charge distribution.
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A conductor is an equipotential.
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Visualizing the potential
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Energy of a system of charges
How to assemble a series of charges?
First one do not cost you any energy
Second one:
k q 1 q2
U2 =
d12
Third one:
k q 1 q3
k q 2 q3
+
U3 =
d13
d23
...
N
N
1 X X k q i qj
UN =
2
dij
i=1 j6=i
Order do not matter!
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More on conductor
Capacitor
Charge on conductor automatically redistribute to keep
zero electric field inside each conductor
each conductor an equipotential surface.
connected conductors have the same potential; isolated conductors do not
have to have the same potential.
In general any systems of charged conductor form a capacitor. The most
usual type involve two “parallel” plates.
Store energy in form of electric field by developing a potential difference
across the two plate proportional to the charge of each plate:
Capacitance
Transfer charge Q from one parallel plate conductor to another:
one plate will end up with charge +Q and the other −Q
cost energy:
d
W =
Q2
20 A
C = Q/V
Q measured in coulomb, V in volt, C in farad.
Where did the energy go? In the electric field!
A capacitor stores charge and energy:
Energy density of electric field
Energy stored in capacitor
1
uE = 0 E 2 .
2
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U =
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The capacitance
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QV
CV 2
Q2
=
=
2
2
2C
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Circuits of capacitors
Parallel plates (-like) capacitor:
Parallel connection:
Capacitance of parallel plates capacitor
C =
0 A
d
A capacitor can have some dielectric material between the plates instead
of empty space
0 → . Usually > 0 .
(screenign) Dielectric responds to external electric field by
charge-polarization and creates a opposing field.
reduced total field reduces energy needed to put charge on capacitor
plates.
capacitance increases
both ends connected/shorted
same voltage across each capacitor: V1 = V2 = V3 = · · ·
charges additive: Qtotal = Q1 + Q2 + Q3 + · · ·
capacitance add:
Ctotal = C1 + C2 + C3 + · · ·
Series connection:
one end connected/shorted
same charge: Q1 = Q2 = Q3 = · · ·
voltage additive: Vtotal = V1 + V2 + V3 + · · ·
inverse of capacitance add:
1
1
1
1
=
+
+
+ ···
Ctotal
C1
C2
C3
What happen when C changes depends on how the capacitor is
connected:
Energy U =
Charge do not change for an isolated capacitor(s).
Voltage do not change for a capacitor connected to a battery
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QV
2
always additive. Check for yourself!
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