Download Lesson #5 – Electric Potential

PHYS105
Work & Energy Review
I.
Review of Concepts From Physics 104
A.
Philosophy
In first semester physics, we showed that we could in theory
solve ANY problem involving a system of particles moving
at speeds less than 10% of the speed of light by determining
the net external force acting upon the system and applying
Newton’s Laws.
However, we found that many problems were difficult to
solve by this method because either the individual forces
acting upon the system were unknown (collisions) or because
of the vector nature of forces (energy problems).
Thus, we solved many problems using the scalar concepts
of energy, work, and potential energy instead of using the
vector force methods directly!!
B.
Work
Definition: Work is the area under a force vs position graph.
C.
Conservative vs UnConservative Forces
If the work done by a force upon a body depends only on the
bodies starting and ending point and not upon the path taken
then the force is called “conservative” otherwise it is called
“non-conservative.”
D.
Work-Energy Theorem
The net work done on a body (i.e. the work done by all
forces) is equal to the change in the body’s kinetic energy.
This is the key idea involving work and energy. All other
formulas come from it!!!!
E.
Potential Energy – U
The negative of the work done by a conservative force as
an object is moved from point A to point B is defined to be
the difference in the potential energy of the object due to
that force between point A and point B.
Because potential energy is defined in terms of work (an area
under the force vs position graph) only the difference in
potential energy has physical meaning!!!
Thus, when someone talks about the potential energy at
some point or potential energy function, they actually mean
the potential energy difference with respect to some implied
zero potential reference point!!!
C. Gravitational Potential Energy (near the surface of the
Earth) – Ug
Using the surface of the earth as the zero gravitational
potential energy reference point, the gravitational potential
energy of an object of mass M at a height of h above the
Earth is given by
Note: Gravitational potential energy is NOT a scalar field
since its value depends on the mass of the object and not just
the object’s location. This is true of all types of potential
energy. Potential energy is a property of a system of
objects. In this case, the gravitational potential energy
belongs to system of masses composed of the Earth and the
object at height h.
B.
Example of a Scalar Field
We can create a scalar function that depends only upon the
object’s location above the Earth dividing the gravitational
potential energy function by the mass of the test object. This
new function is a scalar gravitational potential field created
by the Earth.
With this new function, we can easily determine the potential
energy stored as object of mass M is moved from the zero
potential energy reference location to a point at a height of h
above the Earth by multiplying our scalar gravitational
potential energy field by the object’s mass.
In other words, we can easily find out how much work we
must do to move the object from the Earth’s surface to a
height h. Although this process may seem unnecessary for
the simple problem we are considering here, it is extremely
helpful for problems involving more complicated geometry.
Furthermore, we will see shortly that a special instrument
called the voltmeter has been constructed to calculate the
difference in the scalar electric potential between two points
in space.
II.
Electric Potential Energy – UE
A.
For non-time varying electric fields, the electric force on a
test charge q is a conservative force.
B.
Using the definition of potential energy from Physics 1224,
we have that the change in electric potential energy for a test
object of charge q as it is moved from point A to point B is
given by the equation:
Note: The electric potential energy is not a scalar field as it
depends on the charge of the test object. The electrical
potential energy is a property of a system of charges (those
charges that set up the electric field and the test charge).
III. Electric Potential Difference (Voltage) - V
A.
Definition – The electric potential difference between two
points in space is defined as the change in the electric
potential energy that a test object would experience as it was
moved between the two points divided by the charge on the
test object.
B.
Units – Volts 
C.
Electric potential difference depends only on the points in
space and not on the test object. Thus, it is a scalar field
created by the charges that created the electric field.
Electric Potential Difference: (Scalar)
Electric Field Vector)
D.
Electric potential can be uniquely defined at a point in space
only after defining a reference point of zero electric potential.
For a point charge, we will choose r =  as our reference
point.
IV. Calculating the Electric Potential Difference From the
Electric Field
A.
