Electric Potential Energy
From last chapter, we saw the form of the electrostatic
force equation is very similar to the gravitational force
equation:
Both of these forces are conservative and thus a
potential energy can be associated with a conservative
force.
Stop: CONCEPT CHECK!
What is a conservative force?
1. If the work done by the force is independent of the
path taken.
2. Does no work on an object if it moves in a closed
path.
3. You can get the energy back easily!
Referring back to mechanics and the gravitational
potential energy:
We can make the same analogy to electric forces:
As in gravity, there is a potential difference between
points A and B.
In gravity, the potential difference would be:
In electrostatics, the electric potential is defined as the
work per-unit-charge or:
And is given the name Volt (V) where 1 V = 1
Joule/Coulomb.
Concepts of Potential
An object of mass will accelerate as it falls to the
ground. We have to be careful when we talk about charges.
In the previous figure, we saw a positive test charge
between two plates. Since point A is at a higher potential
energy than point B, we conclude that:
A positive charge will accelerate from a region of
higher electric potential toward a region of lower electric
potential.
Stop: CONCEPT CHECK!
What do you think negative charges will do?
Example: A light and a battery
Negative charges flow from B to A. As they flow into the
light, almost all of their potential energy is transformed to
heat, which causes the filament to glow and give off light.
Electric Potential caused by a point charge
A positive point charge +q creates an electric
potential. If we introduce a test charge into the system we
can measure:
 The force between 2 points close to the charge
 The work needed to go between these points
 Thus, the electric potential between the two points.
Q. Where do you think the potential will be the greatest?
Near the charge or far away?
In order to find an expression for the electric potential,
we recall that for a conservative force:
F = - grad U
In order to find an expression for the electric potential
energy, we need to integrate the electrostatic force; also
called the Coulomb force:
An application that you may not think about is the
potential or the electric field present inside a conducting
cavity. If we assume that there is no charge present inside
the cavity, the electric field inside the cavity must be zero,
regardless of the charge distribution on the outside surface
of the conductor. Furthermore, the field in the cavity is
zero even if an electric field exists outside the conductor.
To prove this, we can think of at least 2 points inside the
conductor that are at the same potential. If there is an
electric field inside the conductor, the electric potential is
given as:
VB – VA = -
E ds
If E is non-zero, then we can always find a path between A
and B for which the integral is a positive number. However
since we defined them to be at the same potential, this
violates our assumption.
Therefore, the electric field inside a conducting cavity must
be zero regardless of the field outside.
It is for this reason and not because a car has rubber tires
that if lightning strikes your car, you will not be
electrocuted! Getting fried from the heat distributed is
another story…
Equipotential Surfaces
An equipotential surface is a surface on which the
electric potential is the same everywhere.
According to the previous equation, wherever r is the
same, the potential is the same, thus the equipotential
surfaces are spherical surfaces centered on the charge.
Also, from the definition of a conservative force, no
work is done as a charge moves along an equipotential
surface.
How is this related to the E-Field? The electric field is
everywhere perpendicular to the associated equipotential
surfaces and points in the direction of decreasing potential.
Capacitors and Dielectrics
A capacitor is a device that is used to store charge.
It is usually constructed as two parallel conducting
plates separated by an insulating material known as a
dielectric.
Because there is a difference in charge, an electric
field is set up and thus a potential difference.
Experiment showed that as the amount of charge got
larger, so did the potential difference. Thus, the charge was
directly proportional to the potential difference.
C is known as the capacitance and has units of
Coulomb/Volt = Farad (F)
If a dielectric is inserted between the plates of a
capacitor, the capacitance can be greatly increased.
The presence of a dielectric between two plates causes
the electric field to decrease between the plates as
illustrated below:
The ratio of the electric fields with and without the
dielectric is called the dielectric constant and is defined as:
Not only is the capacitance dependent on the presence
of a dielectric, but is also dependent on the geometry of the
set-up.
It can be found from:
For a capacitor:
If Co is the capacitance when = 1, then:
Energy stored in a Capacitor
The work done transferring charge through a potential
is:
W=qV
Where V is the average potential difference.
Since the average = ½ final:
W = ½ qV
Now q = CV so:
Energy = ½ (CV)V
= ½ C V2
If we now use our form for C from above:
Examples:
1. A computer keyboard:
Circuitry detects the change in capacitance
2. EKG & EEG
Measures the potential differences along the body
and the brain.