Electric potential,
Systems of charges
Physics 114
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Lecture IV
1
Concepts
• Primary concepts:
– Electric potential
– Electric energy
– capacitance
• Secondary concepts:
– Equipotentials
– Electonvolt
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Lecture IV
2
Potential electric energy
High PE
High PE
Low PE
Low PE
Just like gravity electric force can do work
work does not depend on the path
it depends only on the initial and final position
 there is a potential energy associated with electric field.
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Lecture IV
3
Electric potential
PE (q )  q
• PE/q is a property of the field
itself – called electric potential V
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Lecture IV
4
Electric potential
PE
V
q
• V – electric potential is the potential energy of a
positive test charge in electric field, divided by the
magnitude of this charge q.
• Electric potential is a scalar (so much nicer!).
• Electric potential is measured in Volts (V=J/C).
• Potential difference between two points DV=Vb-Va
is often called voltage.
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Lecture IV
5
Charges in electric fields
b

F d

E
+
a
E=const
Force on charge q:
F=qE
Work done by the field to move this charge
W=Fd=dqE
W=PEa-PEb=qVa-qVb=-qDV
• d E=DV
• E= DV/d, points from high potential to
low
• Sometimes electric field is measured in
V/m =N/C
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Lecture IV
6
Non-uniform electric field
+



E ( x )  F ( x )  qE ( x )
b
b
a
a
W   Fdx  q  Edx
 
V   Ed r
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Lecture IV
7
Determine E from V
• Think ski slopes
• If V depends on one coordinate x
• E is directed along x from high V to low
dV
E ( x)  
dx
• If V depends on x,y,z
V
V
V
Ex  
; Ey  
; Ez  
x
y
z
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Lecture IV
8
Electric field and potential in
conductors

Eexternal
+
+
+
+
-
-
-
-
 

E  Eexernal  Eint ernal
V  const
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E=0 in good conductors in the
static situation.
E is perpendicular to the surface
+
of conductor.
Metal hollow boxes are used to
shield electric fields.
When charges are not moving (!!)
conductor is entirely at the
 0 same potential.
Lecture IV
9
Electronvolt
Energy  5000eV
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• Energy that one electron
gains when being accelerated
over 1V potential difference
is called electronvolt eV:
• 1eV=1.6x10-19C 1V= 1.6x10-19J
• Yet another unit to measure
energy,
• Commonly used in atomic
and particle physics.
Lecture IV
10
Equipontentials
Equipotentials
• are surfaces at the same
potential;
• are always perpendicular to
field lines;
• Never cross;
• Their density represents the
strength of the electric field
• Potential is higher at points
closer to positive charge
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Lecture IV
11
Potential of a point charge
V ( )  0
Potential V of electric field
created by a point charge Q at
a radius r is

dr
Q
V (r0 )   Edr  kQ 2 k
r
r0
r0
+
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Q>0  V>0
Q<0  V<0
Do not forget the signs!
Potential goes to 0 at infinity.
Equipotentials of a point charge
are concentric spheres.
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12
Superposition of fields
V1  0
V  V1  V2
+ V2  0
+
-
1
2
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Principle of superposition:
Net potential created by a
system of charges is a
scalar (!) sum of
potentials created by
individual charges:
V  V1  V2  V3  ....
Potential is a scalar 
no direction to worry about,
But signs are important.
Lecture IV
13
Work to move a charge
V  V1  V2
a
a
+
a
V  V  V2
b
1
b
+
20cm
15cm
+
Q1=10mC
-
Q2=-20mC
b
How much work has to be
done by an external force
to move a charge
q=+1.5 mC
from point a to point b?
Work-energy principle
W  DKE  DPE  PEb  PEa
PEb  qVb  q(V  V )
b
1
b
2
PEa  qVa  q (V  V )
a
1
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Lecture IV
a
2
14
Capacitance
• Two parallel plates are called a
capacitor. When capacitor is
connected to a battery plates will
charge up.
Note: net charge =0
Q  CV
• C – coefficient, called
capacitance, property of the
capacitor.
• Capacitance is measured in
Farad (F=C/V)
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Lecture IV
15
Electric field in a capacitor
E=const
• E= V/d
• points from high potential to low
• When V is fixed (same battery), E depends
only on the d.
Q V

• Potential
– High next to + plate
– Low – next to - plate

Q
E

0 0 A
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Lecture IV
0 A
d
Q
0 A
d
0 A
C
d
16
V
Capacitance
• Capacitance depends on the
geometry of a capacitor
A
C  0
d
 0 = 8.85. 10-12 C2/N m2A
• permittivity of free space
• A – area of plates (m2)
• Same sign charges want to “spread out”
– to hold more charge need large area
• d – distance between plates (m)
• Opposite sign charges “hold” each
other, attraction is stronger for shorter d
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Lecture IV
d
17
Dielectrics
• Put non-conductive
material (dielectric)
between plates
• Can hold more charge 
capacitance increases
A
C  K 0
d
• K(>1) – dielectric
constant
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Charging up a capacitor
• Find the work needed to charge a
capacitor C to voltage V
• Take small charge dq and move it
across the capacitor, which is at
voltage V at this moment
• dW=Vdq
Q
Q
2
1
Q
W   Vdq   qdq 
C0
2C
0
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Lecture IV
19
Energy storage
• Work to charge a capacitor=
potential energy stored in the
capacitor
2
Q
QV CV
U


2C
2
2
2
• To use the right formula, watch
what is kept constant
• V=const – if C connected to a
battery
• Q=const - if C disconnected
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Inserting dielectric
• Capacitor is connected to a
battery, supplying voltage V.
How will the energy stored in
the capacitor change if we
insert a dielectric (K=2)?
• CKC=2C – capacitance
increases
CV 2
KCV 2
U

 2U
2
2
V stays const – same battery
2
Q changes
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Lecture IV
2
Q
Q
U

U / 2
2C
2 KC
21
Inserting dielectric
• Capacitor is charged to charge
Q and disconnected from a
battery. How will the energy
stored in the capacitor change
if we insert a dielectric (K=2)?
• CKC=2C – capacitance
increases
CV 2
KCV 2
U

 2U
2
2
V can change
2
2
Q
Q
Q stays const – charge conservation U  2C  2 KC  U / 2
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Lecture IV
22
Inserting dielectric
U U / K
U  KU
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Connected battery –
energy increases
Dielectric will be
“pushed out”
Disconnected battery –
energy decreases
Dielectric will be
“sucked in”
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23
Test problem
• Between two very large oppositely charged
parallel plates at which of the three locations A, B
and C electric potential is the greatest?
–
–
–
–
A
B
C
D
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A
B
C
Equal at all three locations.
Lecture IV
24
E near metal sphere
• Find the largest charge Q that a
conductive sphere radius r=1cm
can hold.
• Air breakdown E=3x106V/m
dV
Q
1
E (r )  
  kQ '  k 2
dr
r
r
1 2 3 106
2 2
7
Q  Er 
(
10
)

0
.
33

10
 0.033mC
9
k
9 10
Larger spheres can hold more charge
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Lecture IV
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Test problem
• What is wrong with this picture?
– A Equipotentials must be parallel to field lines
– B Field lines cannot go to infinity
– C Some field lines point away from the negative charge
– D Equipotentials cannot be closed
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Lecture IV
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