If you know the electric field, you can calculate the electric
potential difference between two points by finding the
“negative” of the area under the Electric Field vs Position
garph.
B.
EXAMPLE: What is the electric potential difference
between point (x =3) and (x=0) for a constant electric

field, E 100. N iˆ
C
C.
Although it is possible to calculate the electric potential given
the electric field, this process is rarely done in practices as it
defeats the advantage of avoiding the vector math associated
with electric fields. Instead, we will follow the same process
that we did in PHYS1224 when we calculated the work
integral for a few special forces (gravity, spring, etc) to
develop potential energy functions. We then used the
potential energy functions when solving problems.
V.
Electric Potential For a Point Charge
A.
We already know that any problem involving a distribution
of charge can be solved by breaking the charge distribution
into a series of point charges and then either integrating or
summing the individual point charge contributions.
B.
The electric potential due to a point charge of charge Q at a
distance r from the point charge is given by
where we have chosen r =  as our zero electric potential
reference point.
VI. Electric Potential For Any Charge Distribution
A.
Discrete Set of Point Charges
The total electric potential for N discrete point charges can be
found using our result from part V. The result is
VII. Equipotential Surfaces
Definition – An equipotential surface is defined as a surface
consisting of a
continuous distribution of points having the same electric
potential (voltage).
A.
D
B
C
A
VBA =
VCA =
VDA =
B.
From the definition of electric potential, we know that an
equipotential surface is also a constant potential energy
surface!!!
Thus, in the same way we can walk on the third floor of a
building without being hurt by gravity, we can place our hand
on the Van de Graaff generator and become charged without
being shocked!!! If we step out of a window on the third
floor, then gravity will do work on our mass as we move
between two surfaces of different gravitational potential
energy. If we touch something or someone who is not at the
same electric potential as the Van de Graaff then the electric
field will do work on the electric charges in our body as they
move between the two surfaces of different electric potential.
(i.e. You get shocked!!)
** If you are having trouble following this discussion then
review the material on the “work energy theorem” and the
“conservation of mechanical energy requirement” from
PHYS1224 **
C.
Electrical Engineering Safety Tip –
Electrical engineers and electronic technicians often work on
devices that are capable of severely shocking or even killing
a person. In order to reduce the possibility of this hazard,
they often tie the common (negative) lead to a single
reference point on the circuit called “ground.” The electric
potential at any point in the circuit is then measured by
touching the positive lead of the voltmeter to the desired
point while keeping the other hand in your pocket or behind
your back. This reduces the chance of you reaching across
the circuit and accidentally touching with your hands two
points that greatly differ in electric potential. Although the
point chosen as the ground can theoretically be any
convenient point in the circuit, it is usually specified in the
schematics when one is troubleshooting.
Electric field is perpindicular to an equipotential surface
at every point on the surface.
PROOF:
Consider two neighboring points on an equipotential surface.
The difference in
electric potential between the two points must be zero so we
have that
D.
Electric field lines point in the direction of DECREASING
electric potential.
EXAMPLE 1: Draw the electric field lines and equipotential
surfaces for the positive point charge Q shown below:
+Q
EXAMPLE 2: Draw the electric field lines and equipotential
surfaces for the charge distribution shown below:
+3Q
-Q
LAB: We will experimentally determine equipotential surfaces
and electric field lines for different charge distributions in lab.
Remember to bring graph paper to lab!!!
VIII. Conductors and Electric Potential
A.
All points on the surface of a conductor are at equipotential!
We know from previous work that the electric field is
perpendicular to the surface
of a conductor. From our work in the previous section, it
therefore follows that the conductor’s surface is an
equipotential surface.
B.
All points INSIDE a conductor are at equipotential!
Surface of Conductor
B
A
Since the electric field inside a conductor is
______________, we have
VBA = VB – VA =
C.
All points in an EMPTY CAVITY inside a conductor are at
the SAME electric
potential as the CONDUCTOR!!
Outer Surface of
Conductor
B
A
Cavity
VBA = VB – VA =