Electric potential,
Systems of charges
Physics 122
9/21/12
Lecture V
1
Concepts
•  Primary concepts:
–  Electric potential
–  Electric energy
•  Secondary concepts:
–  Equipotentials
–  Electonvolt
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Lecture V
2
Charges in electric fields

F

E
+
Positive charges experience
force along the direction
of the field
Negative charges – against
the direction of the field.
-
F
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

F = qE
Lecture V
3
1
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
è there is a potential energy associated with electric field.
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Lecture V
4
Electric potential
PE (q) ∝ q
•  PE/q is a property of the field
itself – called electric potential V
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Lecture V
5
Electric potential
V=
PE
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 ΔV=Vb-Va
is often called voltage.
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Lecture V
6
2
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=-qΔV
•  d E=-ΔV
•  E= -ΔV/d, points from high potential to
low
•  Sometimes electric field is measured in
V/m =N/C
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Lecture V
7
Non-uniform electric field
+



E ( x ) ⇒ F ( x ) = qE ( x )
b
b
W = ∫ Fdx = q ∫ Edx
a
a
U b − U a = −Wba
b  
V = − ∫ Edl
a
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Lecture V
8
Determine E from V
•  Think ski slopes
•  If V depends on one coordinate x
•  E is directed along x from high V to low
E ( x) = −
•  If V depends on x,y,z
Ex = −
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dV
dx
∂V
∂V
∂V
; Ey = −
; Ez = −
∂x
∂y
∂z
Lecture V
9
3
Electric field and potential in
conductors

Eexternal
+
+
+
+
+
-
-
-
-
-
 

E = Eexernal + Eint ernal = 0
V =9/21/12
const
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
same potential.
Lecture V
10
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 V
11
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 V
12
4
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
=k
2
r
r0
r0
V (r0 ) = − ∫ Edr = −kQ ∫
Q>0 à V>0
Q<0 à 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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Lecture V
13
Superposition of fields
Principle of superposition:
Net potential created by a
system of charges is a
scalar (!) sum of
potentials created by
individual charges:
V1 > 0
V = V1 + V2
+ V2 < 0
+
-
1
2
V = V1 + V2 + V3 + ....
Potential is a scalar à
no direction to worry about.
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Lecture V
14
Electric Dipole potential
P
•  evaluate potential at point P
V (r ) = k
r+Δ r
Q
Q
Δr
−k
= kQ
r
r + Δr
r (r + Δr )
r
Δ r=lcosθ
-Q
θ

p
l
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• for r >> L,
V (r ) = k
+Q
Lecture V
p cosθ
r2
15
5
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 V
16
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Lecture V
17
The electric potential of
a system of charges
Bunch of Charges
Charge Distribution
V = V1 + V2 + V3 + ...
Vi = k
qi
, ri - distance from charge i to
ri
point in space where V is evaluated
+
+
+
+
-
-
dq
r
V = ∫ dV
dV = k
+ ++ + +
+ ++ + + +
+
+
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Lecture V
18
6
Symmetry and coordinate systems
•  Coordinate systems are there to help you
•  You have a choice of
–  System type
•  Cartesian
•  Cylindrical
•  Spherical
–  Origin (0,0), Direction of axis
•  A good choice (respecting the symmetry of the
system) can help to simplify the calculations
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Lecture V
19
Ring of charge
•  A thing ring of radius a
holds a total charge Q.
Determine the electric field
on its axis, a distance x from
its center.
r = x2 + a2
Ex
a
x
Qx
E = k 2 2 3/ 2
(x + a )
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
E
θ
Lecture V
E⊥
20
Ring of charge
•  A thing ring of radius a
holds a total charge Q.
Determine the electric
potential on its axis, a
distance x from its center.
r = x2 + a2
a
x
Qdϕ / 2π
r
Q
V = k 2 2 1/ 2
(a + x )
dV = k
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Lecture V
dV
21
7
Work to move a charge
a
V a = V1 + V2
+
a
b
30
20cm
cm
V b = V1 + V2
+
15cm
+
cm
25
Q1=10µC
b
How much work has to be
done by an external force
to move a charge
q=+1.5 µC
from point a to point b?
Work-energy principle
W = ΔKE + ΔPE = PEb − PEa
Q2=-20µC
PEb = qVb = q (V1b + V2b )
PEa = qVa = q(V1a + V2a )
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Lecture V
22
E near metal sphere
•  Find the largest charge Q that a
conductive sphere radius r=1cm
can hold.
•  Air breakdown E=3x106V/m
E (r ) = −
Q=
dV
Q
⎛ 1 ⎞
= −kQ⎜ ⎟' = k 2
dr
r
⎝ r ⎠
1 2 3 ×10 6
Er =
(10 −2 ) 2 = 0.33 ×10 −7 = 0.033µC
k
9 ×109
Larger spheres can hold more charge
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Lecture V
23
8