Earnshaw’s theorem
A point charge cannot be in stable equilibrium in electrostatic field of
other charges
(except right on top of another charge – e.g. in the middle of a
distributed charge)
Stable equilibrium with other
constraints
Atom – system of charges with only Coulombic forces in play.
According to Earhshaw’s theorem, charges in atom must move
However, planetary model of atom doesn’t work
Only quantum mechanics explains the existence of an atom
Electric Potential Energy
Concepts of work, potential energy
and conservation of energy
For a conservative force, work can always
be expressed in terms of potential energy difference
b

Wa b   F d l  U  (U b  U a )
a
Energy Theorem
For conservative forces in play,
total energy of the system is conserved
Ka  U a  Kb  U b
Wa b  Fd  q0 Ed
U  q0 Ey
Wa b  U  q0 E ( ya  yb )
Potential energy Uincreases
as the test charge q0 moves in the direction opposite to

the electric force F  q0 E : it decreases as it moves in the same direction as the force
acting on the charge
Electric Potential Energy of Two Point Charges
b
Wa b

rb
qq
  F d l   ke 20 cos  dl
r
a
r
a
1 1
Wa b  ke qq0   
 ra rb 
qq0
U  ke
r
Electric potential energy of two point charges
Example: Conservation of energy with electric forces
A positron moves away from an a – particle
me  9.1  10 31 kg
ma  7000me
qa  2e

0
a-particle
r0  10 10 m
positron
v 0  3  10 6 m /s
What is the speed at the distance r  2r0  2 1010 m ?
What is the speed at infinity?
Suppose, we have an electron instead of positron. What kind of motion we would expect?

Conservation of energy principle
K0  U 0  K1  U1
Electric Potential Energy of the System of Charges
Potential energy of a test charge q0
in the presence of other charges
U
q0
qi

4 0 i r i
Potential energy of the system of charges
(energy required to assembly them together)
U
1

qi q j
4 0 i  j r ij
Potential energy difference can be equivalently described as a work
done by external force required to move charges into the certain
geometry (closer or farther apart).


External force now is opposite to Wa b  (U b  U a )   Fext d l
the electrostatic force
Electric Potential Energy of System
• The potential energy of a system of two point charges
q1q2
U  q2V1  ke
r12
• If more than two charges are present, sum the energies of every
pair of two charges that are present to get the total potential energy
U total  ke 
i, j
qi q j
rij
 q1q2 q1q3 q2 q3 
U total  ke 



r13
r23 
 r12
Electric potential is electric potential energy per unit charge
Finding potential (a scalar) is often much easier than the field
(which is a vector). Afterwards, we can find field from a potential
U
V
q0
Units of potential are Volts [V]
1 Volt=1Joule/Coulomb
If an electric charge is moved by the electric
field, the work done by the field
Wa b
U

 (Va  Vb )
q0
q0
Potential difference if often called voltage
Two equivalent interpretations of voltage:
1.Vab is the potential of a with respect to b, equals the work done
by the electric force when a UNIT charge moves from a to b.
2. Vab is the potential of a with respect to b, equals the work that must
be done to move a UNIT charge slowly from b to a against the
electric force.
q
V  ke
r
V
1
dq
4 0  r
Potential due to the point charges
Potential due to a continuous
distribution of charge
Finding Electric Potential through Electric Field
b


