22' Electric Potential

1
22. Electric Potential
2
Topics

• Recap Potential Energy
• Electric Potential Difference
• The Volt and the Electronvolt
• The Potential of a Point Charge
• Potential Difference from Superposition
• Electric Field from Electric Potential
• Charged Conductors

3
Recap Potential Energy
The work done by a force along a path from
point A to point B is defined by
4
Recap Potential Energy
If the force is conservative we can define the
potential energy difference as follows
The potential energy difference DU depends only
on the end points A and B, that is, it is
independent of the path taken between A and B.
5
Electric Potential Difference
6
Electric Potential Difference
Suppose a charge q is moved from A to B
against a uniform electric field. The
change in potential energy is
The potential energy is increased.
7
Electric Potential Difference
The electric potential difference between two
points A and B is defined as the energy per unit
charge
A
B
For the special case of a uniform electric field,
we can write
8
The Volt and the Electronvolt
The electric potential difference between two
points is such an important idea that it is given
its own unit the volt (V). If a charge moves
over an electric potential difference of
DV volts, the potential energy changes
by DU q DV.Example A 12-V car battery does
12 J of work in moving 1 C of charge from one
battery terminal to the other.
A
B
9
The Volt and the Electronvolt

• Some Electric Potential Differences
• Between arm and leg 1 mV
• Across cell membrane 80 mV
• Car battery 12 V
• Electric outlet 100 240 V
• Between power line and ground 365 kV
• Between base of thundercloud 100 MV
• and ground

10
The Volt and the Electronvolt

• For molecular and atomic systems, it is usually
  more convenient to measure energy in
  electronvolts (eV). (This is not an SI unit!)
• One electronvolt is the energy gained by a
  particle carrying one elementary charge when it
  moves through a potential difference of 1 volt.
  Example the ionization energy of hydrogen is
  13.6 eV.
• Since the value of an elementary charge is 1.6 x
  10-19 C, 1 eV is 1.6 x 10-19 J

11
Example A Power Line
A long straight power-line wire, of radius r
1.0 cm with charge density l 2.6 mC/m, is at a
height h 22 m above the ground. What is the
potential difference DV between the cable and
the ground, assuming that the electric field is
approximately that of a line charge?
h
DV
12
Example A Power Line
We found that the electric field of an
infinitely long line charge is given by
where the unit vector is perpendicular to, and
points away from, the wire. The potential
difference is
h
DV
that is, 360 kV
13
Potential of a Point Charge
14
The Potential of a Point Charge
Consider a positive point charge. The change in
electric potential between points A and B is
given by
This shows that if a positive charge moves from A
to B the potential energy decreases
15
The Potential of a Point Charge
Although only changes in potential are
physically relevant, it is often convenient to
choose the location of the zero of the potential.
For a car battery, this is typically the
cars chassis for an electrical outlet it is the
ground. For an isolated point charge, it is
convenient to choose the potential to be zero at
infinity
16
The Potential of a Point Charge
The potential difference between two points A
and B from a point charge
can be re-written as
When rA infinity the last term vanishes. We are
free to choose V(A) as we please, e.g., V(A) 0.
17
The Potential of a Point Charge
With this choice, the potential of a point charge
becomes
Remember, however, that only differences in this
number are physically relevant because we can
always add to it an arbitrary constant without
altering the physics
18
Potential Differences using Superposition
19
Electric Potential of a Collection of Charges
The potential at a given point is the sum
of the electric potentials due to
every point charge
-


-
-


20
Electric Potential for a Charge Distribution
The electric potential for a charge distribution
is given by a formula similar to that for an
electric field
But unlike the electric field the electric
potential is a scalar
21
Example A Charged Ring
Note for a fixed point P, the distance r is
constant as we integrate around the ring.
Therefore,
22
Example A Charged Disk
The potential for a ring of charge is
Therefore, for a disk we can write
23
Example A Charged Disk
The charge on a ring of radius a is
where s is the surface charge density.
Therefore,
24
Example A Charged Disk
After performing the integral
we obtain
25
Equipotentials

• If one draws a surface through all points with
  the same potential, one obtains an equipotential.
  
• Here, for example, are
• some equipotentials for
• a dipole

26
Electric Field from Electric Potential
27
Electric Field fromElectric Potential
Since
one can compute the electric field from the
potential V using the gradient
This is the gradient in Cartesian coordinates
28
Example Field of A Charged Disk
We found the potential of a charged disk to be
From this we can compute the electric field in
the x direction from the negative gradient
Close to the disk the field is 2pks as we
found using Gausss law.
29
Charged Conductors
The surface of a conductor in electrostatic
equilibrium is an equipotential. Therefore, if
one connects a charged spherical conductor to
a neutral one, via a thin wire, the
charge will migrate rapidly through
the entire system
until the potential on the surface of the system
is the same everywhere.
30
Charged Conductors
If the spheres are very far apart, the
distribution of charge on each will be
essentially unaffected by the charge
distribution on the other. In this case,
the spheres act like point charges with
potentials
V1 kQ1/R1
kQ1/R1 and kQ2/R2
V2 kQ2/R2
Since V1 V2, the smaller sphere will have the
higher surface charge density
31
Dielectric Breakdown of Air near Conductors
Air suffers dielectric breakdown, and becomes
a conductor, if the electric field in air
exceeds Emax 3 x 106 V/m (N/C)

• At the surface of a conductor the
• electric field is E 4pks V/m.
• Therefore, if the surface charge density
• is large enough, an isolated conductor
• in air will trigger its dielectric
• breakdown.

32
Dielectric Breakdown of Air near Conductors
The charge required to trigger dielectric
breakdown can be estimated by equating Emax to E
4pks V/m
(with R in mm)
Example a 1 mm radius ball bearing with about
1/3 nC of charge is enough to cause the air to
spark!
33
Summary

• Electric potential difference
• Since only differences in potential matter, the
  location of the zero of the potential can be
  chosen as we wish
• Electric potential of a point charge
• If we chose the zero to be infinitely far from a
  point charge, we can write its potential as

34
Summary

• Equipotentials
• These are surfaces of equal potential difference.
  
• The surface of a conductor in equilibrium is an
  equipotential.
• Electric field
• It is equal to the negative gradient of the
  electric potential