PHYS105
Module IV (Part 2) - Electrical Potential II
I.
Energy and Electric Potential
Since the electric field can do work on a charged object, it follows from the
“Work-Energy Theorem” in PHYS104 that the electric field can change the
kinetic energy of a charged object.
In the diagram below, point A is at a higher electric potential than point B. This
could be achieved by connecting a battery to metal plates at A and B such that
the positive lead is connected to plate A. If a test object with positive charge q is
then placed at point A, it will be accelerated toward point B by the electric field.
In going from point A to point B, the test object will lose electrical potential
energy while gaining kinetic energy. This process is analogous to a ball falling
toward the Earth.
+
V
-

E

v
q
A
A.
Change in Potential Energy of the System - U
B.
Change in Kinetic Energy of the Test Object - K
B
PROOF:
The “General Conservation of Mechanical Energy Equation” from PHYS104 says
that
The only force doing work on the test object is the electric force, a conservative
force. Therefore, the work done by non-conservative forces is zero and we have
Using the definition of electrical potential difference
we arrive at the final result that
EXAMPLE: A proton initially at rest is accelerated through an electric potential
difference of 500 kV in a Van de Graaff accelerator as shown below. What is the
proton’s kinetic energy upon exiting the accelerator?
+

vi  0 m/s
500 kV
-
SOLUTION:
II.
Electron Volt – eV
From our previous example, we see that for protons, electrons, and other small
charged particles that the potential/kinetic energy transfers are very small even
when the particle is accelerated through very large electric potential differences!!
For this reason, the Joule is not a convenient energy unit when working problems
at the atomic scale. Thus, we define the energy unit called the electron volt.
A.
Definition –
An electron volt is the amount of energy stored as electric potential energy when a
test object of charge e is moved through an electric potential difference of +1 volt.
U = q V = (1.00 e)(+1.00 V) = +1.00 eV
PREVIOUS EXAMPLE:
The final kinetic energy of the proton is
NOTE: When dealing with electron volts, the unit of charge is e and not the
Coulomb. Thus, the value for the charge on a proton, e, never enters into the
calculations.
B.
Conversion to SI
The conversion factor between Joules and electron volts is
C.
Important Energy Ranges in Science and Engineering
eV range - Chemical Bonds, Visible Light, Electronic Bands, etc
keV range - X-rays, Use of Ion Beams to Modify Materials by Ion
Implantation of Materials, Sputtering and Ion Etching
MeV range - Gamma Rays, Rest Energies of Particle, Nuclear Binding, Use of
Ion Beams to Analyze Materials
GeV and TeV range – Cosmic Rays, Ion Beam Energies Needed to Probe For
Subatomic Particles and To Test Theories on the Fundamental Forces
III.
Potential Energy Stored in an Electric Field due to Multiple Charges
To calculate the total electric potential energy stored in an electric field due to a
system of multiple point charges, you add up the electric potential energy
gained by the system as you bring into the system each point charge individually
from r = .
EXAMPLE: What is the total potential energy for the following system of charges?
Assume that a system containing no charges is the zero potential energy reference point.
Q2
r23
r12
Q1
r13
Q3
SOLUTION:
STEP 1: Bring Q1 into the system from infinity
Ui =
Vi =
“No Charges, No Potential”
X
r12
Q1
Uf =
Vf =
“Potential a charge at Point X would experience”
STEP 2: Bring Q2 into the system from infinity
Ui =
Vi =
Q2
r12
r23
Q1
r13
X
Uf =
Vf =
“Potential at Point X”
STEP 3: Bring Q3 into the system from infinity
Ui =
Vi =
Q2
r23
r12
Q1
r13
Q3
Uf =
Uf =
The general mathematical formula for finding the electrical potential energy of a
group of point charges is therefore
IV.
Electric Potential and Corona Points
We previously mentioned in our discussion of electric fields that charge
accumulates (larger charge density) at sharp points of a conductor causing the
electric field strength to be greater at these points. We now have the necessary
tools to fully demonstrate this phenomena.
Consider two conducting spherical shells of different radii connected by a
conducting thread as shown in the figure below:
Conducting
Thread
Sphere A
Sphere B
If we place charge on one of the spheres then the charge will spread out on to
all of the conductors.
The whole surface acts as a single conductor so both spheres must be at the
same electric potential, V, even if we were to remove the conducting thread.
QA
Sphere A of
Radius RA
QB
Sphere B of
Radius RB
From our previous work, we know that from the surface of the shell outward, a
spherical shell behaves like a point charge. Thus the electric potential at the
surface of conductor A is
Likewise the electric potential at the surface of conductor B is
Since both conductors are at the same electric potential, we obtain the following a
relationship between the ratio of the charge on the spheres and the ratio of their
radii.
We should note that the electric field strength at the surface of the conductors is
not the same as shown below:
EA 
k QA
RA
2

V
RA
and
EB 
k QB
RB
2

V
RB
E A RB

EB RA
Thus, we see that less total charge will collect on the smaller sphere when the
spheres are at the same electric potential. However, density  is defined as the
charge divided by area. For our two spheres, we have
B
A
B
2
 Q B  4 π R A 2   Q B  R A 



 4 π R 2  Q    Q  2 
 B  A   A  R B 

A
Thus, we see that in order to have the same electric potential, the smaller sphere
must have a larger surface charge density and that this increased charge density
results in a higher electric field strength.
A sharp pint on a conductor like a corona point as shown below can be thought
of as a sphere of extremely small radius. Thus, as we stated in the lesson on
electric fields, charge accumulates at a sharp point on a conductor causing an
increased electric field at this point.
Corona Point
Corona points can be very useful or harmful in electrical devices.
Spark Gaps – Since the electric field is highest near a sharp point, an engineer
can use a corona point to limit the maximum electrical potential (voltage) on a
conductor. If electric potential attempts to exceed the maximum value, the high
electric field on the corona point causes a spark (charge transfer) between the
corona point and a second safety conductor. The safety conductor is connected to
the earth or in some other way in which the excess charge can be safely removed
from the device. The most famous device employing this technique is Franklin’s
lightning rods that protect buildings. Spark gap devices can also be used to do
other important jobs by adding engineering twists. In a spark plug for instance,
the spark across the gap explodes the gas-air mixture in the engine. The amount
of charge flowing across the spark gap between a set of corona points and the
high voltage terminal in a Van de Graaff accelerator is used to the maintain the
voltage on the accelerator to fixed value. The heat from a spark on the free fall
apparatus in PHYS104 caused a mark on thermal paper. We used these marks to
determine the acceleration due to gravity during lab last semester.
Undesirable Effects – Unwanted sharp points on a conductor in an electrical
device will cause unintentional sparking that can prevent the device from
operating or even damage the device. The electromagnetic wave created by the
sparking can also destroy or effect the operation of other nearby electronic
devices.