Wa b
 Va  Vb   E d l
q0
a
Some Useful Electric Potentials
• For a uniform electric field
r r
r
V    E  dl   E 
• For a point charge
q
V  ke
r
• For a series of point charges
qi
V  ke 
ri
r
r r
 dl  E  l
Potential of a point charge
Moving along the E-field lines means moving in the direction of
decreasing V.
As a charge is moved by the field, it loses potential energy, whereas if
the charge is moved by the external forces against the E-field, it
acquires potential energy
• Negative charges are a potential minimum
• Positive charges are a potential maximum
Positive Electric Charge Facts
• For a positive source charge
– Electric field points away from a positive source charge
– Electric potential is a maximum
– A positive object charge gains potential energy as it moves
toward the source
– A negative object charge loses potential energy as it moves
toward the source
Negative Electric Charge Facts
• For a negative source charge
– Electric field points toward a negative source charge
– Electric potential is a minimum
– A positive object charge loses potential energy as it moves
toward the source
– A negative object charge gains potential energy as it moves
toward the source
Electron Volts
Electron volts – units of energy
U  eVab
1 eV – energy a positron (charge +e) receives when it goes
through the potential difference Vab =1 V
Unit: 1 Volt= 1 Joule/Coulomb (V=J/C)
Field: N/C=V/m
1 eV= 1.6 x 10-19 J
Just as the electric field is the electric force
per unit charge, the electrostatic potential is
the potential energy per unit charge.
Clicker question
There is a 12 V potential difference between the positive and negative ends of a
set of jumper cables, which are a short distance apart. An electron at the
negative end ready to jump to the positive end has a certain amount of potential
energy. On what quantities does this electrical potential energy depend?
a. the distance and the potential difference between the ends of the cables
b. the distance and the charge on the electron
c. the potential difference and the charge
d. the potential difference, charge, and distance
Assume that two of the electrons at the negative terminal have attached
themselves to a nearby neutral atom. There is now a negative ion with a charge
at this terminal. What are the electric potential and electric potential energy of
the negative ion relative to the electron?
a. The electric potential and the electric potential energy are both twice as much.
b. The electric potential is twice as much and the electric potential energy is the
same.
c. The electric potential is the same and the electric potential energy is twice as
much.
d. The electric potential and the electric potential energy are both the same.
e. The electric potential is the same and the electric potential energy is increased
by the mass ratio of the oxygen ion to the electron.
Examples
A small particle has a charge -5.0 mC and mass 2*10-4 kg. It moves
from point A, where the electric potential is a =200 V and its speed
is V0=5 m/s, to point B, where electric potential is b =800 V. What
is the speed at point B? Is it moving faster or slower at B than at A?
E
2
2
A
B
F
mV0
mV
 qa 
 qb
2
2
Vb ~ 7.4 m / s
In Bohr’s model of a hydrogen atom, an electron is considered
moving around a stationary proton in a circle of radius r. Find
electron’s speed; obtain expression for electron’s energy; find total
energy.
11
e2
V2
r

5.3

10
m
U
Fe  ke 2  m
K
r
r
2
T  13.6 eV
T  K U
Calculating Potential from E field
• To calculate potential function from E field
V   
f
i
r r
E  ds

   (E x iˆ  E y ˆj  E z kˆ )  dxiˆ  dyˆj  dzkˆ
f
i


f
i
E x dx  E y dy  E z dz

When calculating potential due to charge distribution, we calculate potential explicitly
if the exact distribution is known.
If we know the electric field as a function of position, we integrate the field.
b

   E d l
a
Generally, in electrostatics it is easier to calculate a potential (scalar) and then find
electric field (vector). In certain situation, Gauss’s law and symmetry consideration
allow for direct field calculations.
Moreover, if applicable, use energy approach
rather than calculating forces directly
(dynamic approach)
Example: Solid conducting sphere
Outside: Potential of the point charge
1
q
V
4 0 r
Inside: E=0, V=const
Potential of Charged Isolated Conductor
• The excess charge on an isolated conductor will distribute itself
so all points of the conductor are the same potential (inside and
surface).
• The surface charge density (and E) is high where the radius of
curvature is small and the surface is convex
• At sharp points or edges  (and thus external E) may reach high
values.
• The potential in a cavity in a conductor is the same as the
potential throughout the conductor and its surface
At the sharp tip (r tends to zero), large
electric field is present even for small
charges.
Lightning rod – has blunt
end to allow larger charge
Corona – glow of air due to gas discharge built-up – higher probability
near the sharp tip. Voltage breakdown of of a lightning strike
the air
Vmax  3  10
6
V /m
Vmax  REmax